logic is everywhere Truth Tables la l ogica est a por todas - PowerPoint PPT Presentation

Hikmat har Jaga Hai Systematic Search logika je svuda � � ��� � � �� ���� � Steffen H¨ olldobler � � �� Mantık her yerde International Center for Computational Logic logika je vˇ sude Technische Universit¨ at Dresden la logica ` e dappertutto Germany logic is everywhere ◮ Truth Tables la l´ ogica est´ a por todas partes ◮ Semantic Trees ��� � �� ��� ◮ DPLL � ��� � �� � ◮ DPLL-NB Logik ist ¨ uberall ◮ DPLL-CDBL Logika ada di mana-mana ◮ Generic DPLL Logica este peste tot ��� � ���� ��� �������� �� � �� � a l´ ogica est´ a em toda parte la logique est partout Steffen H¨ olldobler Systematic Search 1

Truth Tables ◮ How can we compute the value of a formula F under all possible interpretations? ◮ Computing a truth table 1 Let m = |S ( F ) | be the number of subformulae of F . 2 Let R F = { A | A ∈ R and A ∈ S ( F ) } and n = |R F | be the number of propositional variables occurring in F . 3 Form a table TT( F ) with 2 n rows and m columns, where the first n columns are marked by the elements of R F , the last column is marked by F , and the remaining columns are marked by the other subformulas of F . 4 Fill in the first n columns with ⊤ and ⊥ as follows: In the first column fill in alternating downwards ⊤⊥⊤⊥ . . . , in the second column ⊤⊤⊥⊥ . . . , in the third column ⊤⊤⊤⊤⊥⊥⊥⊥ . . . , etc. 5 Calculate the values in the remaining columns using the known functions on the set of truth values. Steffen H¨ olldobler Systematic Search 2

Some Details ◮ For a row ζ in TT( F ) we denote by ζ ( G ) the truth value in the column marked by the formula G ∈ S ( F ) . ◮ With this, step 5 can be reformulated as follows: 5 For each row ζ in TT( F ) and for all F ◦ G, ¬ F ∈ S ( F ) \R F calculate: ζ ( F ◦ G ) = ζ ( F ) ◦ ∗ ζ ( G ) and ζ ( ¬ F ) = ¬ ∗ ζ ( F ) . Steffen H¨ olldobler Systematic Search 3

Some Observations ◮ Let I be an interpretation, F a formula, R F the set of variables occurring in F and n = |R F | . ◮ If we fix A I for all A ∈ R F , then F I is uniquely determined. ◮ There are exactly 2 n different possibilities of assigning truth values to R F . ◮ Each row in TT( F ) corresponds precisely to one of these possibilities. ◮ For each interpretation I of the language L ( R ) exists exactly one row ζ I in TT( F ) with G I = ζ I ( G ) for all G ∈ S ( F ) . ◮ For every row ζ in TT( F ) exists an interpretation I of the language L ( R ) with G I = ζ ( G ) for all G ∈ S ( F ) . ⊲ I is not uniquely determined. Steffen H¨ olldobler Systematic Search 4

Determining Satisfiability using Truth Tables ◮ F is satisfiable iff TT( F ) contains a row ζ with ζ ( F ) = ⊤ . ◮ F is unsatisfiable iff for all rows ζ in TT( F ) we find ζ ( F ) = ⊥ . ◮ F is valid iff for all rows ζ in TT( F ) we find ζ ( F ) = ⊤ . ◮ F is falsifiable iff TT( F ) contains a row ζ with ζ ( F ) = ⊥ . Steffen H¨ olldobler Systematic Search 5

Semantic Trees – Main Characteristics ◮ Optimization of the truth table method. ◮ Stepwise partitioning of interpretations (through branching). ◮ Usually conceived for formulas in clausal form. ◮ Notatation In the sequel nodes are (labeled by) expressions of the form F :: J , where ⊲ F is a formula and ⊲ J is either a partial interpretation for F , SAT or CONFLICT with the following informal meaning: ⊲ If J is a partial interpretation, then F | J is the “remaining” SAT-problem. ⊲ If J is SAT or CONFLICT , then F | J is undefined. ⊲ F :: J has successor F :: SAT iff J | = F (iff F | J = � � ). ⊲ F :: J has successor F :: CONFLICT iff J �| = F (iff [ ] ∈ F | J ). Steffen H¨ olldobler Systematic Search 6

Semantic Trees ◮ A semantic tree for a CNF-formula F is a binary tree satisfying the following conditions: ⊲ The root node is F ::( ) . ⊲ If F :: J is a node and F | J = � � then it has a successor node F :: SAT . ⊲ If F :: J is a node and [ ] ∈ F | J then it has a successor node F :: CONFLICT . ⊲ If F :: J is a node, [ ] �∈ F | J and A ∈ atom ( F | J ) then F :: J has two successor nodes F :: ˙ ˙ A, J and F :: ¬ A, J . ◮ Note The conditions in the three if-statements are mutually exclusive; the corresponding rules (see next slide) do not overlap. Steffen H¨ olldobler Systematic Search 7

The ST Calculus ◮ Given a CNF-formula F . ◮ Computations are initialized by F ::( ) . ◮ The rules of the calculus are F :: J F :: SAT iff F | J = � � . ❀ SAT F :: J F :: CONFLICT iff [ ] ∈ F | J . ❀ CONF F :: ˙ ˙ A, J | F :: ¬ A, J A ∈ atom ( F | J ) and [ ] �∈ F | J . F :: J iff ❀ SPLIT ◮ SPLIT leads to branching. ◮ Computation terminates if ⊲ a node F :: SAT is reached in which case F is satisfiable or ⊲ all leaf nodes are of the form F :: CONFLICT in which case F is unsatisfiable. ◮ F :: J ❀ F ′ :: J ′ iff F :: J ❀ SAT F ′ :: J ′ or F :: J ❀ CONF F ′ :: J ′ or F :: J ❀ SPLIT F ′ :: J ′ . ∗ ❀ is the reflexive and transitive closure of ❀ . ◮ Steffen H¨ olldobler Systematic Search 8

Example ◮ Let F = � [2 , − 3] , [2 , 3] , [ − 1 , − 2] , [1 , − 3] , [1 , − 2 , 3] � in F ::( ) ✏ PPPPPPPP ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ P q F ::(˙ F ::( ˙ 1) − 1) � ❅ � ❅ � ❅ � ❅ ✠ � ❘ ❅ � ✠ ❅ ❘ F ::( ˙ F ::(˙ 2 , ˙ ˙ F ::( ˙ ˙ − 2 , ˙ F ::(˙ 1) 1) 3 , − 1 ) − 3 , − 1) � ❅ ❇ ✂ � ❅ � ❅ ❇ ✂ � ❅ � ✠ ❅ ❘ � ✠ ❅ ❘ ❇ ✂ ❇ ✂ F ::(˙ − 2 , ˙ ˙ F ::( ˙ − 2 , ˙ ˙ F ::(˙ ˙ ˙ F ::( ˙ ˙ ˙ 3 , 1) − 3 , 1) 2 , − 3 , − 1) − 2 , − 3 , − 1) ❇ ✂ ❳❳❳❳❳❳❳❳❳❳❳ ✘ ❇ ✂ ❍❍❍ ✟ ✘ ✘ ✟ ✘ ❇ ◆ ✌ ✂ ✘ ✟ ✘ ❥ ❍ ✙ ✟ ✘ ✘ ✘ ✘ ✘ ❳ ③ ✾ ✘ F :: CONFLICT ◮ F is unsatisfiable. Steffen H¨ olldobler Systematic Search 9

Another Example ◮ Let F = � [2 , 3] , [ − 1 , − 2] , [1 , − 3] , [1 , − 2 , 3] � in F ::( ) ✏ PPPPPPPP ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✏ ✮ P q F ::(˙ F ::( ˙ 1) − 1) � ❅ � ❅ � ❅ � ❅ ✠ � ❘ ❅ � ✠ ❅ ❘ F ::( ˙ F ::(˙ 2 , ˙ ˙ F ::( ˙ ˙ − 2 , ˙ F ::(˙ 1) 1) 3 , − 1 ) − 3 , − 1) � ❅ ❇ ✂ � ❅ � ❅ ❇ ✂ � ❅ � ✠ ❅ ❘ � ✠ ❅ ❘ ❇ ✂ ❇ ✂ F ::(˙ − 2 , ˙ ˙ F ::( ˙ − 2 , ˙ ˙ F ::(˙ ˙ ˙ F ::( ˙ ˙ ˙ 3 , 1) − 3 , 1) 2 , − 3 , − 1) − 2 , − 3 , − 1) ❇ ✂ ✘ ❇ ✂ ❍❍❍ ✟ ✘ ✘ ✟ ✘ ◆ ❇ ✌ ✂ ✘ ✟ ✘ ❥ ❍ ✙ ✟ ✘ ✘ ✘ ✘ ✘ ❄ ✘ ✾ F :: SAT F :: CONFLICT ◮ (3 , − 2 , 1) | = F . Steffen H¨ olldobler Systematic Search 10

Abstract Reduction Systems ◮ The ST calculus is an abstract reduction system (see e.g. Baader, Nipkow: Term Rewriting and All That. Cambridge University Press: 1998). ◮ An abstract reduction system ( R , → ) is said to be ⊲ terminating iff there is no infinite descending chain t 0 → t 1 → . . . , iff t 1 ← ∗ t → ∗ t 2 implies ( ∃ t ′ ) t 1 → ∗ t ′ ← ∗ t 2 , ⊲ confluent iff t 1 ← t → t 2 implies ( ∃ t ′ ) t 1 → ∗ t ′ ← ∗ t 2 , ⊲ locally confluent ⊲ convergent (or canonical) iff is is terminating and confluent. ◮ Newman’s Lemma A terminating relation is confluent if it is locally confluent. (Newman: On theories with a combinatorial definition of ’equivalence’. Annals of Mathematics 43(2), 223-243: 1942) Steffen H¨ olldobler Systematic Search 11

ST Termination ◮ Theorem ST is terminating. ◮ Proof (sketch) ⊲ SAT, CONF and SPLIT do not overlap, i.e., at most one of these rules is applicable to a node F :: J . ⊲ If SAT or CONFLICT are applied then their only successor nodes F :: SAT and F :: CONFLICT are irreducible. ⊲ We turn to SPLIT: ◮ atom ( F ) is finite. ◮ ◮ SPLIT is applied to F :: J yielding two successor nodes ◮ F :: ˙ ˙ A, J and F :: ¬ A, J if A ∈ atom ( F | J ) . ◮ A �∈ atom ( F | ( ˙ A,J ) ) and A �∈ atom ( F | ( ˙ ¬ A,J ) ) . ◮ ◮ There are no infinite sequences of SPLIT. qed ◮ Steffen H¨ olldobler Systematic Search 12

ST Confluency ◮ Proposition ST is confluent. ◮ Proof Because ST is terminating and, thus, Newman’s Lemma is applicable, it suffices to show that ST is locally confluent. ⊲ Because ❀ SAT , ❀ CONF and ❀ SPLIT do not overlap, the only possible overlap is between two different instance of ❀ SPLIT applicable to some node F :: J . ⊲ Let F :: A, J | F :: ¬ A, J and F :: B, J | F :: ¬ B, J , A � = B , be the respective extensions of F :: J . ⊲ Because B ∈ atom ( F | A , J ) ∩ atom ( F | ¬ A , J ) ❀ SPLIT “on B ” can be applied to F :: A, J and F :: ¬ A, J yielding F :: B, A, J | F :: ¬ B, A, J | F :: B, ¬ A, J | F :: ¬ B, ¬ A, J . ⊲ Because A ∈ atom ( F | B , J ) ∩ atom ( F | ¬ B , J ) ❀ SPLIT “on A ” can be applied to F :: B, J and F :: ¬ B, J yielding F :: A, B, J | F :: ¬ A, B, J | F :: A, ¬ B, J | F :: ¬ A, ¬ B, J . ⊲ The result follows from the fact that in partial interpretations literals can be swapped. qed Steffen H¨ olldobler Systematic Search 13