The Suppression Task Steffen H¨ olldobler International Center for Computational Logic Technische Universit¨ at Dresden Germany ◮ The Suppression Task – Part I ◮ Three-Valued Łukasiewicz Logic ◮ Logic Programs ◮ Completion Semantics and Least Models ◮ Semantic Operators and Least Fixed Points ◮ Contractions ◮ The Suppression Task – Part II ◮ Abduction Steffen H¨ olldobler The Suppression Task 1
The Suppression Task – Forward Reasoning ◮ Byrne: Suppressing Valid Inferences with Conditionals. Cognition 31, 61-83: 1989 ◮ If she has an essay to write then she will study late in the library. She has an essay to write. ⊲ Modus Ponens (MP) in classical logic ⊲ 96% of subjects conlude that she will study late in the library. ◮ If she has an essay to write then she will study late in the library. She has an essay to write. If she has a textbook to read she will study late in the library. ⊲ Alternative Arguments ⊲ 96% of subjects conlude that she will study late in the library. ◮ If she has an essay to write then she will study late in the library. She has an essay to write. If the library stays open she will study late in the library ⊲ 38% of subjects conlude that she will study late in the library. ⊲ Additional arguments lead to suppression of earlier conclusions. Steffen H¨ olldobler The Suppression Task 2
Reasoning Towards an Appropriate Logical Form ◮ Context independent rules ⊲ If she has an essay to write and the library is open then she will study late in the library. If the library is open and she has a reason for studying in the library then she will study late in the library. ◮ Context dependent rule plus exception ⊲ If she has an essay to write then she will study late in the library. However, if the library is not open, then she will not study late in the library. ⊲ The last sentence is the contrapositive of the converse of the original sentence! Steffen H¨ olldobler The Suppression Task 3
The Suppression Task – Denial of Antecedent (DA) ◮ Byrne: Suppressing Valid Inferences with Conditionals. Cognition 31, 61-83: 1989 ◮ If she has an essay to write then she will study late in the library. She does not have an essay to write. ⊲ 46% of subjects conlude that she will not study late in the library. ◮ If she has an essay to write then she will study late in the library. She does not have an essay to write. If she has a textbook to read she will study late in the library. ⊲ 4% of subjects conlude that she will not study late in the library. ◮ If she has an essay to write then she will study late in the library. She does not have an essay to write. If the library stays open she will study late in the library. ⊲ 63% of subjects conlude that she will not study late in the library. Steffen H¨ olldobler The Suppression Task 4
Human Reasoning – The Search for Models ◮ Goal Find a logic which adequately models human reasoning. ◮ How about classical two-valued propositional logic? ◮ Let’s consider a direct encoding: { l ← e , e } { l ← e , e , l ← t } { l ← e , e , l ← o } { l ← e , ¬ e } { l ← e , ¬ e , l ← t } { l ← e , ¬ e , l ← o } Steffen H¨ olldobler The Suppression Task 5
Two-Valued Interpretations ◮ Let L be a language of propositional logic. ◮ A (two-valued) interpretation is a mapping L �→ {⊤ , ⊥} represented by I , where I is a set containing all atoms which are mapped to ⊤ . ⊲ All atoms which do not occur in I are mapped to ⊥ . ◮ Let I denote the set of all interpretations. ⊲ ( I , ⊆ ) is a lattice. ◮ An interpretation I is a model for a program P , in symbols I | = P , iff I ( P ) = ⊤ . ∅ �| = { l ← e , e } { e , l } { e } �| = { l ← e , e } { l } �| = { l ← e , e } { e , l } | = { l ← e , e } { e } { l } ∅ Steffen H¨ olldobler The Suppression Task 6
Logical Consequence (1) ◮ A formula G is a logical consequence of a set of formulas F , in symbols F | = G , iff all models for F are also models for G . { l ← e , e } | = l { e , l } { l ← e , e } | = e { e } { l } ∅ Steffen H¨ olldobler The Suppression Task 7
Logical Consequence (2) ◮ A formula G is a logical consequence of a set of formulas F , in symbols F | = G , iff all models for F are also models for G . { l ← e , ¬ e } | = ¬ e { e , l } { l ← e , ¬ e } �| = l { l ← e , ¬ e } �| = ¬ l { e } { l } ∅ Steffen H¨ olldobler The Suppression Task 8
The Suppression Task – A Classical Logic Approach ◮ Recall the examples: { l ← e , e } | = l modus ponens { l ← e , e , l ← t } | = l classical logic is monoton { l ← e , e , l ← o } | = l upps, humans don’t do this { l ← e , ¬ e } �| = ¬ l denial of antecendent { l ← e , ¬ e , l ← t } �| = ¬ l { l ← e , ¬ e , l ← o } �| = ¬ l ◮ Conclusion classical logic is inadequate. ⊲ Often mistakenly generalized to “logic is inadequate”. Steffen H¨ olldobler The Suppression Task 9
The Suppression Task – A Computational Logic Approach ◮ Goal Find a logic which adequately models human reasoning. ◮ Solution I propose the following: ⊲ Logic programs under (weak) completion semantics ◮ Non-monotonicity ◮ ⊲ Reasoning towards an appropriate logical form ◮ Logic programs ◮ ⊲ Three-valued Łukasiewicz logic ◮ Least models ◮ ⊲ An appropriate semantic operator ◮ Least fixed points are least models ◮ ◮ Least fixed points can be computed by iterating the operator ◮ ⊲ Reasoning with respect to the least models ⊲ A connectionist realization Steffen H¨ olldobler The Suppression Task 10
Adequateness ◮ When is a logic adequate? ◮ In this talk If it qualitatively gives the same answers as subjects in the corresponding experiments. Steffen H¨ olldobler The Suppression Task 11
Logic Programs ◮ A (logic) program is a finite set of clauses. ⊲ A (program) clause is an expression of the form A ← B 1 ∧ · · · ∧ B n , where n ≥ 1, A is an atom, and each B i , 1 ≤ i ≤ n , is either a literal, ⊤ or ⊥ . ⊲ A is called head and B 1 ∧ · · · ∧ B n body of the clause. ⊲ A clause of the form A ← ⊤ is called a positive fact. ⊲ A clause of the form A ← ⊥ is called a negative fact. { l ← e , e ← ⊤} { l ← e , e ← ⊤ , l ← t } { l ← e , e ← ⊥} ◮ P is definite if the bodies of all clauses of P consist only of atoms and ⊤ . ◮ Here I consider only propositional programs, but the approach extends to first-order programs. ◮ The language L underlying a program P shall contain precisely the relation symbols occurring in P , and no others. Steffen H¨ olldobler The Suppression Task 12
Program Completion ◮ Let P be a program. Consider the following transformation: 1 All clauses with the same head A ← Body 1 , A ← Body 2 , . . . are replaced by A ← Body 1 ∨ Body 2 ∨ . . . . 2 If an atom A is not the head of any clause in P then add A ← ⊥ . 3 All occurrences of ← are replaced by ↔ . The resulting set is called completion of P or c P . If 2 is omitted then the resulting set is called weak completion of P or wc P . Steffen H¨ olldobler The Suppression Task 13
Program Completion – Example 1 ◮ Consider P 1 = { l ← e , e ← ⊤} c P 1 = { l ↔ e , e ↔ ⊤} wc P 1 = { l ↔ e , e ↔ ⊤} ◮ The only model of c P 1 and wc P 1 is: { e , l } { e } { l } ∅ ◮ Hence, c P 1 | = l and wc P 1 | = l . Steffen H¨ olldobler The Suppression Task 14
Program Completion – Example 2 ◮ Consider P 2 = { l ← e , e ← ⊥} c ( P 2 ) = { l ↔ e , e ↔ ⊥} wc ( P 2 ) = { l ↔ e , e ↔ ⊥} ◮ The only model of c P 2 and wc P 2 is: { e , l } { e } { l } ∅ ◮ Hence, c P 2 | = ¬ l and wc P 2 | = ¬ l . ◮ Remember, P 2 �| = ¬ l . Steffen H¨ olldobler The Suppression Task 15
Program Completion – Example 3 ◮ Consider P 3 = { l ← e , e ← ⊤ , l ← t } c P 3 = { l ↔ e ∨ t , e ↔ ⊤ , t ↔ ⊥} wc P 3 = { l ↔ e ∨ t , e ↔ ⊤} ◮ The only model of c P 3 is: { e , l } ◮ The models of wc P 3 are: { e , l } { e , l , t } ◮ Hence, c P 3 | = ¬ t whereas wc P 3 �| = ¬ t and wc P 3 �| = t . Steffen H¨ olldobler The Suppression Task 16
Monotonicity ◮ Let F and F ′ be sets of formulas and G a formula. A logic is monotonic if the following holds: = G then F ∪ F ′ | If F | = G . ◮ Classical logic is monotonic. ◮ A logic based on completion semantics is non-monotonic. ⊲ Consider P 3 = { l ← e , e ← ⊤ , l ← t } P ′ = P ∪ { t ← ⊤} 3 ⊲ Then c P 3 | = ¬ t c P ′ �| = ¬ t 3 Steffen H¨ olldobler The Suppression Task 17
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