Automated theorem proving by resolution in non-classical logics Viorica Sofronie-Stokkermans Max-Planck-Institut f¨ ur Informatik, Saarbr¨ ucken, Germany JIM’03: Knowledge Discovery and Discrete Mathematics Metz, September 3–6, 2003
Overview • Motivation • Resolution-based theorem proving in non-classical logics 1. Finitely-valued logics (superposition; ordered chaining) 2. Infinitely-valued logics 3. Modal logics and generalizations • logics based on distributive lattices with operators • applications: description logics, terminological reasoning 4. Beyond modal logics: deciding uniform word problems • Conclusions and future work
Motivation • Huge variety of non-classical logics • Huge variety of methods for automated theorem proving in these logics - sequent calculi, semantic tableaux, various extensions of resolution
Motivation • Huge variety of non-classical logics • Huge variety of methods for automated theorem proving in these logics - sequent calculi, semantic tableaux, various extensions of resolution Natural goal find a common framework which applies to large classes of non-classical logics (e.g. in automated theorem proving)
Motivation • Huge variety of non-classical logics • Huge variety of methods for automated theorem proving in these logics - sequent calculi, semantic tableaux, various extensions of resolution Natural goal find a common framework which applies to large classes of non-classical logics (e.g. in automated theorem proving) This talk: Embedding into first-order logic + resolution
Overview 1. finitely-valued logics 2. infinitely-valued logics 3. modal logics and generalizations 4. beyond modal logics
Overview 1. finitely-valued logics (first-order) superposition and ordered chaining 2. infinitely-valued logics 3. modal logics and generalizations 4. beyond modal logics
Many-valued logics Definition: L = ( X, Op , Pred , Σ , Quant), A : set of truth values Interpretations: I = ( D, I, d ) – interpretation of terms: as in classical logic I ( P ) : D a ( P ) → A – P ∈ Pred I ( σ ) = σ A : A a ( σ ) → A – σ ∈ Σ – Q ∈ Quant I ( Q ) = Q A : P ( A ) \∅ → A I ( Qxφ ( x )) = Q A ( {I ( φ )( d ) | d ∈ D } )
Examples A = { 0 , 1 • 3-valued logics 2 , 1 } 1 − � Lukasiewicz 2 : possible 1 − Bochvar 2 : meaningless 1 − Kleene 2 : undefined
Examples A = { 0 , 1 n − 1 . . . n − 1 1 • 3-valued logics 2 , 1 } • n -valued logics A = { 0 , n − 1 } 1 − � Lukasiewicz 2 : possible − � L n � Lukasiewicz logic 1 − Bochvar 2 : meaningless − G n G¨ odel logic 1 − Kleene 2 : undefined − P n Post logic
Examples A = { 0 , 1 n − 1 . . . n − 1 1 • 3-valued logics 2 , 1 } • n -valued logics A = { 0 , n − 1 } 1 − � Lukasiewicz 2 : possible − � L n Lukasiewicz logic � 1 − Bochvar 2 : meaningless − G n G¨ odel logic 1 − Kleene 2 : undefined − P n Post logic • Other examples SH n -logics [Iturrioz, Or� lowska 1996] (1,1) (n−2/n−1, 1) (1,n−2/n−1) (1/n−1,1) (1,1/n−1) (0,1) (1,0) (0,n−2/n−1) (n−2/n−1, 0) (0,1/n−1) (1/n−1,0) (0,0)
Examples n − 1 . . . n − 1 1 A = { 0 , 1 • n -valued logics A = { 0 , n − 1 } • 3-valued logics 2 , 1 } 1 − � L n � Lukasiewicz logic − � Lukasiewicz 2 : possible 1 − G n G¨ odel logic − Bochvar 2 : meaningless 1 − P n Post logic − Kleene 2 : undefined • Other examples SH n -logics [Iturrioz, Or� lowska 1996] (1,1) (n−2/n−1, 1) (1,n−2/n−1) (1/n−1,1) (1,1/n−1) (0,1) (1,0) (0,n−2/n−1) (n−2/n−1, 0) (0,1/n−1) (1/n−1,0) (0,0) • many (propositional) logics are characterized by one single algebra: the Lindenbaum algebra (usually difficult to effectively describe)
Examples: fuzzy logics ∗ Logic Truth values Connectives n − 1 . . . n − 1 1 � L n { 0 , n − 1 } x ◦ y = max(0 , x + y − 1) prop. → : the right residuum1 st order � L ∞ [0 , 1] of ◦ prop. 1 st order n − 1 . . . n − 1 1 G n { 0 , n − 1 } x ◦ y = min( x, y ) prop. → : the right residuum1 st order [0 , 1] of ◦ prop. G ∞ 1 st order Π ∞ [0 , 1] x ◦ y = x · y prop. → : the right residuum1 st order of ◦ * in all cases above for the 1 st order version of the logic the quantifiers are: ∀ = inf ; ∃ = sup
Examples: fuzzy logics Logic Truth values Connectives ∗ Complexity (validity) n − 1 . . . n − 1 1 � L n { 0 , n − 1 } x ◦ y = max(0 , x + y − 1) prop. co-NP → : the right residuum1 st order � L ∞ [0 , 1] of ◦ prop. co-NP 1 st order Π 2 -complete n − 1 . . . n − 1 1 G n { 0 , n − 1 } x ◦ y = min( x, y ) prop. co-NP → : the right residuum1 st order [0 , 1] of ◦ prop. co-NP G ∞ 1 st order Σ 1 -complete x ◦ y = x · y Π ∞ [0 , 1] prop. co-NP → : the right residuum1 st order Π 2 -hard of ◦ * in all cases above for the 1 st order version of the logic the quantifiers are: ∀ = inf ; ∃ = sup
Automated theorem proving (finite-valued logics) • translation to clause normal form (signed literals) • many-valued resolution rules
Automated theorem proving (finite-valued logics) • translation to clause normal form (signed literals) L v ) � ( � [Baaz and Ferm¨ uller 1995] �→ φ L v ) unsatisfiable � ( � φ valid iff • many-valued resolution rules
Automated theorem proving (finite-valued logics) • translation to clause normal form (signed literals) L v ) � ( � [Baaz and Ferm¨ uller 1995] �→ φ L v ) unsatisfiable � ( � φ valid iff φ I ∈ D L v ) � � � ( � b � = a φ = b �→ a ∈ D • many-valued resolution rules
Automated theorem proving (finite-valued logics) • translation to clause normal form (signed literals) L v ) � ( � [Baaz and Ferm¨ uller 1995] �→ φ • many-valued resolution rules C ∨ L v D ∨ L u C ∨ D provided that u � = v
Summarizing Truth values MV-ATP Signed lit. L v A arbitrary MV-resolution [Baaz,Ferm¨ uller’95] ( A, ≤ ) poset ↑ v : L annotated resolution [Kifer,Lozinskii’92] ∼↑ v : L [Lu,Murray,Rosenthal’98] ( A, ≤ ) ↑ v i : L regular resolution total order [H¨ ahnle’94,’96] ↓ v i : L ( v 1 < · · · < v n )
Summarizing C ∨ L v D ∨ L u Truth values MV-ATP Signed lit. C ∨ D L v A arbitrary MV-resolution provided that u � = v [Baaz,Ferm¨ uller’95] ( A, ≤ ) poset ↑ v : L annotated resolution C ∨↑ v 1 : L D ∨∼↑ v 2 : L [Kifer,Lozinskii’92] ∼↑ v : L C ∨ D [Lu,Murray,Rosenthal’98] provided that v 1 ≥ v 2 ( A, ≤ ) ↑ v i : L regular resolution total order [H¨ ahnle’94,’96] ↓ v i : L ( v 1 < · · · < v n )
Summarizing C ∨ L = v D ∨ L = u Truth values MV-ATP Signed lit. C ∨ D L v A arbitrary MV-resolution provided that u � = v [Baaz,Ferm¨ uller’95] ( A, ≤ ) poset ↑ v : L annotated resolution C ∨ L ≥ v 1 D ∨ L �≥ v 2 [Kifer,Lozinskii’92] ∼↑ v : L C ∨ D [Lu,Murray,Rosenthal’98] provided that v 1 ≥ v 2 ( A, ≤ ) ↑ v i : L regular resolution total order [H¨ ahnle’94,’96] ↓ v i : L ( v 1 < · · · < v n )
A simple translation to classical logic Truth values MV-ATP Signed lit.Classical lit.Theory ax. L v A arbitrary MV-resolution L = v Eq [Baaz,Ferm¨ uller’95] Φ A , Fin ( A, ≤ ) poset ↑ v : L v ≤ L Trans ≤ annotated resolution [Kifer,Lozinskii’92] ∼↑ v : L v �≤ L Φ A , Fin [Lu,Murray,Rosenthal’98] (Sup, Min) ( A, ≤ ) ↑ v i : L v i ≤ L Trans ≤ regular resolution total order [H¨ ahnle’94,’96] ↓ v i : L L<v i +1 or Tot ≤ ( v 1 < · · · < v n ) v i +1 �≤ L Φ A , Min Φ c classical clauses A + Φ �→ Φ c classically unsat. ⇔ Φ mv-unsat.
Automated theorem proving (finite-valued logics) [Ganzinger & VS 2000]: Specialize superposition and ordered chaining (calculi that encode inferences with congruence resp. transitivity) to many-valued logics Advantages • direct encoding • reconstruct known completeness results but much more restricted calculi – ordering, selection – simplification/elimination of redundancies • allows use of efficient implementations (SPASS, Saturate)
Comments Limitations • The method relies on a suitable translation to clause form. • May not be applicable: − for infinitely-valued logics − when the semantics is given in terms of a class of algebras.
Overview 1. finitely-valued logics 2. infinitely-valued logics 3. modal logics and generalizations 4. beyond modal logics
Infinitely-valued logics • Propositional � Lukasiewicz and G¨ odel logics [H¨ ahnle 1994, 1996] - CNF translation; reduction to mixed integer programming
Infinitely-valued logics • Propositional � Lukasiewicz and G¨ odel logics [H¨ ahnle 1994, 1996] - CNF translation; reduction to mixed integer programming • � Lukasiewicz logics - propositional: McNaughton’s theorem [Aguzzoli, Ciabattoni 2000] reduction to finitely-valued � Lukasiewicz logics but number of truth values exponential in size of formula - first-order: resolution-like calculus [Mundici, Olivetti 1998]
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