A Computational Logic Approach tothe Abstract and the Social Case of the Selection Task

Emmanuelle Dietz and Steffen HolldoblerInternational Center for Computational Logic

TU Dresden, Germany

Marco RagniCenter for Cognitive Science

Universitat Freiburg, Germany

The Wason Selection Task: Abstract Case (Wason 1968)

Consider four cards where each of them has a letter on one sideand a number on the other side. Given the conditional

If there is a D on one side of the card, then there is a 3 on the other side.

Which cards must be turned to prove that the conditional holds?

D F 3 7

Experimental Results 89% 16% 62% 25%

The Wason Selection Task: Social Case (Griggs and Cox 1982)

Consider four cards, where on one side there is the person’s age andon the other side of the card what the person is drinking. Given the conditional

If a person is drinking beer, then the person must be over 19 years of age.

Which cards must be turned to prove that the conditional holds?

beer coke 22yrs 16yrs

Experimental Results 95% 0.025% 0.025% 80%

Adequate Framework for Human Reasoning

How to adequately formalize human reasoning in computational logic?

Stenning and van Lambalgen (2008) propose a two step process:human reasoning should be modeled by

1. reasoning towards an appropriate representation,→ conceptual adequacy

2. reasoning with respect to this representation.→ inferential adequacy

The adequacy of a computational logic approach that aims a representing humanreasoning should be evaluated based on how humans actually reason.

State of the Art

1. Stenning and van Lambalgen (2008) formalize the Suppression Task.

2. Kowalski (2011) models the Selection Task.

3. Holldobler and Kencana Ramli (2009) found technical mistakes done byStenning and van Lambalgen and propose to model human reasoning by

I logic programsI under weak completion semanticsI based on the three-valued Lukasiewicz (1920) logic.

They show that this approach adequately models the Suppression Task.

Can we adequately model the Selection Task with this approach?

Logic Programs and their Weak Completion

A logic program P is a finite set of clauses of the form

A← B1, . . . ,Bn

I A is an atom called head of the clause, and B1, . . . ,Bn

is a conjunction of (negated) atoms called body of the clause.

I An atom is undefined in P if it is not the head of some clause in P.

I A← > is a positive fact and A← ⊥ is a negative fact.

Consider the following transformation for P:

1. All clauses with the same head A← body1, A← body2, . . .are replaced by A← body1 ∨ body2 ∨ . . ..

2. All occurrences of ← are replaced by ↔.

The resulting set is called the weak completion of P (wcP).

Three-Valued Lukasiewicz Logic

¬> ⊥⊥ >U U

∧ > U ⊥> > U ⊥U U U ⊥⊥ ⊥ ⊥ ⊥

∨ > U ⊥> > > >U > U U⊥ > U ⊥

←L > U ⊥> > > >U U > >⊥ ⊥ U >

↔L > U ⊥> > U ⊥U U > U⊥ ⊥ U >

>, ⊥, and U denote true, false, and unknown, respectively.

I Interpretation I is a mapping from formulas to the set of truth values {>,⊥,U}.I It is represented by pairs 〈I>, I⊥〉 .

I I is a model for P if I maps each clause occurring in P to >.

Reasoning towards and Appropriate Representation

3← D ∧ ¬ab1 represents the conditional in the abstract caseo ← b ∧ ¬ab2 represents the conditional in the social case.

where o means ‘a person older than 19 years’, b means ‘drinking beer’Abnormality predicates are added to the conditionals.

Consider the person drinking beer and nothing abnormal is known.

P Case Facts

Pbeer drinking beer ab2 ← ⊥, b ← >

The weak completion of Pbeer is

wcPbeer = {ab2 ↔ ⊥, b ↔ >}

Computing Least Models

The least model of the weak completion of P (lm LwcP) is identical to theleast fixed point of ΦP , an operator defined by Stenning and van Lambalgen.

Let I be an interpretation in ΦP (I ) = 〈J>, J⊥〉, where

J> = {A | there exists A← body ∈ P with I (body) = >},J⊥ = {A | there exists A← body ∈ P and

for all A← body ∈ P we find I (body) = ⊥}.

Holldobler and Kencana Ramli showed that the model intersection propertyholds for weakly completed programs.This guarantees the existence of a least model for every program.

Reasoning with Respect to this Representation

Consider the first card in the social case of the Selection Task:

Pbeer = {ab2 ← ⊥, b ← >}

and an empty interpretation I0 = 〈∅, ∅〉:

I1 = ΦPbeer(I0) = 〈{b}, {ab2}〉 = ΦPbeer

(I1)

I A ∈ I> if there exists A← body ∈ P with I (body) = >I A ∈ I⊥ if there exists A← body ∈ P and for all A← body : I (body) = ⊥

The least model of the weak completion of Pbeer is 〈{b}, {ab2}〉.

Social Case of the Selection Task

According to Kowalski the social case is understood as a social constraint.

The conditional is encoded as a goal G = {o ← b ∧ ¬ab2},

wc (Pbeer ) = {ab2 ↔ ⊥, b ↔ >}

case program P lm LwcP Griggs& Cox

beer {ab2 ← ⊥, b ← >} 〈{b}, {ab2}〉 6|=3L o ← b ∧ ¬ab2 95%coke {ab2 ← ⊥, b ← ⊥} 〈∅, {b, ab2}〉 |=3L o ← b ∧ ¬ab1 0.025%22yrs {ab2 ← ⊥, o ← >} 〈{o}, {ab2}〉 |=3L o ← b ∧ ¬ab2 0.025%16yrs {ab2 ← ⊥, o ← ⊥} 〈∅, {o, ab2}〉 6|=3L o ← b ∧ ¬ab2 80%

Abduction

According to Kowalski, the abstract case is understood as a belief.

Consider P = {3← D ∧ ¬ab1, ab1 ← ⊥} with lm LwcP = 〈∅, {ab1}〉.〈∅, {ab1}〉 does not explain any letter on a card.

Given an abductive framework 〈P,A, |=lm wc L 〉 and an observation O where

I the set of abducibles A are the positive and negative factsfor all undefined atoms.

I P |=lm wc L F if and only if lm LwcP(F ) = > for formula F .

I O is a literal explained by E iff E ⊆ A,P ∪ E is satisfiable, and P ∪ E |=lm wc L O.

The abstract case of the selection task

P = {3← D ∧ ¬ab1, ab1 ← ⊥}wcP = {3↔ D ∧ ¬ab1, ab1 ↔ ⊥}lm LwcP = 〈∅, {ab1}〉

The set of abducibles A is

{D ← >, D ← ⊥, F ← >, F ← ⊥, 7← >, 7← ⊥}

Abstract Case of the Selection Task

The set of abducibles A is

{D ← >, D ← ⊥, F ← >, F ← ⊥, 7← >, 7← ⊥}

wc (P ∪ ED ) = {D ↔ 3 ∧ ¬ab1,D ↔ >}

O E lm Lwc (P ∪ E) Wason

D {D ← >} 〈{D, 3}, {ab1}〉 turn 89%F {F ← >} 〈{F}, {ab1}〉 no turn 16%3 {D ← >} 〈{D, 3}, {ab1}〉 turn 62%7 {7← >} 〈{7}, {ab1}〉 no turn 25%

Conclusion and Future Work

Conclusion

Weak completion semanticsThis seems to adequately model the selection and the suppression task.

Future Work

Theoretical ApproachExtension of this approach to other human reasoning patterns:How do humans reason with respect to spatial relations?

Conceptual Adequacy – Reasoning towards an Appropriate RepresentationThe first step of Stenning and van Lambalgen is not formalized yet:How to distinguish between the social and the abstract case?

Weak Completion vs Well-founded Semantics

Weak completion semantics seems to adequately solveByrne’s suppression task and Wason’s selection task.

How does our approach relate to well-founded semantics (Van Gelder et al. 1991)?

Both semantics have the following similarities:

1. Given any logic program, there always exists a least model.

2. The least model can be a three-valued interpretationwith unknown values for some atoms.

Program Classes

P1 P2 P3

{p ← q} {p ← q, q ← p} {p ← ¬q, q ← ¬p}

If at least one atom depends on itself, then we have a cycle.

I Acyclic programs do not contain cycles.

I Stratified programs only contain positive cycles (Apt et al. [1988]).

I Tight programs only contain negative cycles (Erdem and Lifschitz [2003]).

Weak completion semantics corresponds towell-founded semantics for modified tight logic programs.

Correspondence to Well-founded Semantics

Weak completion semantics corresponds towell-founded semantics for modified tight logic programs.

1. Undefined atomsP1 = {p ← q}

p and q are false in well-founded, but unknown in weak completion semantics.Adding to P for all undefined atoms A in P the following two clauses

A← ¬n A, n A← ¬A

weak completion and well-founded semantics coincide.

2. Positive Cycles

P2 = {p ← q, q ← p} P3 = {p ← ¬q, q ← ¬p}P3 is tight, whereas P2 is not tight. For P2

I under weak completion semantics, p and q will stay unknown,I under well-founded semantics p and q are mapped to false.

The logic programs modeling the Selection and the Suppression Task are tight.What about non-tight logic programs? How do people reason with cyclic statements?

Psychological Study

Participants were presented with 17 problems consisting of cyclic conditionalsof length 1,2 and 3. Consider the following cyclic conditional of length 1:

If they open the window, then they open the window.

� They open the window. �X They open the window.� They do not open the window.� It is unknown whether they open the window.

Another conditional of length 3:

If they open the window, then it is cold.If it is cold, then they wear their jackets.

If they wear their jackets, then they open the window.

Experimental Results

Length Chosen answer in percentage Mean responseof cycle Positive Negative Unknown times in msec

1 75 0 25 52672 60 3 37 115163 55 4 41 11680

Considering Stenning and van Lambalgen’s first step reasoning towards an appropriaterepresentation, we can identify two kinds of representations:

I participants who give a positive answer understand the statement as fact.

In a logic program, a fact p is represented by p ← >.In both, weak completion and well-founded semantics, p is true.

I participants answering ‘it is unknown...’ see the statements as conditionals.

A cyclic conditional of length 1 is represented by p ← p.In weak completion, p is unknown whereas in well-founded semantics p is false.

Conclusion and Future Work

Conclusion

Weak completion semanticsThis seems to adequately model some aspects of human reasoning.Our study seems to be in favor of weak completion semantics.

Future Work

Practical ExperimentsCarrying out more refined psychological studies:How people reason with negative cycles and cycles with facts?

References

K. R. Apt, H. A. Blair, and A. Walker. Foundations of deductive databases and logic programming. In J. Minker, editor, Towards a theoryof declarative knowledge, pages 89–148. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1988.

Esra Erdem and Vladimir Lifschitz. Tight logic programs. Theory and Practice of Logic Programming, 3:499–518, 2003.

R.A. Griggs and J.R. Cox. The elusive thematic materials effect in the wason selection task. British Journal of Psychology, 73:407–420,1982.

Steffen Holldobler and Carroline Dewi Kencana Ramli. Logic Programs under Three-Valued Lukasiewicz Semantics. In Patricia M. Hill andDavid Scott Warren, editors, International Conference on Logic Programming, LNCS, volume 5649 of LNCS, pages 464–478, Berlin,Heidelberg, 2009. Springer-Verlag. ISBN 978-3-642-02845-8.

Robert Kowalski. Computational Logic and Human Thinking: How to be Artificially Intelligent. Cambridge University Press, 1 edition, jul2011. ISBN 0521123364.

J. Lukasiewicz. O logice trojwartosciowej. Ruch Filozoficzny, 5:169–171, 1920. English translation: On Three-Valued Logic. In: Jan Lukasiewicz Selected Works. (L. Borkowski, ed.), North Holland, 87-88, 1990.

K. Stenning and M. Lambalgen. Human reasoning and cognitive science. Bradford Books. MIT Press, 2008. ISBN 9780262195836.

Allen Van Gelder, Kenneth A. Ross, and John S. Schlipf. The well-founded semantics for general logic programs. Journal of the ACM, 38(3):619–649, 1991.

P. Wason. Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20:273–281, 1968.