Equational Logic Steffen H¨ olldobler International Center for Computational Logic Technische Universit¨ at Dresden Germany ◮ Equational Systems ◮ Paramodulation ◮ Term Rewriting Systems ◮ Unification Theory ◮ Application: Multisets Steffen H¨ olldobler Equational Logic 1
Equational Systems ◮ Consider a first order language with the following precedence hierarchy: {∀ , ∃} > ¬ > ∧ > ∨ > {← , →} > ↔ . ◮ Let ≈ be a binary predicate symbol written infix. ◮ An equation is an atom of the form s ≈ t . ◮ An equational system E is a finite set of universally closed equations. ◮ Notation Universal quantifiers are usually omitted. E 1 : ( X · Y ) · Z ≈ X · ( Y · Z ) , (associativity) 1 · X ≈ X , (left unit) X · 1 ≈ X , (right unit) X − 1 · X ≈ 1 , (left inverse) X · X − 1 ≈ 1 . (right inverse) Steffen H¨ olldobler Equational Logic 2
Axioms of Equality ◮ The equality relation enjoys some typical properties expressed by the following universally closed axioms of equality E ≈ : X ≈ X , ( reflexivity ) X ≈ Y → Y ≈ X , ( symmetry ) X ≈ Y ∧ Y ≈ Z → X ≈ Z , ( transitivity ) � n i = 1 X i ≈ Y i → f ( X 1 , . . . , X n ) ≈ f ( Y 1 , . . . , Y n ) , ( f–substitutivity ) � n i = 1 X i ≈ Y i ∧ r ( X 1 , . . . , X n ) → r ( Y 1 , . . . , Y n ) . ( r–substitutivity ) ◮ Note ⊲ Substitutivity axioms are defined for each function symbol f and each relation symbol r in the underlying alphabet. ⊲ Universal quantifiers have been omitted. Steffen H¨ olldobler Equational Logic 3
Equality and Logical Consequence ◮ We are interested in computing logical consequences of E ∪ E ≈ ⊲ E 1 ∪ E ≈ | = ( ∃ X ) X · a ≈ 1? ⊲ E 1 ∪ E ≈ ∪ { X · X ≈ 1 } | = ( ∀ X , Y ) X · Y ≈ Y · X ? ◮ One possibility is to apply resolution. ⊲ There are 10 21 resolution steps needed to solve the examples. ⊲ E ∪ E ≈ causes an extremely large search space. ◮ Idea Remove troublesome formulas from E ∪ E ≈ and built them into the deductive machinery. ⊲ Use additional rule of inference like paramodulation. ⊲ Build the equational theory into the unification computation. Steffen H¨ olldobler Equational Logic 4
Least Congruence Relation ◮ E ∪ E ≈ is a set of definite clauses. ◮ There exists a least model for E ∪ E ≈ . ◮ Example ⊲ Let the only function symbols be the constants a , b , and the binary g . ⊲ Let E 2 = { a ≈ b } . ⊲ The least model of E 2 ∪ E ≈ is { t ≈ t | t is a ground term } ∪ { a ≈ b , b ≈ a } ∪ { g ( a , a ) ≈ g ( b , a ) , g ( a , a ) ≈ g ( a , b ) , g ( a , a ) ≈ g ( b , b ) , . . . } ◮ Define s ≈ E t iff E ∪ E ≈ | = ∀ s ≈ t . ⊲ g ( a , a ) ≈ E 2 g ( a , b ) , g ( X , a ) ≈ E 2 g ( X , b ) ⊲ ≈ E is the least congruence relation on terms generated by E . Steffen H¨ olldobler Equational Logic 5
Paramodulation ◮ L ⌈ s ⌉ literal which contains an occurrence of the term s ; L ⌈ s / t ⌉ literal obtained from L by replacing s by t . ◮ Paramodulation [ L 1 ⌈ s ⌉ , . . . , L n ] [ l ≈ r , L n + 1 , . . . , L m ] θ = mgu ( s , l ) [ L 1 ⌈ s / r ⌉ , L 2 , . . . , L m ] θ Instead of ¬ s ≈ t we write s �≈ t . ◮ Notation ◮ Remember E ∪ E ≈ | = ∀ s ≈ t E∪E ≈ → ∀ s ≈ t is valid iff � ¬ ( � E∪E ≈ → ∀ s ≈ t ) is unsatisfiable iff E ∪ E ≈ ∪ {¬∀ s ≈ t } is unsatisfiable iff E ∪ E ≈ ∪ {∃ s �≈ t } is unsatisfiable. iff E ∪ E ≈ ∪ {∃ s �≈ t } is unsatisfiable iff there is a refutation of ◮ Theorem 1 E ∪ { X ≈ X } ∪ {∃ s �≈ t } wrt paramodulation, resolution and factoring. Steffen H¨ olldobler Equational Logic 6
An Example E 1 ∪ { X ≈ X , X · X ≈ 1 } | = ( ∀ X , Y ) X · Y ≈ Y · X 1 a · b �≈ b · a initial query · hypothesis 2 1 · X 1 ≈ X 1 left unit a · b �≈ (( X 3 · X 3 ) · b ) · ( a · ( X 4 · X 4 )) 3 X 2 ≈ X 2 reflexivity associativity · 4 X 1 ≈ 1 · X 1 pm(2,3) a · b �≈ ( X 3 · (( X 3 · b ) · ( a · X 4 ))) · X 4 5 a · b �≈ ( 1 · b ) · a pm(1,4) · hypothesis 6 X 3 · X 3 ≈ 1 hypothesis a · b �≈ ( a · 1 ) · b 7 X 4 ≈ X 4 reflexivity · right unit 8 1 ≈ X 3 · X 3 pm(6,7) n a · b �≈ a · b n ′ 9 a · b �≈ (( X 3 · X 3 ) · b ) · a pm(5,8) X 5 ≈ X 5 reflexivity n ′′ res ( n , n ′ ) right unit [ ] · a · b �≈ (( X 3 · X 3 ) · b ) · ( a · 1 ) Steffen H¨ olldobler Equational Logic 7
The Example in Shorthand Notation E 1 ∪ { X ≈ X , X · X ≈ 1 } | = ( ∀ X , Y ) X · Y ≈ Y · X 1 a · b �≈ b · a initial query · hypothesis 2 1 · X 1 ≈ X 1 left unit a · b �≈ (( X 3 · X 3 ) · b ) · ( a · ( X 4 · X 4 )) 3 X 2 ≈ X 2 reflexivity associativity · 4 X 1 ≈ 1 · X 1 pm(2,3) a · b �≈ ( X 3 · (( X 3 · b ) · ( a · X 4 ))) · X 4 5 a · b �≈ ( 1 · b ) · a pm(1,4) · hypothesis 6 X 3 · X 3 ≈ 1 hypothesis a · b �≈ ( a · 1 ) · b 7 X 4 ≈ X 4 reflexivity · right unit 8 1 ≈ X 3 · X 3 pm(6,7) n a · b �≈ a · b n ′ 9 a · b �≈ (( X 3 · X 3 ) · b ) · a pm(5,8) X 5 ≈ X 5 reflexivity n ′′ res ( n , n ′ ) right unit [ ] · a · b �≈ (( X 3 · X 3 ) · b ) · ( a · 1 ) Steffen H¨ olldobler Equational Logic 8
The Example in Shorthand Notation Again b · a ≈ ( 1 · b ) · a left unit ≈ (( X 3 · X 3 ) · b ) · a hypothesis ≈ (( X 3 · X 3 ) · b ) · ( a · 1 ) right unit ≈ (( X 3 · X 3 ) · b ) · ( a · ( X 4 · X 4 )) hypothesis ≈ ( X 3 · (( X 3 · b ) · ( a · X 4 ))) · X 4 associativity ≈ ( a · 1 ) · b hypothesis ≈ a · b right unit ◮ Now, the search space is 10 11 instead of 10 21 steps. ⊲ Symmetry can be simulated, which leads to cycles. ⊲ All π ∈ P L 1 are candidates. ⊲ L 1 ⌈ π ⌉ may be a variable and can be unified with any l . ◮ There are still many redundant and useless steps. ◮ Idea Use equations only from left to right: term rewriting system. Steffen H¨ olldobler Equational Logic 9
Term Rewriting Systems ◮ An expression of the form s → t is called rewrite rule. ◮ A term rewriting system is a finite set of rewrite rules. ◮ In the sequel, R shall denote a term rewriting system. ◮ s ⌈ u ⌉ denotes a term s which contains an occurrence of u , s ⌈ u / v ⌉ denotes the term obtained from s by replacing u with v . ◮ The rewrite relation → R on terms is defined as follows: s ⌈ u ⌉ → R t iff there exist l → r ∈ R and θ such that u = l θ and t = s ⌈ u / r θ ⌉ . R 3 = { append ([ ] , X ) → ◮ Example Let X , append ([ X | Y ] , Z ) → [ X | append ( Y , Z )] } . → R 3 [ 1 | append ([ 2 ] , [ 3 , 4 ])] ⊲ Then, append ([ 1 , 2 ] , [ 3 , 4 ]) → R 3 [ 1 , 2 | append ([ ] , [ 3 , 4 ])] → R 3 [ 1 , 2 , 3 , 4 ] . Steffen H¨ olldobler Equational Logic 10
Matching ◮ Matching problem Given terms u and l , does there exist a substitution θ such that u = l θ ? If such a substitution exists, then it is called a matcher. ◮ If a matching problem is solvable, then there exists a most general matcher. ◮ If can be computed by a variant of the unification algorithm, where variables occurring in u are treated as (different new) constant symbols. ◮ Whereas unification is in the complexity class P , matching is in N C . Steffen H¨ olldobler Equational Logic 11
Closures ∗ → R denotes the reflexive and transitive closure of → R . ◮ ∗ → R 3 [ 1 , 2 , 3 , 4 ] . ⊲ append ([ 1 , 2 ] , [ 3 , 4 ]) ◮ s ↔ R t iff s ← R t or s → R t . ⊲ Let R 4 = { a → b , c → b } , then a → R 4 b ← R 4 c and, consequently, a ↔ R 4 b ↔ R 4 c . ∗ ↔ R denotes the reflexive and transitive closure of ↔ R . ◮ ∗ ⊲ a ↔ R 4 c . ◮ We sometimes simply write → or ↔ instead of → R or ↔ R , respectively. Steffen H¨ olldobler Equational Logic 12
Term Rewriting Systems and Equational Systems ◮ Let R be a term rewriting system. ◮ E R := { l ≈ r | l → r ∈ R} ∪ E ≈ . ⊲ For R 4 = { a → b , c → b } we obtain E R 4 = { a ≈ b , c ≈ b } ∪ E ≈ . ∗ → R t s ≈ E R t . ◮ Theorem 2 (i) s implies ∗ (ii) s ≈ E R t iff s ↔ R t . ◮ Proof � Exercise ⊲ g ( X , a ) → R 4 g ( X , b ) and g ( X , a ) ≈ E R 4 g ( X , b ) . ⊲ g ( X , a ) ≈ E R 4 g ( X , c ) and g ( X , a ) → R 4 g ( X , b ) ← R 4 g ( X , c ) . Steffen H¨ olldobler Equational Logic 13
Reducibility and Normal Forms ◮ s is reducible wrt R iff there exists t such that s → R t ; otherwise it is irreducible. ∗ ◮ t is a normal form of s wrt R iff s → R t and t irreducible. ⊲ [ 1 , 2 , 3 , 4 ] is the normal form of append ([ 1 , 2 ] , [ 3 . 4 ]) wrt R 3 . ◮ Normal forms are not necessarily unique. Consider R 5 = { neg ( neg ( X )) → X , → neg ( or ( X , Y )) & ( neg ( X ) , neg ( Y )) , → neg ( & ( X , Y )) or ( neg ( X ) , neg ( Y )) , → & ( X , or ( Y , Z )) or ( & ( X , Y ) , & ( X , Z )) , → or ( & ( Y , Z ) , & ( Z , X )) } . & ( or ( X , Y ) , Z ) & ( or ( X , Y ) , or ( U , V )) has the normal forms or ( or ( & ( Y , U ) , & ( U , X )) , or ( & ( Y , V ) , & ( V , X ))) and or ( or ( & ( Y , U ) , & ( Y , V )) , or ( & ( V , X ) , & ( X , U ))) wrt R 5 . Steffen H¨ olldobler Equational Logic 14
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