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Unification in modal logic Alt 1 Philippe Balbiani 1 and Tinko Tinchev 2 1 Institut de recherche en informatique de Toulouse CNRS Universit e de Toulouse 2 Department of Mathematical Logic and Applications Sofia University Introduction

  1. Unification in modal logic Alt 1 Philippe Balbiani 1 and Tinko Tinchev 2 1 Institut de recherche en informatique de Toulouse CNRS — Universit´ e de Toulouse 2 Department of Mathematical Logic and Applications Sofia University

  2. Introduction Unification problem in a logical system L ◮ Given a formula ψ ( x 1 , . . . , x n ) ◮ Determine whether there exists formulas ϕ 1 , . . . , ϕ n such that ψ ( ϕ 1 , . . . , ϕ n ) is in L Admissibility problem in a logical system L ◮ Given a rule of inference ϕ 1 ( x 1 ,..., x n ) ,...,ϕ m ( x 1 ,..., x n ) ψ ( x 1 ,..., x n ) ◮ Determine whether for all formulas χ 1 , . . . , χ n , if ϕ 1 ( χ 1 , . . . , χ n ) , . . . , ϕ m ( χ 1 , . . . , χ n ) are in L then ψ ( χ 1 , . . . , χ n ) is in L

  3. Introduction Rybakov (1984) ◮ The admissibility problem in IPL and S 4 is decidable Chagrov (1992) ◮ There exists a decidable normal modal logic with an undecidable admissibility problem Ghilardi (1999, 2000) ◮ IPL , K 4, etc have a finitary unification type Wolter and Zakharyaschev (2008) ◮ The unification problem for any normal modal logic between K U and K 4 U is undecidable

  4. Introduction Chagrov (1992) ◮ There exists a decidable normal modal logic with an undecidable admissibility problem Proof: For all integers m , n , let F ( m , n ) be the frame

  5. Introduction Chagrov (1992) ◮ There exists a decidable normal modal logic with an undecidable admissibility problem Proof: ◮ For all integers m , n , let F ( m , n ) be the frame . . . ◮ For all sets S of pairs of integers, let L ( S ) = Log {F ( m , n ) : ( m − 1 2 , n − 1 2 ) �∈ S } ◮ If S is recursive then L ( S ) -membership is decidable ◮ If Pr 2 S is nonrecursive then L ( S ) -admissibility is undecidable

  6. Introduction Other frames F ( P , a ) associated to a Minsky program P and a configuration a Chagrov, A. Undecidable properties of extensions of the logic of provability. Algebra i Logika 29 (1990) 350–367.

  7. Introduction Other frames F ( P , a ) associated to a Minsky program P and a configuration a Chagrov, A., Zakharyaschev, M. The undecidability of the disjunction property of propositional logics and other related problems. The Journal of Symbolic Logic 58 (1993) 967–1002.

  8. Introduction Other frames F ( P , a ) associated to a Minsky program P and a configuration a Chagrov, A., Chagrova, L. The truth about algorithmic problems in correspondence theory. In: Advances in Modal Logic. Vol. 6. College Publications (2006) 121–138.

  9. Introduction Other frames F ( P ) associated to a Minsky program P Isard, S. A finitely axiomatizable undecidable extension of K. Theoria 43 (1977) 195–202.

  10. Introduction Wolter and Zakharyaschev (2008) ◮ The unification problem for any normal modal logic between K U and K 4 U is undecidable Proof: Let P be a Minsky program, a = ( s , m , n ) be a configuration and F ( P , a ) be the frame

  11. Introduction Wolter and Zakharyaschev (2008) ◮ The unification problem for any normal modal logic between K U and K 4 U is undecidable Proof: ◮ Let P be a Minsky program, a = ( s , m , n ) be a configuration and F ( P , a ) be the frame . . . ◮ Let α , β , etc be formulas characterizing the points in F ( P , a ) ◮ With each configuration b , associate a modal formula ψ ( b ) ◮ If K U ⊆ L ⊆ K 4 U then P : a → b iff ψ ( b ) is unifiable in L

  12. Introduction Unification problem in a logical system L ◮ Given a formula ψ ( x 1 , . . . , x n ) ◮ Determine whether there exists formulas ϕ 1 , . . . , ϕ n such that ψ ( ϕ 1 , . . . , ϕ n ) is in L Example: � x ∨ � ¬ x is unifiable in all normal logics ◮ K (class of all frames) ◮ KD (class of all serial frames) ◮ K 4 (class of all transitive frames) ◮ S 4 (class of all reflexive transitive frames) ◮ S 5 (class of all partitions)

  13. Introduction Computability and type of unification in L L Computability Type K ? Nullary KD NP -complete ? K 4 Decidable Finitary KD 4 NP -complete Finitary K 45 NP -complete Unitary KD 45 NP -complete Unitary S 4 NP -complete Finitary S 5 NP -complete Unitary S 4 . 3 NP -complete Unitary

  14. Introduction Our results ◮ The unification problem in Alt 1 is decidable ( PSPACE ) ◮ Alt 1 has a nullary unification type

  15. Normal logics: syntax and semantics Syntax ◮ ϕ ::= x | ⊥ | ¬ ϕ | ( ϕ ∨ ψ ) | � ϕ Semantics ◮ M = ( W , R , V ) where ◮ W � = ∅ ◮ R ⊆ W × W ◮ for all variables x , V ( x ) ⊆ W Truth-conditions ◮ M , s | = x iff s ∈ V ( x ) ◮ M , s | = � ϕ iff for all t ∈ W , if sRt then M , t | = ϕ

  16. Normal logics: unification in L Substitutions ◮ σ : variable x �→ formula σ ( x ) Composition of substitutions ◮ σ ◦ τ : variable x �→ formula τ ( σ ( x )) Equivalence relation between substitutions ◮ σ ≃ L τ iff for all variables x , σ ( x ) ↔ τ ( x ) ∈ L Partial order between substitutions ◮ σ � L τ iff there exists a substitution µ such that σ ◦ µ ≃ L τ

  17. Normal logics: unification in L Unifiers ◮ A substitution σ is a unifier of a formula ϕ iff σ ( ϕ ) ∈ L Complete sets of unifiers ◮ A set Σ of unifiers of a formula ϕ is complete iff ◮ For all unifiers τ of ϕ , there exists a unifier σ of ϕ in Σ such that σ � L τ Important questions ◮ Given a formula, has it a unifier? ◮ If so, has it a minimal complete set of unifiers? ◮ If so, how large is this set?

  18. Why unification is NP -complete when KD ⊆ L Computability and type of unification in L L Computability Type K ? Nullary KD NP -complete ? K 4 Decidable Finitary KD 4 NP -complete Finitary K 45 NP -complete Unitary KD 45 NP -complete Unitary S 4 NP -complete Finitary S 5 NP -complete Unitary S 4 . 3 NP -complete Unitary

  19. Why unification is NP -complete when KD ⊆ L Proposition: If KD ⊆ L , unification in L is NP -complete Proof: ◮ A substitution σ is ground if it replaces each variable by a variable-free formula ◮ If a formula has a unifier then it has a ground unifier ◮ Since ♦ ⊤ ∈ L , therefore there are only two non-equivalent variable-free formulas: ⊥ and ⊤ ◮ Thus, to decide whether a formula has a unifier, it suffices to check whether any of the ground substitutions makes it equivalent to ⊤ (which can be done in polynomial time)

  20. Why unification is nullary in K Computability and type of unification in L L Computability Type K ? Nullary KD NP -complete ? K 4 Decidable Finitary KD 4 NP -complete Finitary K 45 NP -complete Unitary KD 45 NP -complete Unitary S 4 NP -complete Finitary S 5 NP -complete Unitary S 4 . 3 NP -complete Unitary

  21. Why unification is nullary in K Proposition: The formula ϕ = x → � x has no minimal complete set of unifiers Proof: ◮ The following substitutions are unifiers of ϕ ◮ σ ⊤ ( x ) = ⊤ ◮ σ i ( x ) = � < i x ∧ � i ⊥ ◮ If i ≤ j then σ j � K σ i ◮ If i < j then σ i �� K σ j ◮ If τ is a unifier of ϕ then either σ ⊤ � K τ , or σ i � K τ when deg ( τ ( x )) ≤ i Je˘ r´ abek, E. Blending margins: the modal logic K has nullary unification type. Journal of Logic and Computation 25 (2015) 1231–1240.

  22. Why unification is decidable and finitary in K 4 Computability and type of unification in L L Computability Type K ? Nullary KD NP -complete ? K 4 Decidable Finitary KD 4 NP -complete Finitary K 45 NP -complete Unitary KD 45 NP -complete Unitary S 4 NP -complete Finitary S 5 NP -complete Unitary S 4 . 3 NP -complete Unitary

  23. Why unification is decidable and finitary in K 4 A formula ϕ is projective if it has a unifier σ such that ◮ ϕ ∧ � ϕ → ( σ ( x ) ↔ x ) ∈ K 4 Remark ◮ Such unifier is a most general unifier of ϕ Proposition: The projectivity problem in K 4 is decidable Proposition If the substitution σ is a unifier of the formula ϕ then there exists a projective formula ψ , depth ( ψ ) ≤ depth ( ϕ ) , such that ◮ σ is a unifier of ψ ◮ ψ ∧ � ψ → ϕ ∈ K 4 Ghilardi, S. Best solving modal equations. Annals of Pure and Applied Logic 102 (2000) 183–198.

  24. Why unification is unitary in S 5 Computability and type of unification in L L Computability Type K ? Nullary KD NP -complete ? K 4 Decidable Finitary KD 4 NP -complete Finitary K 45 NP -complete Unitary KD 45 NP -complete Unitary S 4 NP -complete Finitary S 5 NP -complete Unitary S 4 . 3 NP -complete Unitary

  25. Why unification is unitary in S 5 Proposition: If a formula has a unifier then it has a most general unifier Proof: ◮ Let σ be a unifier of ϕ ◮ Let τ be the following “L¨ owenheim” substitution ◮ τ ( x ) = ( � ϕ ∧ x ) ∨ ( ♦ ¬ ϕ ∧ σ ( x )) ◮ � ϕ → ( τ ( ψ ) ↔ ψ ) ∈ S 5 ◮ ♦ ¬ ϕ → ( τ ( ψ ) ↔ σ ( ψ )) ∈ S 5 ◮ τ is a unifier of ϕ ◮ If µ is a unifier of ϕ then τ � S 5 µ ◮ Thus, τ is a most general unifier of ϕ Baader, F., Ghilardi, S. Unification in modal and description logics. Logic Journal of the IGPL 19 (2011) 705–730.

  26. Normal logic Alt 1 : syntax and semantics Syntax ◮ ϕ ::= x | ⊥ | ¬ ϕ | ( ϕ ∨ ψ ) | � ϕ Semantics ◮ Class of all deterministic frames Axiomatization ◮ K + ♦ x → � x Computability ◮ co NP -complete

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