The Complexity of Contextual Abduction in Human Reasoning Tasks

The Complexity of Contextual Abduction
in Human Reasoning Tasks
Emmanuelle-Anna Dietz Saldanha1 Steﬀen H¨olldobler1,2 Tobias Philipp1
1International Center for Computational Logic, TU Dresden, Germany
2North-Caucasus Federal University, Stavropol, Russian Federation

Modeling Human Reasoning Tasks
Car Brake Scenario
Cummins: Naive Theories and Causal Deduction. In: Memory & Cognition.
Stenning et. al.: Adaptive Reasoning: Integrating Fast and Frugal Heuristics with a Logic of
Interpretation. In: Decision.
If the brakes are pressed, then the car slows down
In this talk
1. The Weak Completion Semantics with Contextual Logic Programs
2. Contextual Abduction
3. Complexity of Contextual Abductive Reasoning
1

Car Brake Scenario
If the brakes are not OK, then the car does not slow down
In this talk
1

Car Brake Scenario
If the car accelerates, then the car does not slow down
In this talk
1

Car Brake Scenario
If the road is slippery, then the car does not slow down
In this talk
1

Car Brake Scenario
If the road is icy, then the road is slippery
In this talk
1

Car Brake Scenario
If the road is downhill, then the car accelerate
In this talk
1

Car Brake Scenario
If the car has snow chains on the wheels, then the road
is not slippery for the car
In this talk
1

Car Brake Scenario
If the car has snow chains on the wheels and the brakes
are pressed, then the car does not accelerate when the
road is downhill
In this talk
1

The Weak Completion Semantics (1)
Propositional Variables A
slow down, press, brakes ok
Literal L A or ¬A
¬brakes ok
Logic Program P finite set of
rule A ←L1 ∧ . . . ∧ Lm
fact A ←
assumption A ←⊥
P = {slow down ← press ∧ ¬ab1, ab1 ← ⊥}
Dietz Saldanha, Hölldobler, Pereira: Contextual Reasoning: Usually Birds can Abductively
Fly. In: LPNMR 2017.
Contextual Logic Program P finite set of
rule A ←L1 ∧ . . . ∧ Lm ∧ ctxt(Lm+1) ∧ . . . ∧ ctxt(Lm+p)
2

Weak Completion wcP
1. Replace all clauses with the same head A ← Body1, . . . , A ← Bodyn by
A ← Body1 ∨ Body2 ∨ . . . ∨ Bodyn
2. Replace ← by ↔
wcP1 = {slow down ↔ press ∧ ¬ab1, ab1 ↔ ⊥}
Three-Valued Interpretation I = I , I⊥
F ¬F
⊥
⊥
U U
∧ U ⊥
U ⊥
U U U ⊥
⊥ ⊥ ⊥ ⊥
∨ U ⊥
U U U
⊥ U ⊥
← U ⊥
U U
⊥ ⊥ U
L ctxt(L)
⊥ ⊥
U ⊥
P entails F under the weak completion semantics, P |=WCS F
I1 = {slow down, press}, {ab1}
MP1
= {}, {ab1}
3

Stenning and van Lambalgen Operator Φ ΦP (I) = J , J⊥
J = {A | there is A ← body ∈ P such that I(body) = }
J⊥ = {A | there is A ← body ∈ P
and for all A ← body ∈ P, we find that I(body) = ⊥}
Hölldobler, Kencana Ramli: Logic Programs under Three-Valued Lukasiewicz Semantics.
In: ICLP 2009.
For context-free logic programs:
1. ΦP is monotonic
2. MP is the least fixed point of ΦP
Dietz Saldanha, Hölldobler, Pereira: Contextual Reasoning: Usually Birds can Abductively Fly.
In: LPNMR 2017.
For contextual logic programs:
1. ΦP is not monotonic
2. lfp ΦP exists, if P is acyclic
4

The Car Brake Scenario (1)
Stenning, van Lambalgen: Human Reasoning and Cognitive Science.
1. If the brakes are pressed and nothing is abnormal (¬ab1), then the car slows down
slow down ← press ∧ ¬ab1 ab1 ← ⊥
5

2. If the brakes are not OK, then something is abnormal w.r.t. ab1
ab1 ← ctxt(¬brakes ok)
5

3. If the car accelerates, then something is abnormal w.r.t. ab1
ab1 ← ctxt(accelerate)
5

4. If the road is slippery, then something is abnormal w.r.t. ab1
ab1 ← ctxt(slippery)
5

5. If the road is icy and nothing is abnormal (ab2), then the road is slippery
slippery ← icy road ∧ ¬ab2 ab2 ← ⊥
5

6. If the road is downhill and nothing is abnormal w.r.t. (ab3), then the car accelerates
accelerate ← downhill ∧ ¬ab3 ab3 ← ⊥
5

7. If the car has snow chains, then something is abnormal w.r.t. ab2
ab2 ← ctxt(snow chain)
5

8. If the car has snow chains and the brakes are pressed, then something is abnormal w.r.t. ab3
ab3 ← ctxt(snow chain) ∧ press
5

8. If the car has snow chains and the brakes are pressed, then something is abnormal w.r.t. ab3
ab3 ← ctxt(snow chain) ∧ press
lfp ΦP = ∅, {ab1, ab2, ab3}
5

Contextual Abduction
Abductive Problem X = P, A, O
P is an acyclic contextual program
A ⊆ AP is the set of abducibles
AP = {A ← | A is undefined in P or A is head of an exception clause in P}
∪ {A ← ⊥ | A is undefined in P and ¬A is not assumed in P}
O is a finite set of literals, called observation
E is a contextual explanation of X
1. E ⊆ A
2. P ∪ E |=wcs O
3. for all A ← ∈ E and A ← ⊥ ∈ E there is L ∈ O such that
L strongly depends on A
6

Strongly Depends on Relation
Strongly depends relation smallest transitive relation such that
A ← L1 ∧ · · · ∧ Lm ∧ ctxt(Lm+1) ∧ · · · ∧ ctxt(Lm+p)
A strongly depends on Li for i ∈ {1, . . . , m}
If L strongly depends on L , then L strongly depends on L
If L strongly depends on L , then L strongly depends on L
P = { p ← r, p ← ctxt(q) }
p strongly depends on r, ¬p strongly depends on r, . . .
p does not strongly depend on q, neither on ctxt(q)
7

When it is slippery and the car does not slow down,..
Pctxt
car is as follows:
slow down ← press ∧ ¬ab1 slippery ← icy road ∧ ¬ab2
ab1 ← ctxt(slippery) accelerate ← downhill ∧ ¬ab3
ab1 ← ctxt(¬brakes ok) ab2 ← ctxt(snow chain)
ab1 ← ctxt(accelerate) ab3 ← ctxt(snow chain) ∧ press
ab1 ← ⊥ ab2 ← ⊥
ab3 ← ⊥
Assume that
O = {¬slow down, slippery}
E = {icy road ← } is a contextual explanation for O and is the only minimal one
8

Complexity of
Contextual Abductive Reasoning

Modiﬁcations
H¨olldobler, Philipp, Wernhard: An Abductive Model for Human Reasoning. In: Commonsense
Reasoning 2011.
Contextual Logic Programs
Abducibles
AP = {A ← ⊥, A ← | A does not occur in a head in P}
Strongly depends on relation
9

Complexity Results
EX Does there exists an explanation?
MIN Is E a minimal explanation?
SE F follows skeptically from X
1. X has an explanation
2. for all explanations E for X it holds that P ∪ E |=wcs F
PTIME
N
P
NPC
co-N
P
DPC
D
P
PSPACE
EX
MIN
SE
cEX
cMIN
cSE
10

Conclusion
Contextual programs: usual case vs the exception cases
Theorem
deciding consistency is NP-complete
deciding whether F follows skeptically is DP-complete
deciding whether a contextual explanation is minimal is in PSPACE
Future Work
Better bounds for minimality
11

The Complexity of Contextual Abduction
in Human Reasoning Tasks
Emmanuelle-Anna Dietz Saldanha1 Steﬀen H¨olldobler1,2 Tobias Philipp1
1International Center for Computational Logic, TU Dresden, Germany
2North-Caucasus Federal University, Stavropol, Russian Federation
Thank you for your attention!

References
Keith Stenning and Michiel van Lambalgen. Human Reasoning and Cognitive
Science. A Bradford Book. MIT Press, Cambridge, MA, 2008. ISBN
9780262195836.
Keith Stenning, Laura Martignon, and Alexandra Varga. Adaptive reasoning:
integrating fast and frugal heuristics with a logic of interpretation. Decision, in
press.
Steﬀen H¨olldobler, Tobias Philipp, and Christoph Wernhard. An abductive model for
human reasoning. In Logical Formalizations of Commonsense Reasoning, Papers
from the AAAI 2011 Spring Symposium, AAAI Spring Symposium Series Technical
Reports, pages 135–138, Cambridge, MA, 2011. AAAI Press.
12

The Complexity of Contextual Abduction in Human Reasoning Tasks

More Related Content

Recently uploaded (20)

Featured (20)

The Complexity of Contextual Abduction in Human Reasoning Tasks