Negation Normal Form a formula is in negation normal form (NNF) if - PowerPoint PPT Presentation

Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Negation Normal Form ◮ a formula is in negation normal form (NNF) if it only contains ∧ , Build Your Own First-Order Prover ∨ , ∀ , ∃ , and ¬ only occurs (directly) in front of atomic formulae Part 3a: A Tableau Prover ◮ every formula F can be translated into a (classically) equivalent (model-preserving) formula F ′ that is in negation normal form Jens Otten ◮ translation of a formula into negation normal form: University of Oslo 1. Eliminate all logical operators except ¬ , ∧ , ∨ : A ↔ B ≡ ( A → B ) ∧ ( B → A ) , A → B ≡ ¬ A ∨ B 2. Push negations inward using De Morgan’s laws: ¬ ( A ∧ B ) ≡ ( ¬ A ∨ ¬ B ) , ¬ ( A ∨ B ) ≡ ( ¬ A ∧ ¬ B ) , ¬ ∀ x A ≡ ∃ x ¬ A , ¬ ∃ x A ≡ ∀ x ¬ A 3. Eliminate double negation: ¬¬ A ≡ A Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 1 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 2 / 28 Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Skolemization for NNF Block Tableau Calculus for NNF with Free Variables The tableau calculus for formulae in negation normal form (NNF) ◮ a formula is in skolemized negation normal form, if all its consists of one axiom and an α -rule, a β -rule, and a γ -rule. Eigenvariables have been replaced by Skolem terms. ◮ axiom ◮ let F be a formula in NNF, ∀ xG be a subformula in F , and (with P / P ′ unifiable, i.e., σ ( P ) = σ ( P ′ )) axiom P , ¬ P ′ , ∆ ∃ y 1 , . . . , ∃ y n be the existential quantifiers ”in front” of G ◮ α -rule β -rule γ -rule ◮ then F is valid iff F [ x \ f ∗ ( y 1 , . . . , y n )] is valid, where the A , B , ∆ A , ∆ B , ∆ ∧ A [ x \ x ∗ ] , ∃ x A , ∆ ∃ (new x ∗ ) Eigenvariable x (in G ) is replaced by the skolem term ∨ f ∗ ( y 1 , . . . , y n ) for a new function symbol f ∗ A ∨ B , ∆ A ∧ B , ∆ ∃ x A , ∆ ◮ similar to one-sided sequent calculus with skolemized NNF Example: ∀ x ∃ y p ( x , y ) → ∃ y ∀ x p ( x , y ) ◮ all rules are now invertible; P , P ′ are atomic formulae ◮ formula in NNF: ∃ x ∀ y ¬ p ( x , y ) ∨ ∃ y ∀ x p ( x , y ) ◮ all literals in axioms have to unify under a single substitution σ ◮ skolemized NNF: ∃ x ¬ p ( x , f 1 ( x )) ∨ ∃ y p ( f 2 ( y ) , y ) F is valid iff there is a proof for F in the block tableau calculus. Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 3 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 4 / 28

Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Block Tableau Calculus with Ordered Sets Implementing the Block Tableau Calculus ◮ if an ordered set ∆ is considered, the rule applications can be The main predicate is prove(Fml,UnExp,Lits,FreeV,VarLim) . restricted to its first element; this simplifies the proof search ◮ the words of the calculus are of the form F , ∆ , Lits where Lits is ◮ Fml is the formula on the (current) branch that will be considered next a set of literals (i.e. negated or non-negated atomic formulae) ◮ axiom ◮ UnExp is a list of formulae on the (current) branch not expanded so far (with P / P unifiable, i.e., σ ( P ) = σ ( P )) P , ∆ , { P }∪ Lits ◮ Lits is a list of literals on the (current) branch ◮ α -, β -, and γ -rules ◮ FreeV is a list of free variables on the (current) branch A , { B }∪ ∆ , Lits A , ∆ , Lits B , ∆ , Lits A [ x \ x ∗ ] , ∆ ∪{∃ x A } , Lits ◮ VarLim specifies the maximum number of free variables on the branch A ∨ B , ∆ , Lits A ∧ B , ∆ , Lits ∃ x A , ∆ , Lits (used for iterative deepening on the number of free variables on branch) ◮ next -rule additionally necessary A , ∆ , { P }∪ Lits The translation into skolemized negation normal form is done by the P , { A }∪ ∆ , Lits predicate nnf(F,F1) in the Prolog module nnf pure.pl . ◮ all rules are still invertible; P and its complement P are literals Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 5 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 6 / 28 Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Implementing α -, β - and γ -rule Implementing Axiom and Next-rule ◮ α -, β -, and γ -rules ◮ axiom A , { B }∪ ∆ , Lits A , ∆ , Lits B , ∆ , Lits A [ x \ x 1 ] , ∆ ∪{∃ x A } , Lits (with P / P unifiable, i.e., σ ( P ) = σ ( P )) P , ∆ , { P }∪ Lits A ∧ B , ∆ , Lits A ∨ B , ∆ , Lits ∃ x A , ∆ , Lits ◮ leantap pure.pl : ◮ leantap pure.pl : prove(Lit,_,Lits,_,_) :- prove((A|B),UnExp,Lits,FreeV,VarLim) :- !, (Lit = -Neg; -Lit = Neg) -> prove(A,[B|UnExp],Lits,FreeV,VarLim). member(L,Lits), unify1(Neg,L). prove((A&B),UnExp,Lits,FreeV,VarLim) :- !, ◮ next -rule prove(A,UnExp,Lits,FreeV,VarLim), prove(B,UnExp,Lits,FreeV,VarLim). A , ∆ , { P }∪ Lits P , { A }∪ ∆ , Lits prove((?[X]:Fml),UnExp,Lits,FreeV,VarLim) :- !, \+ length(FreeV,VarLim), ◮ leantap pure.pl : copy_term((X,Fml,FreeV),(X1,Fml1,FreeV)), prove(Lit,[Next|UnExp],Lits,FreeV,VarLim) :- append(UnExp,[(?[X]:Fml)],UnExp1), prove(Next,UnExp,[Lit|Lits],FreeV,VarLim). prove(Fml1,UnExp1,Lits,[X1|FreeV],VarLim). Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 7 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 8 / 28

Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary leanTAP – A Minimal Tableau Prover Term Unification prove((E,F),A,B,C,D) :- !,prove(E,[F|A],B,C,D). ◮ algorithm for term unification according to [Robinson 1965] prove((E;F),A,B,C,D) :- !,prove(E,A,B,C,D),prove(F,A,B,C,D). prove(all(I,J),A,B,C,D) :- !, ◮ unify ( s , t ) – unification of the terms s and t \+length(C,D),copy_term((I,J,C),(G,F,C)), σ – represents most general unifier append(A,[all(I,J)],E),prove(F,E,B,[G|C],D). ◮ unify ( t , t ) → σ remains unchanged prove(A,_,[C|D],_,_) :- unify ( x , t ) → σ ( x ) = t if x does not occur in t ((A= -(B);-(A)=B) -> (unify(B,C);prove(A,[],D,_,_))). prove(A,[E|F],B,C,D) :- prove(E,F,[A|B],C,D). unify ( t , x ) → σ ( x ) = t if x does not occur in t unify ( f ( s 1 , ..., s n ) , f ( t 1 , ..., t n )) → unify ( s 1 , t 1 ) , . . . , unify ( s n , t n ) ◮ first popular lean prover [Beckert/Posegga 1995] unify ( f ( s 1 , . . . , s n ) , g ( t 1 , . . . , t n )) → fail ◮ based on block tableau calculus for NNF with free variables ◮ this algorithm has an exponential worst-case time complexity ◮ source code size of minimal version only 360 bytes ◮ there exist algorithms with polynomial time complexity ◮ performance good on rather easy problems Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 9 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 10 / 28 Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Tableau Calculus A Tableau Prover Term Unification Connection Calculus A Connection Prover Summary Implementing Term Unification Hands-On: Run the Tableau Prover ◮ $> swipl % start SWI-Prolog ◮ unify1(A,B) :- unify([A],[B]). succeeds iff A and B are unifiable ◮ [leanseq v5]. % load the sequent prover ◮ if lists are empty ◮ [ex quant]. % load the quant formula unify([],[]). % and try to prove it fof(quant, ,F), prove(F). ◮ if A and B are identical terms (e.g. identical variables) % load the f12 formula [ex f12]. % and try to prove it fof(f12, ,F), prove(F). unify([A|A1],[B|B1]) :- A==B, !, unify(A1,B1). ◮ [leantap pure]. % load the tableau prover ◮ if A / B is a variable and A / B does not occur in B / A : assign B / A to A / B ◮ [ex barber]. % load the barber puzzle unify([A|A1],[B|B1]) :- var(A), !, not_in(A,B), A=B, unify(A1,B1). % solve puzzle unify([A|A1],[B|B1]) :- var(B), !, not_in(B,A), A=B, unify(A1,B1). fof(barber, ,F), prove(F). [ not_in(A,B) :- term_variables(B,L), \+ (member(X,L), X==A). ] ◮ [ex quant]. % load the quant formula ◮ otherwise, if A = f ( s 1 , ..., s n ) and B = f ( t 1 , ..., t n ), unify s i and t i for 1 ≤ i ≤ n % and try to prove it fof(quant, ,F), prove(F). unify([A|A1],[B|B1]) :- A=..[F|ArgA], B=..[F|ArgB], % load the f12 formula [ex f12]. length(ArgA,N), length(ArgB,N), unify(ArgA,ArgB), unify(A1,B1). % and try to prove it fof(f12, ,F), prove(F). % load the f20 formula [ex f20]. There is also a Prolog built-in predicate: unify_with_occurs_check(A,B) . % and try to prove it fof(f20, ,F), prove(F). Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 11 / 28 Jens Otten (UiO) Build Your Own First-Order Prover — Part 3 CADE Tutorial, August ’19 12 / 28