SlideShare a Scribd company logo

Like Alice in Wonderland: Unraveling Reasoning and Cognition Using Analogies and Concept Blending - Tarek R. Besold

Facultad de Informática UCM, profile picture
Facultad de Informática UCM

The document discusses computational models of analogy and concept blending, focusing on the intersections of cognitive science and artificial intelligence. It highlights collaborative research under the European FP7 Coinvent project, which explores theory blending, analogy-making, and the complexity of cognitive capacities. The work aims to develop formal models to enhance understanding and computational realizations in fields like mathematics and music.

Engineering
1 of 90
Download to read offline
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Like Alice in Wonderland: Unraveling Reasoning and Cognition Using Analogies and Concept Blending Tarek R. Besold KRDB, Faculty of Computer Science, Free University of Bozen-Bolzano 16. June 2016 Tarek R. Besold Computational Models of Analogy and Concept Blending
Honour to whom honour is due... The following is joint work with many people, most notably: Robert Robere, Department of Computer Science, University of Toronto (Canada). Enric Plaza, IIIA-CSIC, Barcelona (Spain). Kai-Uwe K¨uhnberger, Institute of Cognitive Science, University of Osnabr¨uck (Germany). Tarek R. Besold Computational Models of Analogy and Concept Blending
And Who is Paying (Some of) the Bills? The described work on concept blending has been conducted as part of the European FP7 Concept Invention Theory (COINVENT) project (FET-Open grant number 611553). Consortium members are: Free University of Bozen-Bolzano (S¨udtirol-Alto Adige, Italy) University of Osnabr¨uck (Germany) University of Magdeburg (Germany) University of Dundee (Scotland, UK) University of Edinburgh (Scotland, UK) Goldsmiths, University of London (UK) IIIA-CSIC, Barcelona (Catalunya, Spain) Aristotle University of Thessaloniki (Greece) Tarek R. Besold Computational Models of Analogy and Concept Blending
Non-Classical and Cross-Domain Reasoning Tarek R. Besold Computational Models of Analogy and Concept Blending
Back in the Day (1) Tarek R. Besold Computational Models of Analogy and Concept Blending
Back in the Day (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
Back in the Day (3) Rutherford analogy (underlying the Bohr-Rutherford model of the atom): Analogy between solar system and hydrogen atom: ...nucleus is more massive than electrons, sun is more massive than planets. ...nucleus attracts electrons (Coulomb’s law), sun attracts planets (Newton’s law of gravity). ...attraction plus mass relation causes electrons to revolve around nucleus, similarly planets revolve around sun. Tarek R. Besold Computational Models of Analogy and Concept Blending
Analogy Tarek R. Besold Computational Models of Analogy and Concept Blending
Intermezzo: Analogy (1) Analogy “ànalog–a” - analogia, “proportion”. Informally: Claims of similarity, often used in argumentation or when explaining complex situations. A bit more formal: Analogy-making is the human ability of perceiving dissimilar domains as similar with respect to certain aspects based on shared commonalities in relational structure or appearance. (Incidental remark: In less complex forms also to be found in some other primates.) Tarek R. Besold Computational Models of Analogy and Concept Blending
Intermezzo: Analogy (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
Non-Classical and Cross-Domain Reasoning Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (1) Heuristic-Driven Theory Projection (HDTP) Computing analogical relations and inferences (domains given as many-sorted first-order logic representation/many-sorted term algebras) using a generalisation-based approach. Base and target of analogy defined in terms of axiomatisations, i.e., given by a finite set of formulae. Aligning pairs of formulae by means of anti-unification (extending classical Plotkin-style first-order anti-unification to a restricted form of higher-order anti-unification). Proof-of-concept applications in modelling mathematical reasoning and concept blending in mathematics. Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (2) Figure: Analogy-making in HDTP. Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (3) Anti-Unification Dual to the unification problem (see, e.g., logic programming or automated theorem proving). Generalising terms in a meaningful way, yielding for each term an anti-instance (distinct subterms replaced by variables). Goal: Finding the most specific anti-unifier. Plotkin: For a proper definition of generalisation, for a given pair of terms there always is exactly one least general generalisation (up to renaming of variables). Problem: Structural commonalities embedded in different contexts possibly not accessible by first-order anti-unification. Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (4) Restricted Higher-Order Anti-Unification First-order terms extended by introducing variables taking arguments (first-order variables become variables with arity 0), making a term either a first-order or a higher-order term. Class of substitutions restricted to (compositions of) the following four cases: 1 Renamings rF,F⇤ : F(t1,...,tn) rF,F⇤ ! F⇤(t1,...,tn). 2 Fixations fF c : F(t1,...,tn) fF f ! f(t1,...,tn). 3 Argument insertions iF,F⇤ G,i : F(t1,...,tn) i F,F⇤ G,i ! F⇤(t1,...,ti ,G(ti+1,...,ti+k ),ti+k+1,...,tn). 4 Permutations pF,F⇤ a : F(t1,...,tn) p F,F⇤ a ! F⇤(ta(1),...,ta(n)). Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (5) Examples of higher-order anti-unifications: Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (6) Tarek R. Besold Computational Models of Analogy and Concept Blending
Heuristic-Driven Theory Projection (7) Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity and Tractability in Cognitive Models and Systems Tarek R. Besold Computational Models of Analogy and Concept Blending
Computer Metaphor and Church-Turing Thesis Famous ideas at the heart of many endeavours in computational cognitive modelling and/or AI: 1 “Computer metaphor” of the mind (i.e. the concept of a computational theory of mind). 2 Church-Turing thesis. 1 Bridges gap between humans and computers: Human mind and brain can be seen as information processing system. Reasoning and thinking corresponds to computation as formal symbol manipulation. 2 Gives account of the nature and limitations of the computational power of such a system. Tarek R. Besold Computational Models of Analogy and Concept Blending
Computer Metaphor and Church-Turing Thesis Famous ideas at the heart of many endeavours in computational cognitive modelling and/or AI: 1 “Computer metaphor” of the mind (i.e. the concept of a computational theory of mind). 2 Church-Turing thesis. 1 Bridges gap between humans and computers: Human mind and brain can be seen as information processing system. Reasoning and thinking corresponds to computation as formal symbol manipulation. 2 Gives account of the nature and limitations of the computational power of such a system. Tarek R. Besold Computational Models of Analogy and Concept Blending
P-Cognition Thesis Significant impact on cognitive science and cognitive psychology: Explain human cognitive capacities modelled in terms of computational-level theories (i.e., as precise characterisations of hypothesised inputs and outputs of respective capacities together with functional mappings between them). Problem: Computational-level theories often underconstrained by available empirical data! ) Use mathematical complexity theory as assisting tool: NP-completeness! P-Cognition thesis Human cognitive capacities hypothesised to be of the polynomial-time computable type. (Interpretation: “Humans can comfortably solve non-trivial instances of this problem, where the exact size depends on the problem at hand”.) Tarek R. Besold Computational Models of Analogy and Concept Blending
P-Cognition Thesis Significant impact on cognitive science and cognitive psychology: Explain human cognitive capacities modelled in terms of computational-level theories (i.e., as precise characterisations of hypothesised inputs and outputs of respective capacities together with functional mappings between them). Problem: Computational-level theories often underconstrained by available empirical data! ) Use mathematical complexity theory as assisting tool: NP-completeness! P-Cognition thesis Human cognitive capacities hypothesised to be of the polynomial-time computable type. (Interpretation: “Humans can comfortably solve non-trivial instances of this problem, where the exact size depends on the problem at hand”.) Tarek R. Besold Computational Models of Analogy and Concept Blending
“polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
“polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
“polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of HDTP (1) HDTP is naturally split into two mechanisms: Analogical matching of input theories. Re-representation of input theories by deduction in FOL. ) Re-representation is undecidable (undecidability of FOL). ) Focus on mechanism for analogical matching. Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of HDTP (2) Problem 1. F Anti-Unification Input: Two terms f,g, and a natural k 2 N Problem: Is there an anti-unifier h, containing at least k variables, using only renamings and fixations? Problem 2. FP Anti-Unification Input: Two terms f,g, and naturals l,m,p 2 N. Problem: Is there an anti-unifier h, containing at least l 0-ary variables and at least m higher arity variables, and two substitutions s,t using only renamings, fixations, and at most p permutations such that h s ! f and h t ! g? Problem 3. FPA Anti-Unification Input: Two terms f,g and naturals l,m,p,a 2 N. Problem: Is there an anti-unifier h, containing at least l 0-ary variables, at least m higher arity variables, and two substitutions s,t using renamings, fixations, at most p permutations, and at most a argument insertions such that h s ! f and h t ! g? Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of HDTP (3) ...a fair share of formal magic involving a Canadian and some “Subgraph Isomorphism to Clique” reductions later... Complexity of HDTP (Higher-Order Anti-Unification) 1 F Anti-Unification is solvable in polynomial time. 2 Let m denote the minimum number of higher arity variables and let p be the maximum number of permutations applied. Then FP Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p}. 3 Let r be the maximum arity and s be the maximum number of subterms of the input terms. Then FP Anti-Unification is in FPT w.r.t. parameter set {s,r,p}. 4 FPA Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p,a}. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of HDTP (3) ...a fair share of formal magic involving a Canadian and some “Subgraph Isomorphism to Clique” reductions later... Complexity of HDTP (Higher-Order Anti-Unification) 1 F Anti-Unification is solvable in polynomial time. 2 Let m denote the minimum number of higher arity variables and let p be the maximum number of permutations applied. Then FP Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p}. 3 Let r be the maximum arity and s be the maximum number of subterms of the input terms. Then FP Anti-Unification is in FPT w.r.t. parameter set {s,r,p}. 4 FPA Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p,a}. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
Concept Blending Tarek R. Besold Computational Models of Analogy and Concept Blending
Concept blending: A + B = ? (1) Tarek R. Besold Computational Models of Analogy and Concept Blending
Concept blending: A + B = ? (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
Concept blending: A + B = ? (3) Tarek R. Besold Computational Models of Analogy and Concept Blending
Concept blending: A + B = ? (4) Tarek R. Besold Computational Models of Analogy and Concept Blending
Foundations of Theory Blending (1) Concept Blending Given two domain theories I1 and I2, representing two conceptualisations... ...look for a generalisation G... ...construct the blend space B in such a way as to preserve the correlations between I1 and I2 established by G. Tarek R. Besold Computational Models of Analogy and Concept Blending
Foundations of Theory Blending (2) Example: Houseboat vs. boathouse Concept blends of HOUSE and BOAT into BOATHOUSE and HOUSEBOAT. I1 = {HOUSE v 8LIVES IN.RESIDENT} I2 = {BOAT v 8RIDES ON.PASSENGER} HOUSEBOAT: Aligning parts of the conceptual spaces... RESIDENT $ PASSENGER LIVES IN $ RIDES ON HOUSE $ BOAT BOATHOUSE: Aligning parts of the conceptual spaces... RESIDENT $ BOAT Tarek R. Besold Computational Models of Analogy and Concept Blending
The Concept Invention Theory (COINVENT) Project (1) To develop a novel, computationally feasible, formal model of conceptual blending based on Fauconnier and Turner’s theory. To gain a deeper understanding of conceptual blending and its role in computational creativity. To design a generic, creative computational system capable of serendipitous invention and manipulation of novel abstract concepts. To validate our model and its computational realisation in two representative working domains: mathematics and music. Tarek R. Besold Computational Models of Analogy and Concept Blending
The Concept Invention Theory (COINVENT) Project (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
Amalgamation 101 (1) I1 I2 ¯I2 ¯I1 G = I1 u I2 A = ¯I1 t ¯I2 v v v vvv v v Amalgam A description A 2 L is an amalgam of two inputs I1 and I2 (with anti-unification G = I1 uI2) if there exist two generalisations ¯I1 and ¯I2 such that (1) G v¯I1 v I1, (2) G v¯I2 v I2, and (3) A =¯I1 t¯I2 Tarek R. Besold Computational Models of Analogy and Concept Blending
Amalgamation 101 (2) v vv v v v A = S t T S S T G = S u T Asymmetric Amalgam An asymmetric amalgam A 2 L of two inputs S (source) and T (target) satisfies that A = S0 tT for some generalisation of the source S0 v S. Tarek R. Besold Computational Models of Analogy and Concept Blending
COINVENT’s Blending Schema 1.) Compute shared generalisation G from S and T with fS(G) = Sc. 2.) Re-use fS in generalisation of S into S0. 3.) Combine S0 in asymmetric amalgam with T into proto-blend T0 = S0 tT. 4.) By application of fT , complete T0 into blended output theory TB. (✓: element-wise subset relationship between sets of axioms. v: subsumption between theories in direction of respective arrows.) Tarek R. Besold Computational Models of Analogy and Concept Blending
...and the Implementation? Use HDTP for computation of generalisation(s) and substitution chains/higher-order anti-unifications. Currently: Restrict HDTP to using only renamings and fixations. ) Possibility to use “classical” semantic consequence |= as ordering relationship. (Also preserved by later unifications and addition of axioms.) Use HDTP’s heuristics for selecting least general generalisation G (among several options). Currently: Naive consistency/inconsistency check with final blend (both internally and against world knowledge). ) Clash resolution by re-start with reduced set of input axioms. Tarek R. Besold Computational Models of Analogy and Concept Blending
...and the Implementation? Use HDTP for computation of generalisation(s) and substitution chains/higher-order anti-unifications. Currently: Restrict HDTP to using only renamings and fixations. ) Possibility to use “classical” semantic consequence |= as ordering relationship. (Also preserved by later unifications and addition of axioms.) Use HDTP’s heuristics for selecting least general generalisation G (among several options). Currently: Naive consistency/inconsistency check with final blend (both internally and against world knowledge). ) Clash resolution by re-start with reduced set of input axioms. Tarek R. Besold Computational Models of Analogy and Concept Blending
Example: Brillo, the Foldable Toothbrush Tarek R. Besold Computational Models of Analogy and Concept Blending
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Stereotypical characterization for a pocketknife: Stereotypical characterization for a toothbrush:
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Computing a shared generalization: Applied substitutions:
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Applied substitutions:
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Shared generalization: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Computing the generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Computing the generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Computing the proto-blend/ asymmetric amalgam:
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Proto-blend/ asymmetric amalgam:
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Computing the final blend: has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold) Final blend:
Conclusion Tarek R. Besold Computational Models of Analogy and Concept Blending
(Definitely Not) The End! If you are interested in non-classical reasoning, tractability, approximability and similar topics in A(G)I and/or cognitive science, you are happily invited to... 1 ...talk to me after the presentation. 2 ...get in touch by e-mail: TarekRichard.Besold@unibz.it. 3 ...occasionally have a look at our publications.2 2 For instance: Besold, T. R., and Robere, R.. When Thinking Never Comes to a Halt: Using Formal Methods in Making Sure Your AI Gets the Job Done Good Enough. In V. C. M¨uller (ed.), Fundamental Issues of Artificial Intelligence (Synthese Library, vol. 376). Springer, 2016. Besold, T. R., and Plaza, E. Generalize and Blend: Concept Blending Based on Generalization, Analogy, and Amalgams. In H. Toivonen, S. Colton, M. Cook, and D. Ventura, Proceedings of the Sixth International Conference on Computational Creativity (ICCC) 2015. Brigham Young University Press, 2015. Tarek R. Besold Computational Models of Analogy and Concept Blending
Postludium Tarek R. Besold Computational Models of Analogy and Concept Blending
Disclaimer Frequent criticism: Demanding for cognitive systems and models to work within certain complexity limits overly restrictive. Maybe: Human mental activities actually performed as exponential-time procedures, but never noticed as exponent for some reason always very small. Reply: Possibility currently cannot be excluded. Instead: No claim that cognitive processes without exception within FPT, APX, FPA, or what-have-you... ...but staying within boundaries makes cognitive systems and models plausible candidates for application in resource-bounded general-purpose cognitive agents. Tarek R. Besold Computational Models of Analogy and Concept Blending
Disclaimer Frequent criticism: Demanding for cognitive systems and models to work within certain complexity limits overly restrictive. Maybe: Human mental activities actually performed as exponential-time procedures, but never noticed as exponent for some reason always very small. Reply: Possibility currently cannot be excluded. Instead: No claim that cognitive processes without exception within FPT, APX, FPA, or what-have-you... ...but staying within boundaries makes cognitive systems and models plausible candidates for application in resource-bounded general-purpose cognitive agents. Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of the First HDTP Generation As an aside: Once upon a time, there was HDTP-old based on reducing certain higher-order to first-order anti-unifications by introduction of subterms built from “admissible sequences” over equational theories (i.e., conjunctions of FOL formulae with equality over a term algebra). Complexity of HDTP-old 1 HDTP-old is NP-complete. 2 HDTP-old is W[2]-hard with respect to a minimal bound on the cardinality of the set of all subterms of the term against which admissibility is checked. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
Complexity of the First HDTP Generation As an aside: Once upon a time, there was HDTP-old based on reducing certain higher-order to first-order anti-unifications by introduction of subterms built from “admissible sequences” over equational theories (i.e., conjunctions of FOL formulae with equality over a term algebra). Complexity of HDTP-old 1 HDTP-old is NP-complete. 2 HDTP-old is W[2]-hard with respect to a minimal bound on the cardinality of the set of all subterms of the term against which admissibility is checked. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
A Primer on Approximation Theory Approximability Classes In the following, let... ...PTAS denote the class of all NP optimisation problems that admit a polynomial-time approximation scheme. ...APX be the class of NP optimisation problems allowing for constant-factor approximation algorithms. ...APX-poly be the class of NP optimisation problems allowing for polynomial-factor approximation algorithms. Please note that PTAS ✓ APX ✓ APX-poly (with each inclusion being proper in case P 6= NP). Tarek R. Besold Computational Models of Analogy and Concept Blending
Approximability Analysis of HDTP (1) FP Anti-Unification W[1]-hard to compute for parameter set m,p (m number of higher-arity variables, p number of permutations). ) No polynomial-time algorithm computing “sufficiently complex” generalisations (i.e., with lower bound on number of higher-arity variables), upper bounding number of permutations (W[1]-hardness for single permutation). What if one considers generalisations which merely approximate the “optimal” generalisation in some sense? Tarek R. Besold Computational Models of Analogy and Concept Blending
Approximability Analysis of HDTP (2) Complexity of a Substitution The complexity of a basic substitution s is defined as C(s) = 8 >< >: 0, if s is a renaming. 1, if s is a fixation or permutation. k +1, if s is a k-ary argument insertion. The complexity of a restricted substitution s = s1 ··· sn (i.e., the composition of any sequence of unit substitutions) is the sum of the composed substitutions: C(s) = Ân i=1 C(si ). Tarek R. Besold Computational Models of Analogy and Concept Blending
Approximability Analysis of HDTP (3) Consider problem of finding generalisation which maximises complexity over all generalisations: Complex generalisation would contain “most information” present over all of the generalisations chosen (i.e., maximising the “information load”). Using approximability results on MAXCLIQUE: Approximation Complexity of HDTP Analogy-Making FP anti-unification is not in APX (i.e., does not allow for constant-factor approximation algorithms) and is hard for APX-poly. (For proofs: T. R. Besold and R. Robere. When Almost Is Not Even Close: Remarks on the Approximability of HDTP. In Artificial General Intelligence - 6th International Conference (AGI 2013), LNCS. Springer, 2013.) Tarek R. Besold Computational Models of Analogy and Concept Blending

More Related Content

PDF
Logics of Context and Modal Type Theories
Valeria de Paiva
 
PPTX
Lecture 2: From Semantics To Semantic-Oriented Applications
Marina Santini
 
PDF
Extending the knowledge level of cognitive architectures with Conceptual Spac...
Antonio Lieto
 
PDF
Categorical Explicit Substitutions
Valeria de Paiva
 
PDF
J79 1063
Jahanzeb Jahan
 
PPT
How the philosophy of mathematical practice can be logic by other means (bris...
Brendan Larvor
 
PDF
Eswcsummerschool2010 ontologies final
Elena Simperl
 
PDF
Supermathematics and Artificial General Intelligence
Jordan Bennett
 
Logics of Context and Modal Type Theories
Valeria de Paiva
 
Lecture 2: From Semantics To Semantic-Oriented Applications
Marina Santini
 
Extending the knowledge level of cognitive architectures with Conceptual Spac...
Antonio Lieto
 
Categorical Explicit Substitutions
Valeria de Paiva
 
J79 1063
Jahanzeb Jahan
 
How the philosophy of mathematical practice can be logic by other means (bris...
Brendan Larvor
 
Eswcsummerschool2010 ontologies final
Elena Simperl
 
Supermathematics and Artificial General Intelligence
Jordan Bennett
 

What's hot (20)

PPTX
The logic(s) of informal proofs (vub)
Brendan Larvor
 
ODP
How to Ground A Language for Legal Discourse In a Prototypical Perceptual Sem...
L. Thorne McCarty
 
PDF
Constructive Description Logics 2006
Valeria de Paiva
 
PDF
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Antonio Lieto
 
PDF
An Approach to Automated Learning of Conceptual Graphs from Text
Fulvio Rotella
 
PDF
Basic review on topic modeling
Hiroyuki Kuromiya
 
PDF
Rethinking Critical Editions of Fragments by Ontologies
Matteo Romanello
 
PDF
Improving Robustness and Flexibility of Concept Taxonomy Learning from Text
University of Bari (Italy)
 
PPTX
Introduction to Distributional Semantics
Andre Freitas
 
PDF
Combining text and pattern preprocessing in an adaptive dna pattern matcher
IAEME Publication
 
PDF
Constructive Hybrid Logics
Valeria de Paiva
 
PDF
Parsing techniques
Latchezar Tzvetkoff
 
PDF
postersimbolos
Apostolos Syropoulos
 
PDF
ESR10 Joachim Daiber - EXPERT Summer School - Malaga 2015
RIILP
 
PDF
Ai 2up-05-fol
skasuv
 
PDF
A Constructive Mathematics approach for NL formal grammars
Federico Gobbo
 
PDF
Negation in the Ecumenical System
Valeria de Paiva
 
PDF
Annotating Rhetorical and Argumentative Structures in Mathematical Knowledge
Christoph Lange
 
PDF
FUZZY LOGIC IN NARROW SENSE WITH HEDGES
ijcsit
 
PDF
A Natural Logic for Artificial Intelligence, and its Risks and Benefits
gerogepatton
 
The logic(s) of informal proofs (vub)
Brendan Larvor
 
How to Ground A Language for Legal Discourse In a Prototypical Perceptual Sem...
L. Thorne McCarty
 
Constructive Description Logics 2006
Valeria de Paiva
 
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Antonio Lieto
 
An Approach to Automated Learning of Conceptual Graphs from Text
Fulvio Rotella
 
Basic review on topic modeling
Hiroyuki Kuromiya
 
Rethinking Critical Editions of Fragments by Ontologies
Matteo Romanello
 
Improving Robustness and Flexibility of Concept Taxonomy Learning from Text
University of Bari (Italy)
 
Introduction to Distributional Semantics
Andre Freitas
 
Combining text and pattern preprocessing in an adaptive dna pattern matcher
IAEME Publication
 
Constructive Hybrid Logics
Valeria de Paiva
 
Parsing techniques
Latchezar Tzvetkoff
 
postersimbolos
Apostolos Syropoulos
 
ESR10 Joachim Daiber - EXPERT Summer School - Malaga 2015
RIILP
 
Ai 2up-05-fol
skasuv
 
A Constructive Mathematics approach for NL formal grammars
Federico Gobbo
 
Negation in the Ecumenical System
Valeria de Paiva
 
Annotating Rhetorical and Argumentative Structures in Mathematical Knowledge
Christoph Lange
 
FUZZY LOGIC IN NARROW SENSE WITH HEDGES
ijcsit
 
A Natural Logic for Artificial Intelligence, and its Risks and Benefits
gerogepatton
 
Ad

Similar to Like Alice in Wonderland: Unraveling Reasoning and Cognition Using Analogies and Concept Blending - Tarek R. Besold (20)

PDF
On the problems of interface: explainability, conceptual spaces, relevance
Giovanni Sileno
 
PDF
Cognitive Agents with Commonsense - Invited Talk at Istituto Italiano di Tecn...
Antonio Lieto
 
PPT
Concepts as Action-Oriented as 'Search'
mahmad
 
PDF
Artificial Thinking: can machines reason with analogies?
Federico Bianchi
 
PDF
Ai lecture 1
Ali Sufyan
 
PPT
A myth or a vision for interoperability: can systems communicate like humans do?
Milan Zdravković
 
PDF
A DEVELOPMENT FRAMEWORK FOR A CONVERSATIONAL AGENT TO EXPLORE MACHINE LEARNIN...
mlaij
 
PPTX
Artifical intelligence first lecture notes.pptx
jemwamakanaka
 
PPT
Survey of Analogy Reasoning
Sang-Kyun Kim
 
PPTX
Semantic, Cognitive and Perceptual Computing -Human mental representation
Artificial Intelligence Institute at UofSC
 
PDF
Artificial intelligence
mailmerk
 
PPT
Introduction to AI this is test ppt to download other one because it
YujanRanjitkar
 
PPTX
Sicsds
Devdatt Dubhashi
 
PPTX
l1.pptx
AnujaBeatriceB
 
PPTX
cycLearning: Cyc and Currious Cat as learning companion
The Open Education Consortium
 
DOC
Introduction to Artificial Intelligence.doc
butest
 
PPT
history_ of_ai.ppt
yordiatlaw
 
PPT
history_of_ai.ppt
Joueswant
 
PPT
history_of_ai.ppt
SriHarsha775064
 
PPT
history_of_ai.ppt
MichaelShenouda11
 
On the problems of interface: explainability, conceptual spaces, relevance
Giovanni Sileno
 
Cognitive Agents with Commonsense - Invited Talk at Istituto Italiano di Tecn...
Antonio Lieto
 
Concepts as Action-Oriented as 'Search'
mahmad
 
Artificial Thinking: can machines reason with analogies?
Federico Bianchi
 
Ai lecture 1
Ali Sufyan
 
A myth or a vision for interoperability: can systems communicate like humans do?
Milan Zdravković
 
A DEVELOPMENT FRAMEWORK FOR A CONVERSATIONAL AGENT TO EXPLORE MACHINE LEARNIN...
mlaij
 
Artifical intelligence first lecture notes.pptx
jemwamakanaka
 
Survey of Analogy Reasoning
Sang-Kyun Kim
 
Semantic, Cognitive and Perceptual Computing -Human mental representation
Artificial Intelligence Institute at UofSC
 
Artificial intelligence
mailmerk
 
Introduction to AI this is test ppt to download other one because it
YujanRanjitkar
 
Sicsds
Devdatt Dubhashi
 
l1.pptx
AnujaBeatriceB
 
cycLearning: Cyc and Currious Cat as learning companion
The Open Education Consortium
 
Introduction to Artificial Intelligence.doc
butest
 
history_ of_ai.ppt
yordiatlaw
 
history_of_ai.ppt
Joueswant
 
history_of_ai.ppt
SriHarsha775064
 
history_of_ai.ppt
MichaelShenouda11
 
Ad

More from Facultad de Informática UCM (20)

PDF
¿Por qué debemos seguir trabajando en álgebra lineal?
Facultad de Informática UCM
 
PDF
TECNOPOLÍTICA Y ACTIVISMO DE DATOS: EL MAPEO COMO FORMA DE RESILIENCIA ANTE L...
Facultad de Informática UCM
 
PDF
DRAC: Designing RISC-V-based Accelerators for next generation Computers
Facultad de Informática UCM
 
PDF
uElectronics ongoing activities at ESA
Facultad de Informática UCM
 
PDF
Tendencias en el diseño de procesadores con arquitectura Arm
Facultad de Informática UCM
 
PDF
Formalizing Mathematics in Lean
Facultad de Informática UCM
 
PDF
Introduction to Quantum Computing and Quantum Service Oriented Computing
Facultad de Informática UCM
 
PPTX
Computer Design Concepts for Machine Learning
Facultad de Informática UCM
 
PDF
Inteligencia Artificial en la atención sanitaria del futuro
Facultad de Informática UCM
 
PDF
Design Automation Approaches for Real-Time Edge Computing for Science Applic...
Facultad de Informática UCM
 
PDF
Estrategias de navegación para robótica móvil de campo: caso de estudio proye...
Facultad de Informática UCM
 
PPTX
Fault-tolerance Quantum computation and Quantum Error Correction
Facultad de Informática UCM
 
PDF
Cómo construir un chatbot inteligente sin morir en el intento
Facultad de Informática UCM
 
PDF
Automatic generation of hardware memory architectures for HPC
Facultad de Informática UCM
 
PDF
Type and proof structures for concurrency
Facultad de Informática UCM
 
PDF
Hardware/software security contracts: Principled foundations for building sec...
Facultad de Informática UCM
 
PDF
Jose carlossancho slidesLa seguridad en el desarrollo de software implementad...
Facultad de Informática UCM
 
PDF
Do you trust your artificial intelligence system?
Facultad de Informática UCM
 
PDF
Redes neuronales y reinforcement learning. Aplicación en energía eólica.
Facultad de Informática UCM
 
PDF
Challenges and Opportunities for AI and Data analytics in Offshore wind
Facultad de Informática UCM
 
¿Por qué debemos seguir trabajando en álgebra lineal?
Facultad de Informática UCM
 
TECNOPOLÍTICA Y ACTIVISMO DE DATOS: EL MAPEO COMO FORMA DE RESILIENCIA ANTE L...
Facultad de Informática UCM
 
DRAC: Designing RISC-V-based Accelerators for next generation Computers
Facultad de Informática UCM
 
uElectronics ongoing activities at ESA
Facultad de Informática UCM
 
Tendencias en el diseño de procesadores con arquitectura Arm
Facultad de Informática UCM
 
Formalizing Mathematics in Lean
Facultad de Informática UCM
 
Introduction to Quantum Computing and Quantum Service Oriented Computing
Facultad de Informática UCM
 
Computer Design Concepts for Machine Learning
Facultad de Informática UCM
 
Inteligencia Artificial en la atención sanitaria del futuro
Facultad de Informática UCM
 
Design Automation Approaches for Real-Time Edge Computing for Science Applic...
Facultad de Informática UCM
 
Estrategias de navegación para robótica móvil de campo: caso de estudio proye...
Facultad de Informática UCM
 
Fault-tolerance Quantum computation and Quantum Error Correction
Facultad de Informática UCM
 
Cómo construir un chatbot inteligente sin morir en el intento
Facultad de Informática UCM
 
Automatic generation of hardware memory architectures for HPC
Facultad de Informática UCM
 
Type and proof structures for concurrency
Facultad de Informática UCM
 
Hardware/software security contracts: Principled foundations for building sec...
Facultad de Informática UCM
 
Jose carlossancho slidesLa seguridad en el desarrollo de software implementad...
Facultad de Informática UCM
 
Do you trust your artificial intelligence system?
Facultad de Informática UCM
 
Redes neuronales y reinforcement learning. Aplicación en energía eólica.
Facultad de Informática UCM
 
Challenges and Opportunities for AI and Data analytics in Offshore wind
Facultad de Informática UCM
 

Recently uploaded (20)

PDF
PRIZ Academy - 9 Windows Thinking Where to Invest Today to Win Tomorrow.pdf
PRIZ Guru
 
PDF
Mohammad Mahdi Farshadian CV - Prospective PhD Student 2026
Educational Group Mohammad Farshadian
 
PPTX
FINAL REVIEW FOR COPD DIANOSIS FOR PULMONARY DISEASE.pptx
rubeshbme
 
PPTX
Geodesy 1.pptx...............................................
abhi1361yadav
 
PPTX
CYBER-CRIMES AND SECURITY A guide to understanding
Davies Chacko
 
PPTX
Foundation to blockchain - A guide to Blockchain Tech
Davies Chacko
 
PPT
Project quality management in manufacturing
BarMusaTetik
 
DOCX
ASol_English-Language-Literature-Set-1-27-02-2023-converted.docx
AYUSHMANAV
 
PPTX
Engineering Ethics, Safety and Environment [Autosaved] (1).pptx
MehrabTaseenTalukder
 
PDF
composite construction of structures.pdf
AinieButt1
 
PPTX
UNIT-1 - COAL BASED THERMAL POWER PLANTS
Chandra Kumar S
 
PPTX
M Tech Sem 1 Civil Engineering Environmental Sciences.pptx
ShriNRPrasad
 
PPT
Mechanical Engineering MATERIALS Selection
arsalanahmad705384
 
PPTX
IOT PPTs Week 10 Lecture Material.pptx of NPTEL Smart Cities contd
ssusera400e8
 
PPTX
Welding lecture in detail for understanding
sahilpoonia10
 
PPTX
Lesson 3_Tessellation.pptx finite Mathematics
quakeplayz54
 
PPTX
Lecture Notes Electrical Wiring System Components
eedgecbhojpur
 
PDF
Well-logging-methods_new................
ZafriFarid
 
PPTX
Unit 5 BSP.pptxytrrftyyydfyujfttyczcgvcd
ghousebhasha2007
 
PPTX
MET 305 MODULE 1 KTU 2019 SCHEME 25.pptx
VinayB68
 
PRIZ Academy - 9 Windows Thinking Where to Invest Today to Win Tomorrow.pdf
PRIZ Guru
 
Mohammad Mahdi Farshadian CV - Prospective PhD Student 2026
Educational Group Mohammad Farshadian
 
FINAL REVIEW FOR COPD DIANOSIS FOR PULMONARY DISEASE.pptx
rubeshbme
 
Geodesy 1.pptx...............................................
abhi1361yadav
 
CYBER-CRIMES AND SECURITY A guide to understanding
Davies Chacko
 
Foundation to blockchain - A guide to Blockchain Tech
Davies Chacko
 
Project quality management in manufacturing
BarMusaTetik
 
ASol_English-Language-Literature-Set-1-27-02-2023-converted.docx
AYUSHMANAV
 
Engineering Ethics, Safety and Environment [Autosaved] (1).pptx
MehrabTaseenTalukder
 
composite construction of structures.pdf
AinieButt1
 
UNIT-1 - COAL BASED THERMAL POWER PLANTS
Chandra Kumar S
 
M Tech Sem 1 Civil Engineering Environmental Sciences.pptx
ShriNRPrasad
 
Mechanical Engineering MATERIALS Selection
arsalanahmad705384
 
IOT PPTs Week 10 Lecture Material.pptx of NPTEL Smart Cities contd
ssusera400e8
 
Welding lecture in detail for understanding
sahilpoonia10
 
Lesson 3_Tessellation.pptx finite Mathematics
quakeplayz54
 
Lecture Notes Electrical Wiring System Components
eedgecbhojpur
 
Well-logging-methods_new................
ZafriFarid
 
Unit 5 BSP.pptxytrrftyyydfyujfttyczcgvcd
ghousebhasha2007
 
MET 305 MODULE 1 KTU 2019 SCHEME 25.pptx
VinayB68
 

Like Alice in Wonderland: Unraveling Reasoning and Cognition Using Analogies and Concept Blending - Tarek R. Besold

  • 1. Like Alice in Wonderland: Unraveling Reasoning and Cognition Using Analogies and Concept Blending Tarek R. Besold KRDB, Faculty of Computer Science, Free University of Bozen-Bolzano 16. June 2016 Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 2. Honour to whom honour is due... The following is joint work with many people, most notably: Robert Robere, Department of Computer Science, University of Toronto (Canada). Enric Plaza, IIIA-CSIC, Barcelona (Spain). Kai-Uwe K¨uhnberger, Institute of Cognitive Science, University of Osnabr¨uck (Germany). Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 3. And Who is Paying (Some of) the Bills? The described work on concept blending has been conducted as part of the European FP7 Concept Invention Theory (COINVENT) project (FET-Open grant number 611553). Consortium members are: Free University of Bozen-Bolzano (S¨udtirol-Alto Adige, Italy) University of Osnabr¨uck (Germany) University of Magdeburg (Germany) University of Dundee (Scotland, UK) University of Edinburgh (Scotland, UK) Goldsmiths, University of London (UK) IIIA-CSIC, Barcelona (Catalunya, Spain) Aristotle University of Thessaloniki (Greece) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 4. Non-Classical and Cross-Domain Reasoning Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 5. Back in the Day (1) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 6. Back in the Day (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 7. Back in the Day (3) Rutherford analogy (underlying the Bohr-Rutherford model of the atom): Analogy between solar system and hydrogen atom: ...nucleus is more massive than electrons, sun is more massive than planets. ...nucleus attracts electrons (Coulomb’s law), sun attracts planets (Newton’s law of gravity). ...attraction plus mass relation causes electrons to revolve around nucleus, similarly planets revolve around sun. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 8. Analogy Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 9. Intermezzo: Analogy (1) Analogy “ànalog–a” - analogia, “proportion”. Informally: Claims of similarity, often used in argumentation or when explaining complex situations. A bit more formal: Analogy-making is the human ability of perceiving dissimilar domains as similar with respect to certain aspects based on shared commonalities in relational structure or appearance. (Incidental remark: In less complex forms also to be found in some other primates.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 10. Intermezzo: Analogy (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 11. Non-Classical and Cross-Domain Reasoning Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 12. Heuristic-Driven Theory Projection (1) Heuristic-Driven Theory Projection (HDTP) Computing analogical relations and inferences (domains given as many-sorted first-order logic representation/many-sorted term algebras) using a generalisation-based approach. Base and target of analogy defined in terms of axiomatisations, i.e., given by a finite set of formulae. Aligning pairs of formulae by means of anti-unification (extending classical Plotkin-style first-order anti-unification to a restricted form of higher-order anti-unification). Proof-of-concept applications in modelling mathematical reasoning and concept blending in mathematics. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 13. Heuristic-Driven Theory Projection (2) Figure: Analogy-making in HDTP. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 14. Heuristic-Driven Theory Projection (3) Anti-Unification Dual to the unification problem (see, e.g., logic programming or automated theorem proving). Generalising terms in a meaningful way, yielding for each term an anti-instance (distinct subterms replaced by variables). Goal: Finding the most specific anti-unifier. Plotkin: For a proper definition of generalisation, for a given pair of terms there always is exactly one least general generalisation (up to renaming of variables). Problem: Structural commonalities embedded in different contexts possibly not accessible by first-order anti-unification. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 15. Heuristic-Driven Theory Projection (4) Restricted Higher-Order Anti-Unification First-order terms extended by introducing variables taking arguments (first-order variables become variables with arity 0), making a term either a first-order or a higher-order term. Class of substitutions restricted to (compositions of) the following four cases: 1 Renamings rF,F⇤ : F(t1,...,tn) rF,F⇤ ! F⇤(t1,...,tn). 2 Fixations fF c : F(t1,...,tn) fF f ! f(t1,...,tn). 3 Argument insertions iF,F⇤ G,i : F(t1,...,tn) i F,F⇤ G,i ! F⇤(t1,...,ti ,G(ti+1,...,ti+k ),ti+k+1,...,tn). 4 Permutations pF,F⇤ a : F(t1,...,tn) p F,F⇤ a ! F⇤(ta(1),...,ta(n)). Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 16. Heuristic-Driven Theory Projection (5) Examples of higher-order anti-unifications: Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 17. Heuristic-Driven Theory Projection (6) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 18. Heuristic-Driven Theory Projection (7) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 19. Complexity and Tractability in Cognitive Models and Systems Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 20. Computer Metaphor and Church-Turing Thesis Famous ideas at the heart of many endeavours in computational cognitive modelling and/or AI: 1 “Computer metaphor” of the mind (i.e. the concept of a computational theory of mind). 2 Church-Turing thesis. 1 Bridges gap between humans and computers: Human mind and brain can be seen as information processing system. Reasoning and thinking corresponds to computation as formal symbol manipulation. 2 Gives account of the nature and limitations of the computational power of such a system. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 21. Computer Metaphor and Church-Turing Thesis Famous ideas at the heart of many endeavours in computational cognitive modelling and/or AI: 1 “Computer metaphor” of the mind (i.e. the concept of a computational theory of mind). 2 Church-Turing thesis. 1 Bridges gap between humans and computers: Human mind and brain can be seen as information processing system. Reasoning and thinking corresponds to computation as formal symbol manipulation. 2 Gives account of the nature and limitations of the computational power of such a system. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 22. P-Cognition Thesis Significant impact on cognitive science and cognitive psychology: Explain human cognitive capacities modelled in terms of computational-level theories (i.e., as precise characterisations of hypothesised inputs and outputs of respective capacities together with functional mappings between them). Problem: Computational-level theories often underconstrained by available empirical data! ) Use mathematical complexity theory as assisting tool: NP-completeness! P-Cognition thesis Human cognitive capacities hypothesised to be of the polynomial-time computable type. (Interpretation: “Humans can comfortably solve non-trivial instances of this problem, where the exact size depends on the problem at hand”.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 23. P-Cognition Thesis Significant impact on cognitive science and cognitive psychology: Explain human cognitive capacities modelled in terms of computational-level theories (i.e., as precise characterisations of hypothesised inputs and outputs of respective capacities together with functional mappings between them). Problem: Computational-level theories often underconstrained by available empirical data! ) Use mathematical complexity theory as assisting tool: NP-completeness! P-Cognition thesis Human cognitive capacities hypothesised to be of the polynomial-time computable type. (Interpretation: “Humans can comfortably solve non-trivial instances of this problem, where the exact size depends on the problem at hand”.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 24. “polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 25. “polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 26. “polynomial-time computable” = “efficient”? Humans able to solve problems which may be hard in general but feasible if certain parameters of the problem restricted. Parametrised complexity theory: “tractability” captured by FPT.1 FPT-Cognition thesis (van Rooij, 2008) Human cognitive capacities hypothesised to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., computational-level theories have to be in FPT). Tractable AGI thesis (Besold & Robere, 2013) Models of cognitive capacities in artificial intelligence and computational cognitive systems have to be fixed-parameter tractable for one or more input parameters that are small in practice (i.e., have to be in FPT). 1 A problem P is in FPT if P admits an O(f(k)nc ) algorithm, where n is the input size, k is a parameter of the input constrained to be “small”, c is an independent constant, and f is some computable function. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 27. Complexity of HDTP (1) HDTP is naturally split into two mechanisms: Analogical matching of input theories. Re-representation of input theories by deduction in FOL. ) Re-representation is undecidable (undecidability of FOL). ) Focus on mechanism for analogical matching. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 28. Complexity of HDTP (2) Problem 1. F Anti-Unification Input: Two terms f,g, and a natural k 2 N Problem: Is there an anti-unifier h, containing at least k variables, using only renamings and fixations? Problem 2. FP Anti-Unification Input: Two terms f,g, and naturals l,m,p 2 N. Problem: Is there an anti-unifier h, containing at least l 0-ary variables and at least m higher arity variables, and two substitutions s,t using only renamings, fixations, and at most p permutations such that h s ! f and h t ! g? Problem 3. FPA Anti-Unification Input: Two terms f,g and naturals l,m,p,a 2 N. Problem: Is there an anti-unifier h, containing at least l 0-ary variables, at least m higher arity variables, and two substitutions s,t using renamings, fixations, at most p permutations, and at most a argument insertions such that h s ! f and h t ! g? Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 29. Complexity of HDTP (3) ...a fair share of formal magic involving a Canadian and some “Subgraph Isomorphism to Clique” reductions later... Complexity of HDTP (Higher-Order Anti-Unification) 1 F Anti-Unification is solvable in polynomial time. 2 Let m denote the minimum number of higher arity variables and let p be the maximum number of permutations applied. Then FP Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p}. 3 Let r be the maximum arity and s be the maximum number of subterms of the input terms. Then FP Anti-Unification is in FPT w.r.t. parameter set {s,r,p}. 4 FPA Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p,a}. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 30. Complexity of HDTP (3) ...a fair share of formal magic involving a Canadian and some “Subgraph Isomorphism to Clique” reductions later... Complexity of HDTP (Higher-Order Anti-Unification) 1 F Anti-Unification is solvable in polynomial time. 2 Let m denote the minimum number of higher arity variables and let p be the maximum number of permutations applied. Then FP Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p}. 3 Let r be the maximum arity and s be the maximum number of subterms of the input terms. Then FP Anti-Unification is in FPT w.r.t. parameter set {s,r,p}. 4 FPA Anti-Unification is NP-complete and W[1]-hard w.r.t. parameter set {m,p,a}. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 31. Concept Blending Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 32. Concept blending: A + B = ? (1) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 33. Concept blending: A + B = ? (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 34. Concept blending: A + B = ? (3) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 35. Concept blending: A + B = ? (4) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 36. Foundations of Theory Blending (1) Concept Blending Given two domain theories I1 and I2, representing two conceptualisations... ...look for a generalisation G... ...construct the blend space B in such a way as to preserve the correlations between I1 and I2 established by G. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 37. Foundations of Theory Blending (2) Example: Houseboat vs. boathouse Concept blends of HOUSE and BOAT into BOATHOUSE and HOUSEBOAT. I1 = {HOUSE v 8LIVES IN.RESIDENT} I2 = {BOAT v 8RIDES ON.PASSENGER} HOUSEBOAT: Aligning parts of the conceptual spaces... RESIDENT $ PASSENGER LIVES IN $ RIDES ON HOUSE $ BOAT BOATHOUSE: Aligning parts of the conceptual spaces... RESIDENT $ BOAT Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 38. The Concept Invention Theory (COINVENT) Project (1) To develop a novel, computationally feasible, formal model of conceptual blending based on Fauconnier and Turner’s theory. To gain a deeper understanding of conceptual blending and its role in computational creativity. To design a generic, creative computational system capable of serendipitous invention and manipulation of novel abstract concepts. To validate our model and its computational realisation in two representative working domains: mathematics and music. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 39. The Concept Invention Theory (COINVENT) Project (2) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 40. Amalgamation 101 (1) I1 I2 ¯I2 ¯I1 G = I1 u I2 A = ¯I1 t ¯I2 v v v vvv v v Amalgam A description A 2 L is an amalgam of two inputs I1 and I2 (with anti-unification G = I1 uI2) if there exist two generalisations ¯I1 and ¯I2 such that (1) G v¯I1 v I1, (2) G v¯I2 v I2, and (3) A =¯I1 t¯I2 Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 41. Amalgamation 101 (2) v vv v v v A = S t T S S T G = S u T Asymmetric Amalgam An asymmetric amalgam A 2 L of two inputs S (source) and T (target) satisfies that A = S0 tT for some generalisation of the source S0 v S. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 42. COINVENT’s Blending Schema 1.) Compute shared generalisation G from S and T with fS(G) = Sc. 2.) Re-use fS in generalisation of S into S0. 3.) Combine S0 in asymmetric amalgam with T into proto-blend T0 = S0 tT. 4.) By application of fT , complete T0 into blended output theory TB. (✓: element-wise subset relationship between sets of axioms. v: subsumption between theories in direction of respective arrows.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 43. ...and the Implementation? Use HDTP for computation of generalisation(s) and substitution chains/higher-order anti-unifications. Currently: Restrict HDTP to using only renamings and fixations. ) Possibility to use “classical” semantic consequence |= as ordering relationship. (Also preserved by later unifications and addition of axioms.) Use HDTP’s heuristics for selecting least general generalisation G (among several options). Currently: Naive consistency/inconsistency check with final blend (both internally and against world knowledge). ) Clash resolution by re-start with reduced set of input axioms. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 44. ...and the Implementation? Use HDTP for computation of generalisation(s) and substitution chains/higher-order anti-unifications. Currently: Restrict HDTP to using only renamings and fixations. ) Possibility to use “classical” semantic consequence |= as ordering relationship. (Also preserved by later unifications and addition of axioms.) Use HDTP’s heuristics for selecting least general generalisation G (among several options). Currently: Naive consistency/inconsistency check with final blend (both internally and against world knowledge). ) Clash resolution by re-start with reduced set of input axioms. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 45. Example: Brillo, the Foldable Toothbrush Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 46. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Stereotypical characterization for a pocketknife: Stereotypical characterization for a toothbrush:
  • 47. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Computing a shared generalization: Applied substitutions:
  • 48. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) Applied substitutions:
  • 49. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
  • 50. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
  • 51. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) Applied substitutions: pocketknife, toothbrush => E
  • 52. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
  • 53. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
  • 54. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
  • 55. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P
  • 56. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Shared generalization: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 57. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Computing the generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 58. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, F) Computing the generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 59. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 60. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 61. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 62. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Generalized source theory: Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 63. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Computing the proto-blend/ asymmetric amalgam:
  • 64. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 65. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 66. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 67. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 68. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(E, P) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 69. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 70. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 71. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(E, P) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F
  • 72. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Proto-blend/ asymmetric amalgam:
  • 73. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F Computing the final blend: has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
  • 74. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
  • 75. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(E, hinge) has_functionality(E, fold)
  • 76. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
  • 77. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
  • 78. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold)
  • 79. has_part(pocketknife, handle) has_part(pocketknife, blade) has_functionality(pocketknife, cut) has_part(pocketknife, hinge) has_functionality(pocketknife, fold) has_part(toothbrush, handle) has_part(toothbrush, brush_head) has_functionality(toothbrush, brush) has_part(toothbrush, handle) has_part(toothbrush, brush_head) Applied substitutions: pocketknife, toothbrush => E blade, brush_head => P cut, brush => F has_functionality(toothbrush, brush) has_part(toothbrush, hinge) has_functionality(E, fold) Final blend:
  • 80. Conclusion Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 81. (Definitely Not) The End! If you are interested in non-classical reasoning, tractability, approximability and similar topics in A(G)I and/or cognitive science, you are happily invited to... 1 ...talk to me after the presentation. 2 ...get in touch by e-mail: TarekRichard.Besold@unibz.it. 3 ...occasionally have a look at our publications.2 2 For instance: Besold, T. R., and Robere, R.. When Thinking Never Comes to a Halt: Using Formal Methods in Making Sure Your AI Gets the Job Done Good Enough. In V. C. M¨uller (ed.), Fundamental Issues of Artificial Intelligence (Synthese Library, vol. 376). Springer, 2016. Besold, T. R., and Plaza, E. Generalize and Blend: Concept Blending Based on Generalization, Analogy, and Amalgams. In H. Toivonen, S. Colton, M. Cook, and D. Ventura, Proceedings of the Sixth International Conference on Computational Creativity (ICCC) 2015. Brigham Young University Press, 2015. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 82. Postludium Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 83. Disclaimer Frequent criticism: Demanding for cognitive systems and models to work within certain complexity limits overly restrictive. Maybe: Human mental activities actually performed as exponential-time procedures, but never noticed as exponent for some reason always very small. Reply: Possibility currently cannot be excluded. Instead: No claim that cognitive processes without exception within FPT, APX, FPA, or what-have-you... ...but staying within boundaries makes cognitive systems and models plausible candidates for application in resource-bounded general-purpose cognitive agents. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 84. Disclaimer Frequent criticism: Demanding for cognitive systems and models to work within certain complexity limits overly restrictive. Maybe: Human mental activities actually performed as exponential-time procedures, but never noticed as exponent for some reason always very small. Reply: Possibility currently cannot be excluded. Instead: No claim that cognitive processes without exception within FPT, APX, FPA, or what-have-you... ...but staying within boundaries makes cognitive systems and models plausible candidates for application in resource-bounded general-purpose cognitive agents. Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 85. Complexity of the First HDTP Generation As an aside: Once upon a time, there was HDTP-old based on reducing certain higher-order to first-order anti-unifications by introduction of subterms built from “admissible sequences” over equational theories (i.e., conjunctions of FOL formulae with equality over a term algebra). Complexity of HDTP-old 1 HDTP-old is NP-complete. 2 HDTP-old is W[2]-hard with respect to a minimal bound on the cardinality of the set of all subterms of the term against which admissibility is checked. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 86. Complexity of the First HDTP Generation As an aside: Once upon a time, there was HDTP-old based on reducing certain higher-order to first-order anti-unifications by introduction of subterms built from “admissible sequences” over equational theories (i.e., conjunctions of FOL formulae with equality over a term algebra). Complexity of HDTP-old 1 HDTP-old is NP-complete. 2 HDTP-old is W[2]-hard with respect to a minimal bound on the cardinality of the set of all subterms of the term against which admissibility is checked. (For proofs: R. Robere and T. R. Besold. Complex Analogies: Remarks on the Complexity of HDTP. In Proceedings of the 25th Australasian Joint Conference on Artificial Intelligence (AI 2012), LNCS 7691. Springer, 2012.) Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 87. A Primer on Approximation Theory Approximability Classes In the following, let... ...PTAS denote the class of all NP optimisation problems that admit a polynomial-time approximation scheme. ...APX be the class of NP optimisation problems allowing for constant-factor approximation algorithms. ...APX-poly be the class of NP optimisation problems allowing for polynomial-factor approximation algorithms. Please note that PTAS ✓ APX ✓ APX-poly (with each inclusion being proper in case P 6= NP). Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 88. Approximability Analysis of HDTP (1) FP Anti-Unification W[1]-hard to compute for parameter set m,p (m number of higher-arity variables, p number of permutations). ) No polynomial-time algorithm computing “sufficiently complex” generalisations (i.e., with lower bound on number of higher-arity variables), upper bounding number of permutations (W[1]-hardness for single permutation). What if one considers generalisations which merely approximate the “optimal” generalisation in some sense? Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 89. Approximability Analysis of HDTP (2) Complexity of a Substitution The complexity of a basic substitution s is defined as C(s) = 8 >< >: 0, if s is a renaming. 1, if s is a fixation or permutation. k +1, if s is a k-ary argument insertion. The complexity of a restricted substitution s = s1 ··· sn (i.e., the composition of any sequence of unit substitutions) is the sum of the composed substitutions: C(s) = Ân i=1 C(si ). Tarek R. Besold Computational Models of Analogy and Concept Blending
  • 90. Approximability Analysis of HDTP (3) Consider problem of finding generalisation which maximises complexity over all generalisations: Complex generalisation would contain “most information” present over all of the generalisations chosen (i.e., maximising the “information load”). Using approximability results on MAXCLIQUE: Approximation Complexity of HDTP Analogy-Making FP anti-unification is not in APX (i.e., does not allow for constant-factor approximation algorithms) and is hard for APX-poly. (For proofs: T. R. Besold and R. Robere. When Almost Is Not Even Close: Remarks on the Approximability of HDTP. In Artificial General Intelligence - 6th International Conference (AGI 2013), LNCS. Springer, 2013.) Tarek R. Besold Computational Models of Analogy and Concept Blending