Projective unification in modal logic II Projective unification in - PowerPoint PPT Presentation

Projective unification in modal logic II

Projective unification in modal logic II Piotr Wojtylak Institute of Mathematics, Silesian University, Katowice, Poland

Projective unification in modal logic II Piotr Wojtylak Institute of Mathematics, Silesian University, Katowice, Poland This talk is based on a paper by Wojciech Dzik and Piotr Wojtylak, Projective Unification in Modal Logic , submitted to the Logic Journal of the IGPL.

Projective unification in modal logic II Piotr Wojtylak Institute of Mathematics, Silesian University, Katowice, Poland This talk is based on a paper by Wojciech Dzik and Piotr Wojtylak, Projective Unification in Modal Logic , submitted to the Logic Journal of the IGPL. Utrecht, 26-28th May 2011

Modal Logic

Modal Logic Let Var = { x , y , z , . . . } be the set of propositional variables and Fm be the set of modal formulas . For each formula A , let Var ( A ) denote the (finite) set of variables occurring in A .

Modal Logic Let Var = { x , y , z , . . . } be the set of propositional variables and Fm be the set of modal formulas . For each formula A , let Var ( A ) denote the (finite) set of variables occurring in A . By a modal logic we mean any consistent and normal extension of S 4. More specifically, a modal logic is a proper subset of Fm closed under substitutions, closed under MP : A → B , A RG : A and � A , B containing all classical tautologies, and � A → A �� A → � A � ( A → B ) → ( � A → � B ) .

Modal Logic Let Var = { x , y , z , . . . } be the set of propositional variables and Fm be the set of modal formulas . For each formula A , let Var ( A ) denote the (finite) set of variables occurring in A . By a modal logic we mean any consistent and normal extension of S 4. More specifically, a modal logic is a proper subset of Fm closed under substitutions, closed under MP : A → B , A RG : A and � A , B containing all classical tautologies, and � A → A �� A → � A � ( A → B ) → ( � A → � B ) . Given a modal logic L , we define its global entailment relation ⊢ L . Thus, X ⊢ L A means that A can be derived from X ∪ L using the rules MP and RG .

Unifiers By a substitution we mean any finite mapping ε : Var → Fm : ε := x 1 / B 1 · · · x n / B n and A [ ε ] or A [ x 1 / B 1 · · · x n / B n ] is the result of the substitution.

Unifiers By a substitution we mean any finite mapping ε : Var → Fm : ε := x 1 / B 1 · · · x n / B n and A [ ε ] or A [ x 1 / B 1 · · · x n / B n ] is the result of the substitution. A substitution ε is called a unifier for a formula A in L if ⊢ L A [ ε ]. If ε is an L -unifier for A , then εδ is also an L -unifier for A for each substitution δ .

Unifiers By a substitution we mean any finite mapping ε : Var → Fm : ε := x 1 / B 1 · · · x n / B n and A [ ε ] or A [ x 1 / B 1 · · · x n / B n ] is the result of the substitution. A substitution ε is called a unifier for a formula A in L if ⊢ L A [ ε ]. If ε is an L -unifier for A , then εδ is also an L -unifier for A for each substitution δ . Unifiers of the form v : Var ( A ) → {⊥ , ⊤} are called ground unifiers for A . They can be identified with valuations in the two-element topological Boolean algebra 2 which satisfy the formula A . Given a unifier ε for A and any substitution δ : Var → {⊥ , ⊤} we get the ground unifier εδ for the formula A .

Projectivity A substitution ε is said to be a projective unifier of a formula A if (i) ⊢ L A [ ε ]; (ii) A ⊢ L x [ ε ] ↔ x , for each variable x .

Projectivity A substitution ε is said to be a projective unifier of a formula A if (i) ⊢ L A [ ε ]; (ii) A ⊢ L x [ ε ] ↔ x , for each variable x . The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until • Baader, F.,Ghilardi, S., Unification in modal and description logics , Logic Journal of the IGPL.

Projectivity A substitution ε is said to be a projective unifier of a formula A if (i) ⊢ L A [ ε ]; (ii) A ⊢ L x [ ε ] ↔ x , for each variable x . The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until • Baader, F.,Ghilardi, S., Unification in modal and description logics , Logic Journal of the IGPL. Let ε 0 be a unifier for A in L . Then ε 0 is said to be a most general unifier (or, in short, an MGU ) for A , on the ground of L , if each L -unifier is an instantiation of ε 0 , that is if for each L -unifier ε there is a substitution δ such that ε = L ε 0 δ .

Projectivity A substitution ε is said to be a projective unifier of a formula A if (i) ⊢ L A [ ε ]; (ii) A ⊢ L x [ ε ] ↔ x , for each variable x . The concept of a projective unifier (formula, substitution) is due to S.Ghilardi. He used them extensively throughout his papers though the term ‘projective unifier’ did not appear until • Baader, F.,Ghilardi, S., Unification in modal and description logics , Logic Journal of the IGPL. Let ε 0 be a unifier for A in L . Then ε 0 is said to be a most general unifier (or, in short, an MGU ) for A , on the ground of L , if each L -unifier is an instantiation of ε 0 , that is if for each L -unifier ε there is a substitution δ such that ε = L ε 0 δ . Theorem Each projective unifier for A is an MGU for A.

Intuitionistic logic The problem of projective unification for intermediate logics was solved by A.Wro´ n ski • Wro´ n ski A., Transparent verifiers in intermediate logics , Abstracts of the 54-th Conference in History of Mathematics, Cracow (2008). Theorem An intermediate logic L INT has projective unification iff LC ⊆ L INT .

Intuitionistic logic The problem of projective unification for intermediate logics was solved by A.Wro´ n ski • Wro´ n ski A., Transparent verifiers in intermediate logics , Abstracts of the 54-th Conference in History of Mathematics, Cracow (2008). Theorem An intermediate logic L INT has projective unification iff LC ⊆ L INT . One implication of the above theorem follows from • Minari P. , Wro´ n ski A., The property (HD) in intuitionistic Logic. A Partial Solution of a Problem of H. Ono , Reports on mathematical logic 22 (1988), 21–25. They defined a substitution ε (in {⇒ , ∧ , ¬} ) by putting � A ⇒ B if B [ v ] = ⊤ B [ ε ] = B [ v ] = ⊥ . ¬¬ A ∧ ( A ⇒ B ) if

Intutionistic Logic Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as (( q → r ) → r ) ∧ (( r → q ) → q ).

Intutionistic Logic Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as (( q → r ) → r ) ∧ (( r → q ) → q ). To prove the reverse implication, let us assume that ε is a projective unifier for ( q ⇒ r ) ∨ ( r ⇒ q ), denoted by A , which is an axiom for LC . Then (by A ⊢ L x [ ε ] ↔ x ) A ∧ x ⊢ x [ ε ] and x [ ε ] ⊢ A ⇒ x , for each variable x .

Intutionistic Logic Now, to prove that LC has projective unification, it suffices to know that the disjunction q ∨ r is defined in LC as (( q → r ) → r ) ∧ (( r → q ) → q ). To prove the reverse implication, let us assume that ε is a projective unifier for ( q ⇒ r ) ∨ ( r ⇒ q ), denoted by A , which is an axiom for LC . Then (by A ⊢ L x [ ε ] ↔ x ) A ∧ x ⊢ x [ ε ] and x [ ε ] ⊢ A ⇒ x , for each variable x . Now, we need to know that (in intuitionistic logic) ( A ⇒ q ) = q = ( A ∧ q ) and ( A ⇒ r ) = r = ( A ∧ r ) which is sufficient to show that x [ ε ] = x for each variable x ∈ { q , r } . Thus, we get A [ ε ] = A . Since ε is an L INT -unifier for A , we conclude that L INT ⊢ A and hence LC ⊆ L INT .

Problem The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems.

Problem The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems. Theorem If a modal logic L has projective unification, then � ( � y → � z ) ∨ � ( � z → � y ) ∈ L. Thus, to characterize all modal logics with projective unification, one should consider the modal logic S 4 . 3, which is obtained by extending S 4 with � ( � A → � B ) ∨ � ( � B → � A ) .

Problem The question arises if the same characterization of logics with projective unification, as for intermediate logics, can be proved for modal systems. Theorem If a modal logic L has projective unification, then � ( � y → � z ) ∨ � ( � z → � y ) ∈ L. Thus, to characterize all modal logics with projective unification, one should consider the modal logic S 4 . 3, which is obtained by extending S 4 with � ( � A → � B ) ∨ � ( � B → � A ) . However, it is not an easy matter to show that S 4 . 3 enjoys projective unification. It turns out, in particular, that the method of ground unifiers, as used for intermediate logics, does not work for modal systems.

Example Example Suppose that a formula A is unifiable, let v : Var ( A ) → {⊤ , ⊥} be a ground unifier for A . Let us define a substitution ε as follows: � � A → x if x [ v ] = ⊤ x [ ε ] = � A ∧ x if x [ v ] = ⊥ for each variable x ∈ Var ( A ).