SlideShare a Scribd company logo
Contexts for Quantification
Valeria de Paiva
Nuance Communications, Inc.
May, 2013
1 / 35
Natural Language inspired KR and Inference
How? Natural Logic Program (80’s but here especially Moss and
Pratt-Hartmann)
Goal: Show that significant parts of natural language inference can be
carried out in simple, mathematically characterized logical systems.
Whenever possible, to obtain complete axiomatizations.
Work using all tools of fields like
proof theory, categorical logic, (finite) model theory
and perhaps complexity theory...
2 / 35
Natural Logic
Familiar logic concepts of satisfiability and entailment can be defined
for any subset of a natural language that has been translated into a
formal system such as first-order logic FOL
For such a fragment of English it makes sense to ask the
computational complexity of determining satisfiability and entailment
within this fragment
It also makes sense to ask how complete is the fragment
3 / 35
The System TIL
This work presents the logical system TIL (for textual inference logic),
a formalization of the AI system Bridge developed in PARC.
Constraints on Bridge:
robustness (any sentence gets a representation)
as expressive as possible, without compromising
computability/tractability and ease of use
4 / 35
The System TIL
Previous presentations of TIL described it as a stand alone system,
synthesized from what is produced by the software, when processing
sentences in English.
Much work has gone and has still has to go into the process of
creating sensible logical representations from sentences.
Not worrying about this process (or how to improve it) now.
Goal is to place TIL in the context of the work of Natural Logic, as an
extension of traditional syllogistic logic.
5 / 35
Which logic?
Bridge reads in a sentence in English and produces a logical
knowledge representation R for it.
The collection of all representations plus some primitive means of
manipulating them form the system TIL.
TIL was meant to be kept as close as it was sensible to FOL
(first-order logic) but from the start we knew that to model natural
language sentences we wanted intensional ‘concepts’ and ‘contexts’
(cf. ‘Preventing Existence’, FOIS’01).
TIL comes from event semantics, AI and description logic well
tested intuitions.
6 / 35
From linguists: Event Semantics
Because it makes easier to infer from
Zach Quinto played Spock in Star Trek in 2009.
that:
Quinto played Spock.
Quinto played in Star Trek in 2009.
Quinto played Spock in 2009.
Etc..
All other variations (Lasersohn 2006)
7 / 35
From linguists: Intensionality is Necessary
Because one needs e.g. to deal with
1. non-existent entities
Negotiations prevented a strike.
Cannot simply render this as
∃n : Negotiation, ∃s : Strike. prevented(n, s)
2. Propositional Attitudes:
i want to know if the red sox game is going to be showed on the east
coast time
3. Negation:
record the ones that are not repeats
all interesting films that are not quentin tarantino
not so loud
8 / 35
From AI: McCarthy contexts
John McCarthy: “contexts as abstract mathematical entities with
properties useful in artificial intelligence"
ist(c, p) means proposition p is true in context c
Which properties?
Fencing-off propositions, “lifting relations" between contexts..
Because contexts are small “possible worlds” that seem just enough for
the phenomena at hand.
Modal logic carves out a tractable subset of FOL.
9 / 35
Description-Logic like concepts?
Description Logics:
carving out a subset of FOL computationally tractable
Similar concepts in TIL, but not quite the same: relational DL
Franconi (early 90s): DLs used in NLP mainly for representing case
frames attached to world knowledge.
He wanted to extend ‘description logics in the direction of capturing
NL phenomena’.
We capture the NL phenomena in the grammar.
10 / 35
An Example: “Three boys ate pizzas"
Conceptual Structure:
subconcept(boy-1,[List1])
subconcept(eat-3,[List2])
subconcept(pizza-5,[List3])
role(ob,eat-3,pizza-5)
role(sb,eat-3,boy-1)
role(cardinality-restriction,boy-1,3)
role(cardinality-restriction,pizza-5,pl)
Contextual Structure:
context(t)
instantiable(boy-1,t)
instantiable(eat-3,t)
instantiable(pizza-5,t)
top-context(t)
11 / 35
TIL has concepts, contexts and roles...
Instead of constants and variables like FOL,
TIL has concepts and subconcepts.
The concept represented by boy-1 is a subconcept of one of the
concepts in the list of concepts [List1].
These can be the synsets in WordNet for the word ‘boy’ or a list of
Cyc concepts, or ...
Concepts come from a parameter ontology:
Cyc, WN, UL, KM, SUMO, etc.
You choose the ontology!
12 / 35
TIL has concepts, roles and contexts...
Concepts in TIL are similar to Description Logic concepts.
They are similar to predicates in FOL,
but are not always unary predicates.
Think of ‘eat-3’ above as a collection of ‘eating’ events,
in which other concepts in our domain participate.
Have two kinds of concepts, primitive concepts from the ontology
and constructed concepts, which are always sub-concepts of some
primitive concept.
Sub-concepts can be refined as necessary, so they are
underspecified, a good thing.
13 / 35
TIL has concepts, contexts and roles...
Concepts are related to others via roles. Like
role(ob,eat-3,pizza-5)
role(sb,eat-3,boy-1)
The name of the role in question (ob, sb, agent, patient, theme,
arg-1, arg-2, etc...) will not matter for us here. They come from a
fixed vocabulary associated to the ontology.
Deciding which roles will be used with which concepts are hard
problems in computational linguistics.
We assume roles are assigned in a consistent, coherent and maximally
informative way.
14 / 35
TIL has concepts, contexts and roles...
One main difference is contexts and how we use them for e.g.
negation. For example, for the sentence No boys hopped.
Conceptual Structure:
role(cardinality-restriction,boy-5,no)
role(sb,hop-6,boy-5)
subconcept(boy-5,[List1])
subconcept(hop-6,[List2])
Contextual Structure:
context(ctx(hop-6)), context(t)
context-lifting-relation(antiveridical,t,ctx(hop-6))
context-relation(t,ctx(hop-6),not-29)
instantiable(boy-5,ctx(hop-6))
instantiable(hop-6,ctx(hop-6))
top-context(t)
uninstantiable(hop-6,t)
15 / 35
TIL has concepts, contexts and roles...
Notice that context relations are:
veridical
antiveridical
averidical
Negation is antiveridical,
while a context introduced by say is averidical.
16 / 35
TIL has concepts, contexts and roles...
John knew that Mary arrived.
Conceptual Structure:
role(cardinality-restriction,john-1,sg)
role(cardinality-restriction,mary-3,sg)
role(sb,know-2,john-1)
role(topic,know-2,ctx(arrive-4))
role(sb,arrive-4,mary-3)
subconcept(john-1,[List1]) subconcept(mary-3,[List3])
subconcept(know-2,[List2]) subconcept(arrive-4,[List4])
Contextual Structure:
context(ctx(arrive-4)) context(t) top-context(t)
context-lifting-relation(veridical,t,ctx(arrive-4))
context-relation(t,ctx(arrive-4),know-2)
instantiable(john-1,t) instantiable(mary-2,t)
instantiable(arrive-4,ctx(arrive-4)) instantiable(know-2, t)
instantiable(arrive-4,t)
17 / 35
TIL has concepts, contexts and roles...
Instead of formulas we have a clausal representation R
R::= conceptual stx | contextual strux | (temporal strux)
CpA ::= subconcept (Sk-1, PrimitiveConcept) |
role(rolename, Sk-1, Sk-2) |
role(rolename, Sk-1, CONST1)
PrimitiveConcepts from Ontology O, role names from SetOfRoles
CtxA ::= t | ctx(Sk-1) | context-relation(cxt-1,cxt-2, Sk)
context-lifting-relation(CONST2, cxt-1, cxt-2) |
instantiable (Sk, cxt) | uninstantiable (Sk, cxt)
where CONST2= veridical, averidical, antiveridical
+ syntactic sugar
18 / 35
Inference
Conceptual structure can be translated back into FOL.
What about contextual structure?
Contexts look like possible worlds. Just Modal Logic?
19 / 35
Just Modal Logic?
Not really. Standard Propositional modal logic is not enough
Knowjohn(Arrived(Mary)) is OK.
but what about:
John and Bob knew that Mary arrived,
John but not Bob knew that Mary arrived, etc..
Quantified modal logic looks too much.
and theorem proving becomes an issue.
How much Modal Logic?
20 / 35
Inference in TIL?
Inference in TIL is very rudimentary.
We can ‘drop clauses’ like in most event semantics.
From the sentence Ed walked and Mary talked
we are able to infer both Ed walked and Mary talked
by simply forgetting the respective clauses in the original
representation.
We can do trivial inferences like identity and we can compose
derivations:
s → s
s → t t → r
s → r
21 / 35
How does TIL deal with syllogisms?...
Example: All boys are mammals
Conceptual Structure:
role(cardinality-restriction,boy-2,all(pl))
role(cardinality-restriction,mammal-4,pl)
role(copula-pred,be-3,mammal-4)
role(copula-subj,be-3,boy-2)
subconcept(boy-2,[List1])
subconcept(mammal-4,[List2])
Contextual Structure:
context(t)
instantiable(be-3,t)
instantiable(boy-2,t)
instantiable(mammal-4,t)
top-context(t)
22 / 35
Syllogistic Logic of All
Syntax: Start with a collection of unary atoms (for nouns).
The sentences are the expressions All p are q
Semantics: A model M is a collection of sets M,
and for each noun p we have an interpretation [[p]] ⊆ M.
M |= All p are q iff [[p]] ⊆ [[q]]
Proof system:
All p are p
All p are n All n are q
All p are q
23 / 35
TIL satisfies the rules for All
All p are p
All p are n All n are q
All p are q
Semantically just transitivity of subset containment.
All boys are boys is odd, but not problematic...
For TIL the transitive inference is simply climbing up the concept
hierarchy.
All boys are mammals All mammals are animals
All boys are animals
24 / 35
Syllogistic Logic of Some
The sentences are the expressions Some p are q
Semantics: A model M is a collection of sets M,
and for each noun p,q we have an interpretation [[p,q]] ⊆ M.
M |= Some p are q iff [[p]] ∩ [[q]] = ∅
Proof system:
Some p are q
Some q are p
Some p are q
Some p are p
All q are n Some p are q
Some p are n
25 / 35
TIL satisfies the rules for Some
Some p are q
Some q are p
Some p are q
Some p are p
All q are n Some p are q
Some p are n
The semantics is the same as Moss’, intersection of subsets.
The inference relation between ‘all’ and ‘some’ is outsourced.
Our ‘poor man’s inference system’ called Entailment and
Contradiction Detection (ECD), has a table of relationships between
‘cardinality-restrictions’ postulated.
This is useful if you want to keep your logic options open, no need to decide if empty domains...
26 / 35
The languages S and S†
with noun-level negation
If one adds complemented atoms on top of
the language of All and Some,
with interpretation via set complement: [[p]] = M  [[p]],
So if one has
S



All p are q
Some p are q
All p are q ≡ No p are q
Some p are q ≡ Some p aren’t q
Some non-p are non-q



S†
Things can get strange, as explained by Moss-Hartmann: no finite
axiomatizability....
27 / 35
The logical system for Sp
+ names
The language Sp
is the language of All and Some, with no negation, p
for positive syllogisms.
All p are p
Some p are q
Some p are p
Some p are q
Some q are p
All p are n All n are q
All p are q
All n are p Some n are q
Some p are q
J is J
J is M M is F
J is F
J is a p J is a q
Some p are q
All p are q J is a p
J is a q
M is a p J is M
J is a p
28 / 35
TIL satisfies Sp
+ names
This corresponds to sentences like
Jon is Jon,
Jon is Mary and Mary is Fred entails Fred is Jon,
Jon is a man and Jon is a doctor entails Some men are doctors,
All cats are mammals and Jon is a cat entails Jon is a mammal
Mary is a cat and Jon is Mary entails Jon is a cat.
29 / 35
TIL is not about syllogisms or copula...
Event semantics in general is about transitive,
intransitive, ditransitive, etc. verbs.
TIL should be able to deal with Sp
and Rp
and more...
Noun negation should not present problems,
(outsourced again)
but sentential negation is dealt via contexts.
Contexts look like the higher parts of the next picture...
Modal and Hybrid logic!
30 / 35
A picture stolen from Larry Moss...
S≥
FOmon
Peano-Frege
Church-Turing
S
S†
R
R†
Sp
Rp
FOL
Kn
FO2 2 variable FO logic
† adds full N-negation
relational syllogistic
31 / 35
Conclusions
Described a system TIL, arising from the confluence of intuitions from
linguistics and AI.
TIL draws from event semantics, McCarthy notions of context and
description logics hopes of tractability.
Showed that the system satisfies some of the fragments investigated
computationally by Moss and Pratt-Hartmann, but since we have a
constructive basis, while they have a classical one, more work is
required for full comparison.
32 / 35
Further Work
Climbing up the syllogistic hierarchy of Moss and Pratt-Hartmann
seems to work, but negation (constructive in TIL) takes us to a
different kind of landscape.
Need to check relational syllogistic.
Need to check how picture works for constructive logics.
As pure proof theory (applied to KR) several pieces of the puzzle have
been put in place.
An orthogonal issue is a practical implementation suitable for
computational semantics in the large, the original motivation.
33 / 35
Some References
Preventing Existence (C. Condoravdi, D. Crouch, V de Paiva, J. Everett, R.
Stolle, D. Bobrow, M. van den Berg), In Proceedings of Formal Ontology in
Information Systems, FOIS’01, October 2001
Computing relative polarity for textual inference (Rowan Nairn, Lauri
Karttunen and Cleo Condoravdi) Computing relative polarity for textual
inference. In the Proceedings of ICoS-5 (Inference in Computational
Semantics). April 20-21, 2006. Buxton, UK.. In the Proceedings of ICoS-5
(Inference in Computational Semantics). April 20-21, 2006. Buxton, UK.
Textual Inference Logic: Take Two, (V. de Paiva, D. G. Bobrow, C.
Condoravdi, R. Crouch, L. Karttunen, T. H. King, R. Nairn and A. Zaenen)
Proceedings of the Workshop on Contexts and Ontologies, Representation
and Reasoning, CONTEXT 2007.
Precision-focused Textual Inference (Bobrow, D. G., C. Condoravdi, L.
Karttunen, T. H. King, V. de Paiva, L. Price, R. Nairn, L.Price, A. Zaenen)
Proceedings of ACL-PASCAL Workshop on Textual Entailment and
Paraphrasing, pp. 16-21, 2007.
34 / 35
Nesting "modalities"
Ed did not forget to force Dave to leave.
⇒ Dave left.
35 / 35

More Related Content

PDF
Lean Logic for Lean Times: Varieties of Natural Logic
PDF
Lean Logic for Lean Times: Entailment and Contradiction Revisited
PDF
Portuguese Linguistic Tools: What, Why and How
PDF
Constructive Description Logics 2006
PDF
Artificial intelligence and first order logic
PPTX
Lecture 2: From Semantics To Semantic-Oriented Applications
PDF
Semantics and Computational Semantics
PDF
Constructive Hybrid Logics
Lean Logic for Lean Times: Varieties of Natural Logic
Lean Logic for Lean Times: Entailment and Contradiction Revisited
Portuguese Linguistic Tools: What, Why and How
Constructive Description Logics 2006
Artificial intelligence and first order logic
Lecture 2: From Semantics To Semantic-Oriented Applications
Semantics and Computational Semantics
Constructive Hybrid Logics

What's hot (20)

PDF
A Bridge Not too Far
PDF
Contexts for Quantification
PPTX
First order predicate logic(fopl)
PDF
Cognitive plausibility in learning algorithms
PDF
Lecture 2: Computational Semantics
PDF
First order logic
PPTX
PDF
Contexts for Quantification
PPT
Syntax and semantics of propositional logic
PPTX
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...
PDF
MORPHOLOGICAL SEGMENTATION WITH LSTM NEURAL NETWORKS FOR TIGRINYA
PPTX
Language models
PDF
Dependent Types in Natural Language Semantics
PDF
Composing (Im)politeness in Dependent Type Semantics
PDF
Languages, Ontologies and Automatic Grammar Generation - Prof. Pedro Rangel H...
PDF
Mathematics rules and scientific representations
PDF
FUZZY LOGIC IN NARROW SENSE WITH HEDGES
PDF
artificial intelligence
PDF
Ai lecture 07(unit03)
PPTX
First order predicate logic (fopl)
A Bridge Not too Far
Contexts for Quantification
First order predicate logic(fopl)
Cognitive plausibility in learning algorithms
Lecture 2: Computational Semantics
First order logic
Contexts for Quantification
Syntax and semantics of propositional logic
Artificial Intelligence (AI) | Prepositional logic (PL)and first order predic...
MORPHOLOGICAL SEGMENTATION WITH LSTM NEURAL NETWORKS FOR TIGRINYA
Language models
Dependent Types in Natural Language Semantics
Composing (Im)politeness in Dependent Type Semantics
Languages, Ontologies and Automatic Grammar Generation - Prof. Pedro Rangel H...
Mathematics rules and scientific representations
FUZZY LOGIC IN NARROW SENSE WITH HEDGES
artificial intelligence
Ai lecture 07(unit03)
First order predicate logic (fopl)
Ad

Similar to Contexts 4 quantification (CommonSense2013) (20)

PDF
Little engines of inference: contexts for quantification
DOC
jan2016jnkjsjksadksdkjnkjuojjhsiodsdsodih.doc
PPT
Inteligencia artificial
PPT
Jarrar.lecture notes.aai.2011s.ch8.fol.introduction
PPT
IKL presentation for Ontolog
DOCX
AI Lab Manual.docx
PPT
PDF
A course on mathematical logic
PDF
Annotations Propp S Functions
PDF
Ai lecture 09(unit03)
PDF
Ontology learning from interpretations in lightweight description logics
PPT
The law of non-contradiction in the combined calculus of sentences, situation...
PDF
Commonsense reasoning as a key feature for dynamic knowledge invention and co...
PPTX
Lecture 1-3-Logics-In-computer-science.pptx
PDF
Prolog,Prolog Programming IN AI.pdf
PPTX
Minimalism.pptx
PPT
INFO-2950-Languages-and-Grammars.ppt
PPTX
First Order Logic and Qualtifiers in AI.pptx
PPT
Predicate logic_2(Artificial Intelligence)
Little engines of inference: contexts for quantification
jan2016jnkjsjksadksdkjnkjuojjhsiodsdsodih.doc
Inteligencia artificial
Jarrar.lecture notes.aai.2011s.ch8.fol.introduction
IKL presentation for Ontolog
AI Lab Manual.docx
A course on mathematical logic
Annotations Propp S Functions
Ai lecture 09(unit03)
Ontology learning from interpretations in lightweight description logics
The law of non-contradiction in the combined calculus of sentences, situation...
Commonsense reasoning as a key feature for dynamic knowledge invention and co...
Lecture 1-3-Logics-In-computer-science.pptx
Prolog,Prolog Programming IN AI.pdf
Minimalism.pptx
INFO-2950-Languages-and-Grammars.ppt
First Order Logic and Qualtifiers in AI.pptx
Predicate logic_2(Artificial Intelligence)
Ad

More from Valeria de Paiva (20)

PDF
Dialectica Comonoids
PDF
Dialectica Categorical Constructions
PDF
Logic & Representation 2021
PDF
Constructive Modal and Linear Logics
PDF
Dialectica Categories Revisited
PDF
PLN para Tod@s
PDF
Networked Mathematics: NLP tools for Better Science
PDF
Going Without: a modality and its role
PDF
Problemas de Kolmogorov-Veloso
PDF
Natural Language Inference: for Humans and Machines
PDF
Dialectica Petri Nets
PDF
The importance of Being Erneast: Open datasets in Portuguese
PDF
Negation in the Ecumenical System
PDF
Constructive Modal and Linear Logics
PDF
Semantics and Reasoning for NLP, AI and ACT
PDF
NLCS 2013 opening slides
PDF
Dialectica Comonads
PDF
Categorical Explicit Substitutions
PDF
Logic and Probabilistic Methods for Dialog
PDF
Intuitive Semantics for Full Intuitionistic Linear Logic (2014)
Dialectica Comonoids
Dialectica Categorical Constructions
Logic & Representation 2021
Constructive Modal and Linear Logics
Dialectica Categories Revisited
PLN para Tod@s
Networked Mathematics: NLP tools for Better Science
Going Without: a modality and its role
Problemas de Kolmogorov-Veloso
Natural Language Inference: for Humans and Machines
Dialectica Petri Nets
The importance of Being Erneast: Open datasets in Portuguese
Negation in the Ecumenical System
Constructive Modal and Linear Logics
Semantics and Reasoning for NLP, AI and ACT
NLCS 2013 opening slides
Dialectica Comonads
Categorical Explicit Substitutions
Logic and Probabilistic Methods for Dialog
Intuitive Semantics for Full Intuitionistic Linear Logic (2014)

Recently uploaded (20)

PDF
Computing-Curriculum for Schools in Ghana
PDF
RMMM.pdf make it easy to upload and study
PPTX
Cell Structure & Organelles in detailed.
PDF
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
PPTX
master seminar digital applications in india
PDF
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
PPTX
Renaissance Architecture: A Journey from Faith to Humanism
PPTX
Pharmacology of Heart Failure /Pharmacotherapy of CHF
PPTX
Microbial diseases, their pathogenesis and prophylaxis
PPTX
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
PDF
Microbial disease of the cardiovascular and lymphatic systems
PDF
Supply Chain Operations Speaking Notes -ICLT Program
PDF
Sports Quiz easy sports quiz sports quiz
PPTX
GDM (1) (1).pptx small presentation for students
PDF
Basic Mud Logging Guide for educational purpose
PDF
2.FourierTransform-ShortQuestionswithAnswers.pdf
PPTX
Cell Types and Its function , kingdom of life
PDF
TR - Agricultural Crops Production NC III.pdf
PDF
Pre independence Education in Inndia.pdf
PDF
01-Introduction-to-Information-Management.pdf
Computing-Curriculum for Schools in Ghana
RMMM.pdf make it easy to upload and study
Cell Structure & Organelles in detailed.
The Lost Whites of Pakistan by Jahanzaib Mughal.pdf
master seminar digital applications in india
Physiotherapy_for_Respiratory_and_Cardiac_Problems WEBBER.pdf
Renaissance Architecture: A Journey from Faith to Humanism
Pharmacology of Heart Failure /Pharmacotherapy of CHF
Microbial diseases, their pathogenesis and prophylaxis
school management -TNTEU- B.Ed., Semester II Unit 1.pptx
Microbial disease of the cardiovascular and lymphatic systems
Supply Chain Operations Speaking Notes -ICLT Program
Sports Quiz easy sports quiz sports quiz
GDM (1) (1).pptx small presentation for students
Basic Mud Logging Guide for educational purpose
2.FourierTransform-ShortQuestionswithAnswers.pdf
Cell Types and Its function , kingdom of life
TR - Agricultural Crops Production NC III.pdf
Pre independence Education in Inndia.pdf
01-Introduction-to-Information-Management.pdf

Contexts 4 quantification (CommonSense2013)

  • 1. Contexts for Quantification Valeria de Paiva Nuance Communications, Inc. May, 2013 1 / 35
  • 2. Natural Language inspired KR and Inference How? Natural Logic Program (80’s but here especially Moss and Pratt-Hartmann) Goal: Show that significant parts of natural language inference can be carried out in simple, mathematically characterized logical systems. Whenever possible, to obtain complete axiomatizations. Work using all tools of fields like proof theory, categorical logic, (finite) model theory and perhaps complexity theory... 2 / 35
  • 3. Natural Logic Familiar logic concepts of satisfiability and entailment can be defined for any subset of a natural language that has been translated into a formal system such as first-order logic FOL For such a fragment of English it makes sense to ask the computational complexity of determining satisfiability and entailment within this fragment It also makes sense to ask how complete is the fragment 3 / 35
  • 4. The System TIL This work presents the logical system TIL (for textual inference logic), a formalization of the AI system Bridge developed in PARC. Constraints on Bridge: robustness (any sentence gets a representation) as expressive as possible, without compromising computability/tractability and ease of use 4 / 35
  • 5. The System TIL Previous presentations of TIL described it as a stand alone system, synthesized from what is produced by the software, when processing sentences in English. Much work has gone and has still has to go into the process of creating sensible logical representations from sentences. Not worrying about this process (or how to improve it) now. Goal is to place TIL in the context of the work of Natural Logic, as an extension of traditional syllogistic logic. 5 / 35
  • 6. Which logic? Bridge reads in a sentence in English and produces a logical knowledge representation R for it. The collection of all representations plus some primitive means of manipulating them form the system TIL. TIL was meant to be kept as close as it was sensible to FOL (first-order logic) but from the start we knew that to model natural language sentences we wanted intensional ‘concepts’ and ‘contexts’ (cf. ‘Preventing Existence’, FOIS’01). TIL comes from event semantics, AI and description logic well tested intuitions. 6 / 35
  • 7. From linguists: Event Semantics Because it makes easier to infer from Zach Quinto played Spock in Star Trek in 2009. that: Quinto played Spock. Quinto played in Star Trek in 2009. Quinto played Spock in 2009. Etc.. All other variations (Lasersohn 2006) 7 / 35
  • 8. From linguists: Intensionality is Necessary Because one needs e.g. to deal with 1. non-existent entities Negotiations prevented a strike. Cannot simply render this as ∃n : Negotiation, ∃s : Strike. prevented(n, s) 2. Propositional Attitudes: i want to know if the red sox game is going to be showed on the east coast time 3. Negation: record the ones that are not repeats all interesting films that are not quentin tarantino not so loud 8 / 35
  • 9. From AI: McCarthy contexts John McCarthy: “contexts as abstract mathematical entities with properties useful in artificial intelligence" ist(c, p) means proposition p is true in context c Which properties? Fencing-off propositions, “lifting relations" between contexts.. Because contexts are small “possible worlds” that seem just enough for the phenomena at hand. Modal logic carves out a tractable subset of FOL. 9 / 35
  • 10. Description-Logic like concepts? Description Logics: carving out a subset of FOL computationally tractable Similar concepts in TIL, but not quite the same: relational DL Franconi (early 90s): DLs used in NLP mainly for representing case frames attached to world knowledge. He wanted to extend ‘description logics in the direction of capturing NL phenomena’. We capture the NL phenomena in the grammar. 10 / 35
  • 11. An Example: “Three boys ate pizzas" Conceptual Structure: subconcept(boy-1,[List1]) subconcept(eat-3,[List2]) subconcept(pizza-5,[List3]) role(ob,eat-3,pizza-5) role(sb,eat-3,boy-1) role(cardinality-restriction,boy-1,3) role(cardinality-restriction,pizza-5,pl) Contextual Structure: context(t) instantiable(boy-1,t) instantiable(eat-3,t) instantiable(pizza-5,t) top-context(t) 11 / 35
  • 12. TIL has concepts, contexts and roles... Instead of constants and variables like FOL, TIL has concepts and subconcepts. The concept represented by boy-1 is a subconcept of one of the concepts in the list of concepts [List1]. These can be the synsets in WordNet for the word ‘boy’ or a list of Cyc concepts, or ... Concepts come from a parameter ontology: Cyc, WN, UL, KM, SUMO, etc. You choose the ontology! 12 / 35
  • 13. TIL has concepts, roles and contexts... Concepts in TIL are similar to Description Logic concepts. They are similar to predicates in FOL, but are not always unary predicates. Think of ‘eat-3’ above as a collection of ‘eating’ events, in which other concepts in our domain participate. Have two kinds of concepts, primitive concepts from the ontology and constructed concepts, which are always sub-concepts of some primitive concept. Sub-concepts can be refined as necessary, so they are underspecified, a good thing. 13 / 35
  • 14. TIL has concepts, contexts and roles... Concepts are related to others via roles. Like role(ob,eat-3,pizza-5) role(sb,eat-3,boy-1) The name of the role in question (ob, sb, agent, patient, theme, arg-1, arg-2, etc...) will not matter for us here. They come from a fixed vocabulary associated to the ontology. Deciding which roles will be used with which concepts are hard problems in computational linguistics. We assume roles are assigned in a consistent, coherent and maximally informative way. 14 / 35
  • 15. TIL has concepts, contexts and roles... One main difference is contexts and how we use them for e.g. negation. For example, for the sentence No boys hopped. Conceptual Structure: role(cardinality-restriction,boy-5,no) role(sb,hop-6,boy-5) subconcept(boy-5,[List1]) subconcept(hop-6,[List2]) Contextual Structure: context(ctx(hop-6)), context(t) context-lifting-relation(antiveridical,t,ctx(hop-6)) context-relation(t,ctx(hop-6),not-29) instantiable(boy-5,ctx(hop-6)) instantiable(hop-6,ctx(hop-6)) top-context(t) uninstantiable(hop-6,t) 15 / 35
  • 16. TIL has concepts, contexts and roles... Notice that context relations are: veridical antiveridical averidical Negation is antiveridical, while a context introduced by say is averidical. 16 / 35
  • 17. TIL has concepts, contexts and roles... John knew that Mary arrived. Conceptual Structure: role(cardinality-restriction,john-1,sg) role(cardinality-restriction,mary-3,sg) role(sb,know-2,john-1) role(topic,know-2,ctx(arrive-4)) role(sb,arrive-4,mary-3) subconcept(john-1,[List1]) subconcept(mary-3,[List3]) subconcept(know-2,[List2]) subconcept(arrive-4,[List4]) Contextual Structure: context(ctx(arrive-4)) context(t) top-context(t) context-lifting-relation(veridical,t,ctx(arrive-4)) context-relation(t,ctx(arrive-4),know-2) instantiable(john-1,t) instantiable(mary-2,t) instantiable(arrive-4,ctx(arrive-4)) instantiable(know-2, t) instantiable(arrive-4,t) 17 / 35
  • 18. TIL has concepts, contexts and roles... Instead of formulas we have a clausal representation R R::= conceptual stx | contextual strux | (temporal strux) CpA ::= subconcept (Sk-1, PrimitiveConcept) | role(rolename, Sk-1, Sk-2) | role(rolename, Sk-1, CONST1) PrimitiveConcepts from Ontology O, role names from SetOfRoles CtxA ::= t | ctx(Sk-1) | context-relation(cxt-1,cxt-2, Sk) context-lifting-relation(CONST2, cxt-1, cxt-2) | instantiable (Sk, cxt) | uninstantiable (Sk, cxt) where CONST2= veridical, averidical, antiveridical + syntactic sugar 18 / 35
  • 19. Inference Conceptual structure can be translated back into FOL. What about contextual structure? Contexts look like possible worlds. Just Modal Logic? 19 / 35
  • 20. Just Modal Logic? Not really. Standard Propositional modal logic is not enough Knowjohn(Arrived(Mary)) is OK. but what about: John and Bob knew that Mary arrived, John but not Bob knew that Mary arrived, etc.. Quantified modal logic looks too much. and theorem proving becomes an issue. How much Modal Logic? 20 / 35
  • 21. Inference in TIL? Inference in TIL is very rudimentary. We can ‘drop clauses’ like in most event semantics. From the sentence Ed walked and Mary talked we are able to infer both Ed walked and Mary talked by simply forgetting the respective clauses in the original representation. We can do trivial inferences like identity and we can compose derivations: s → s s → t t → r s → r 21 / 35
  • 22. How does TIL deal with syllogisms?... Example: All boys are mammals Conceptual Structure: role(cardinality-restriction,boy-2,all(pl)) role(cardinality-restriction,mammal-4,pl) role(copula-pred,be-3,mammal-4) role(copula-subj,be-3,boy-2) subconcept(boy-2,[List1]) subconcept(mammal-4,[List2]) Contextual Structure: context(t) instantiable(be-3,t) instantiable(boy-2,t) instantiable(mammal-4,t) top-context(t) 22 / 35
  • 23. Syllogistic Logic of All Syntax: Start with a collection of unary atoms (for nouns). The sentences are the expressions All p are q Semantics: A model M is a collection of sets M, and for each noun p we have an interpretation [[p]] ⊆ M. M |= All p are q iff [[p]] ⊆ [[q]] Proof system: All p are p All p are n All n are q All p are q 23 / 35
  • 24. TIL satisfies the rules for All All p are p All p are n All n are q All p are q Semantically just transitivity of subset containment. All boys are boys is odd, but not problematic... For TIL the transitive inference is simply climbing up the concept hierarchy. All boys are mammals All mammals are animals All boys are animals 24 / 35
  • 25. Syllogistic Logic of Some The sentences are the expressions Some p are q Semantics: A model M is a collection of sets M, and for each noun p,q we have an interpretation [[p,q]] ⊆ M. M |= Some p are q iff [[p]] ∩ [[q]] = ∅ Proof system: Some p are q Some q are p Some p are q Some p are p All q are n Some p are q Some p are n 25 / 35
  • 26. TIL satisfies the rules for Some Some p are q Some q are p Some p are q Some p are p All q are n Some p are q Some p are n The semantics is the same as Moss’, intersection of subsets. The inference relation between ‘all’ and ‘some’ is outsourced. Our ‘poor man’s inference system’ called Entailment and Contradiction Detection (ECD), has a table of relationships between ‘cardinality-restrictions’ postulated. This is useful if you want to keep your logic options open, no need to decide if empty domains... 26 / 35
  • 27. The languages S and S† with noun-level negation If one adds complemented atoms on top of the language of All and Some, with interpretation via set complement: [[p]] = M [[p]], So if one has S    All p are q Some p are q All p are q ≡ No p are q Some p are q ≡ Some p aren’t q Some non-p are non-q    S† Things can get strange, as explained by Moss-Hartmann: no finite axiomatizability.... 27 / 35
  • 28. The logical system for Sp + names The language Sp is the language of All and Some, with no negation, p for positive syllogisms. All p are p Some p are q Some p are p Some p are q Some q are p All p are n All n are q All p are q All n are p Some n are q Some p are q J is J J is M M is F J is F J is a p J is a q Some p are q All p are q J is a p J is a q M is a p J is M J is a p 28 / 35
  • 29. TIL satisfies Sp + names This corresponds to sentences like Jon is Jon, Jon is Mary and Mary is Fred entails Fred is Jon, Jon is a man and Jon is a doctor entails Some men are doctors, All cats are mammals and Jon is a cat entails Jon is a mammal Mary is a cat and Jon is Mary entails Jon is a cat. 29 / 35
  • 30. TIL is not about syllogisms or copula... Event semantics in general is about transitive, intransitive, ditransitive, etc. verbs. TIL should be able to deal with Sp and Rp and more... Noun negation should not present problems, (outsourced again) but sentential negation is dealt via contexts. Contexts look like the higher parts of the next picture... Modal and Hybrid logic! 30 / 35
  • 31. A picture stolen from Larry Moss... S≥ FOmon Peano-Frege Church-Turing S S† R R† Sp Rp FOL Kn FO2 2 variable FO logic † adds full N-negation relational syllogistic 31 / 35
  • 32. Conclusions Described a system TIL, arising from the confluence of intuitions from linguistics and AI. TIL draws from event semantics, McCarthy notions of context and description logics hopes of tractability. Showed that the system satisfies some of the fragments investigated computationally by Moss and Pratt-Hartmann, but since we have a constructive basis, while they have a classical one, more work is required for full comparison. 32 / 35
  • 33. Further Work Climbing up the syllogistic hierarchy of Moss and Pratt-Hartmann seems to work, but negation (constructive in TIL) takes us to a different kind of landscape. Need to check relational syllogistic. Need to check how picture works for constructive logics. As pure proof theory (applied to KR) several pieces of the puzzle have been put in place. An orthogonal issue is a practical implementation suitable for computational semantics in the large, the original motivation. 33 / 35
  • 34. Some References Preventing Existence (C. Condoravdi, D. Crouch, V de Paiva, J. Everett, R. Stolle, D. Bobrow, M. van den Berg), In Proceedings of Formal Ontology in Information Systems, FOIS’01, October 2001 Computing relative polarity for textual inference (Rowan Nairn, Lauri Karttunen and Cleo Condoravdi) Computing relative polarity for textual inference. In the Proceedings of ICoS-5 (Inference in Computational Semantics). April 20-21, 2006. Buxton, UK.. In the Proceedings of ICoS-5 (Inference in Computational Semantics). April 20-21, 2006. Buxton, UK. Textual Inference Logic: Take Two, (V. de Paiva, D. G. Bobrow, C. Condoravdi, R. Crouch, L. Karttunen, T. H. King, R. Nairn and A. Zaenen) Proceedings of the Workshop on Contexts and Ontologies, Representation and Reasoning, CONTEXT 2007. Precision-focused Textual Inference (Bobrow, D. G., C. Condoravdi, L. Karttunen, T. H. King, V. de Paiva, L. Price, R. Nairn, L.Price, A. Zaenen) Proceedings of ACL-PASCAL Workshop on Textual Entailment and Paraphrasing, pp. 16-21, 2007. 34 / 35
  • 35. Nesting "modalities" Ed did not forget to force Dave to leave. ⇒ Dave left. 35 / 35