Projective Unification in Modal Logic I

Projective Unification in Modal Logic I Wojciech Dzik Instytut Matematyki, Uniwersytet Sl¸ aski Katowice

Workshop on Admissible Rules and Unification UTRECHT

26-28.05.2011

• Unification (E-unification) basics • Unification in logic (modal l. extending S4), splittings of Ext S4 • Projective unification in Ext S4.3 (W.Dzik, P.Wojtylak, 2011) • Applications: structural completeness and almost structural completeness • Unitary unification versus structural completeness in Ext S4

Unification, E-Unification

E - an equational theory, x - variables x1 , . . . , xn ; t1 (x), t2 (x) two terms in variables x1 , . . . , xn , Is there any substitution ε such that (?)

`E εt1 = εt2 ?

(E-unification problem `E t1 = t2 ? ),

A substitution ε satisfying (?) is a E-unifier for (t1 , t2 ) or a unifier in E t1 , t2 are called unifiable in E if such a unifier exists. If E = ∅, then - syntactic unification

Given two (E-)unifiers σ, τ for t1 , t2 , σ is more general than τ , τ  σ, if there is a θ such that, `E θ(σ(x)) = τ (x). xC C

CC τ CC CC  ! T y _ _ _/ T z

σ

θ

 is a preorder (reflexive, transitive). σ is a mgu, most general unifier for t1 , t2 in E if σ is more general then any E-unifier for (t1 , t2 ). mgu is usually not unique,

For a theory E and unifiable terms s, t one of the following holds: 1. There is a mgu for s, t . 2. There are finitely many maximal E-unifiers ρ1 , . . . , ρm for s, t such that for any unifier τ for s, t there is ρi which is more general than τ . 3. There are infinitely many maximal E-unifiers ρ1 , ρ2 , . . . for s and t such that for any unifier τ for s, t there is ρi which is more general than τ . 4. There is a unifier τ for s, t such that there does not exists a maximal unifier for s and t which is more general than τ .

Unification Types of equational theories (varieties): unitary (or 1), each unifiable (t1 , t2 ) has mgu (best), ω or finitary, each unifiable (t1 , t2 ) - finitely max unifiers, and not the unitary type, infinitary (or ∞), each unifiable (t1 , t2 ) - infinitely many max unifiers, and not the unitary or finitary type, nullary, (or 0), for some unifiable (t1 , t2 ) max unifiers does not exist (worst)

semigroups infinitary Plotkin 1972 commutative semigroups finitary Livesey, Siekmann 1976 groups, rings infinitary Lawrence 1989, 1987 Abelian groups finitary Lankford 1979 commutat. rings nullary or infinit. Burris, Lawrence 1989 lattices nullary Willard 1991 semilattices finitary Livesey, Siekmann 1976 distributive lattices nullary Willard 1991 Boolean algebras unitary B¨ utner, Simonis 1987 discriminator algebras unitary Burris 1987 Hilbert algebras unitary Prucnal 1972 Heyting alg. closure alg. finitary Ghilardi 1999, 2000 See Stan Burris http://www.math.uwaterloo.ca/ ∼ snburris/

Algebraic/Category theory approach (Ghilardi 97) An algebra B ∈ VE is finitely presented, if there is a finite set of variables, x1 , . . . , xk = x and a finite set S of equations of terms in x such that B is iso to FE (x)/∼ , where ∼ is a congruence: t1 ∼ t2 iff S `E t1 = t2 An algebra P in a variety V is projective in V if for every A, B of V and f : P → B, g : A → B (g - epi) there is a h : P → A s. t. P @

@@ @@f @@   A g −epi / B

h



P is projective in V iff P is a retract of a free algebra in V, P E -unification problem

m

/ FV (x) q

/P

a finitely presented algebra A ∈ VE .

-

A unifier for A is a pair given by: a projective algebra P and a homomorphism u : A → P. Given two unifiers u1 and u2 for A, u1 : A → P1 is more general then u2 : A → P2 , u2  u1 , if there is a homomorphism g s. t. AB

BB BBu2 BB B!  _ _ _ / P2 P1

u1

g

Unification types are defined in the same way as for symbolic unification, roughly, according to the number (1, finite, infinite or 0) of maximal (w.r.t. ) unifiers of ,,the worst” case. Theorem (S.Ghilardi) For any equational theory E the ,,symbolic” and the ,,algebraic” unification type coincide. Corollary. The unification type is a categorical invariant. Corollary. Boolean algebras have unitary unification. Drawback: no explicit form of any unifier is given.

Unification in LOGIC L - finitely equivalential logic, ↔ Fm the set of propositional formulas built from Var = {x, y , z, . . . } A logic L - given by the inference relation `L ; derivability X `L α A substitution τ is a unifier for a formula α in a logic L, if `L τ α. α is called unifiable in L if such a τ exists. σ, τ : x → F two unifiers for α in L, σ is more general than τ , τ  σ, if there is a substitution θ such that, for x ∈ x, `L θ(σ(x)) ↔ τ (x). A mgu, most general unifier for a formula α in L is a unifier σ for α such that σ is more general then any unifier for α (not unique). Unification Types for a logic L: unitary (or 1), each unifiable α has a mgu (best), ω or finitary, each unifiable α - finitely many max unifiers, infinitary (or ∞), each unifiable α - infinit. max unifiers, nullary, (or 0), for some unifiable α max unif. does not exist

• Classical logic CL (Boolean), S5 (monadic)- unitary unification. • S4 - unification type ω, finitary (Ghilardi), not unitary: α := x ∨ ¬x has 2 maximal unifiers: σ0 (x) =⊥, σ1 (x) = >. Let τ be a unifier for α, `S4 τ (x ∨ ¬x), then `S4 τ x, i.e. τ  σ1

or

`S4 ¬τ x i.e. τ  σ0

and hence τ is not a mgu. • S4.2 - unitary unification (Ghilardi, Sacchetti). NOTE: n-potent BL-algebras (Basic Fuzzy Logic):BL + x n+1 = x n and extensions; Lukasiewicz Ln - unitary unification (WD 2007); Lω (MV-algebras) - nullary (V. Marra, L. Spada 2011) Unification is not preserved by extensions/weakenings.

Splitting of the lattice of logics (theories), R.McKenzie. L1 splits the lattice ExtL0 , if there exists a logic L2 such that the lattice ExtL0 is divided into two disjoint parts, the filter ExtL2 = L2 ↑ and the ideal SubL1 = L1 ↓, i.e. for every logic L ∈ ExtL0 , either L ⊆ L1 or L2 ⊆ L.

L1

L2

L0 A splitting pair (L2 , L1 ). L1 is given by a finite s.i. algebra.

The splitting of the lattice ExtS4 f2 - the ”fork”, the frame ({a, b, c}, ≤), ≤ - a reflexive and trans relation such that a ≤ b, a ≤ c; L(f2 ) - the logic of f2 . •b aB B

BB BB BB

•a

=• || | | || ||

c

L1 = L(f2 ) = S4 + Grz + BD2 + BW2 Grz := ((p → p) → p) → p, Grzegorczyk ax. BD2 := (¬p ∧ ♦p) → ♦p depth ≤ 2 BW2 := ¬(p ∧ q ∧ ♦(p ∧ ¬q) ∧ ♦(¬p ∧ q) ∧ ♦(¬p ∧ ¬q)) width ≤ 2 L2 = S4.2 = S4 + 2 : ♦α → ♦α

Geach ax.

Splitting pair for ExtS4: (S4.2, L(f2 )): for each L ∈ ExtS4 (I ) S4.2 ⊆ L or (II ) L ⊆ L(f2 )

L(f2 )

S4.2

S4 Theorem(WD 2006). In [S4.2) all logics with unitary (1) unification are placed, with S4.2 the least, and some with type 0, in (L(f2 )] all with type ω, some with type (0 or ∞). Conjecture: extension of S4 with type ∞ does not exist.

PROJECTIVE unifiers and formulas

A unifier ε for α is projective in a modal logic L (over S4) , if `L εα and α `L ε(x) ↔ x, for x ∈ Var (α); α is then called a projective formula (Ghilardi 1999). Each projective unifier is a mgu: `L τ α ⇒ `L τ ε(x) ↔ τ x. L has projective unification if every unifiable formula has a projective unifier. • Projective unification (unifier) is preserved under extensions.

projective unification ⇒ unitary unification but 6 ⇐ S4.2 has unitary unification (Ghilardi, Sacchetti) but Example. The unifiable formula (x → y ) → x ∨ z has an mgu in S4.2 but can not have a projective unifier in S4.2. • Projective unifiers in Algebra Theorem(WD 2010) Every discriminator variety has projective unification. (modifying Burris ) Examples: BA, Monadic alg., Cylindric alg. of finite dimension

Method of ground unifiers A ground unifier is a unifier such that τ0 : Fm → {⊥, >} Examples: let τ0 be a ground unifier for a unifiable α. • Classical logic has projective unification, ε(x) = (α → x) ∧ (α ∨ τ0 (x)), for x ∈ Var (α), is a projective unifier for α. • S5 has projective unification, ε(x) = (α → x) ∧ (α ∨ τ0 (x)), for x ∈ Var (α), is a projective unifier for α. • G¨odel logic LC = INT + (α → β) ∨ (β → α) - projective unif. S4.3 = S4 + (α → β) ∨ (β → α) unif.?

Projective unification over S4 Theorem (Dzik, Wojtylak 2011, LJ IGPL). A modal logic L containing S4 has projective unification iff S4.3 ⊆ L.

S4.3 S4.2 Note: For L ⊇ S4.3 one formula for ε(x) usually does not exist. Theorem (Wro´ nski 1995 - 2008). An intermediate logic L has projective unification iff LC ⊆ L.

Applications: Structural Completeness (40 years) W.A.Pogorzelski, Bulletin de LAcademie Polonaise des Sciences, Ser. Sci. Math. vol.19, 349–351 (1971). A logic L is Structurally Complete, SC, if every structural and admissible rule in L is derivable in L. rules : r ⊆ Fmn × Fm, (α1 , . . . , αn ; β) ∈ r , only structural: (α1 , . . . , αn ; β) ∈ r ⇒ (εα1 , . . . , εαn ; εβ) ∈ r for every substitution ε; r : α1 , . . . , αn /β, we mean αi , β formula schemata r : α1 , . . . , αn /β is admissible in L if `L τ α1 , . . . , `L τ αn ⇒ `L τ β , for every τ . r : α1 , . . . , αn /β is derivable in L if α1 , . . . , αn `L β

Structural Completeness • Classical logic (Pogorzelski 1971), • Intuitionistic logic - No, T. Prucnal and A. Wro´ nski: admissible, but not derivable ¬x → y ∨ z / (¬x → y ) ∨ (¬x → z) (Kreisel-Putnam rule) • Intuitionistic Implication - T. Prucnal (St. Logica, 30, 1972), • G¨odel - Dummett’s LC and logics Gn , n = 2, . . . , W.Dzik, A.Wro´ nski, (St. Logica 32, 1973) • Modal logic S5 - No, P2 is admissible, but not derivable P2 : ♦ x ∧ ♦ ¬ x/ ⊥ overflow or passive rule, there does not exists ε such that `S5 ε(♦ x ∧ ♦ ¬ x).

Almost Structural Completeness r : α1 , . . . , αn /β is passive or overflow in L if {α1 , . . . , αn } is not unifiable in L. e.g. P2 : ♦x ∧ ♦¬ x/ ⊥ or ♦x ∧ ♦¬ x/ y is passive in S5. A logic L is Passively Structurally Complete, PSC, if each passive rule is derivable in L. (Cintula, Metcalfe 2008; Wro´ nski 2005: Overflow completeness: Intuitionist.) A logic L is Almost Structurally Complete, ASC, if every (structural) admissible rule which is not passive (or: rule with unifiable premises in L) is derivable in L. SC = ASC + PSC Examples: Ext S5 (monadic alg.), n-valued Lukasiewicz (MVn -alg.) Basic Logic with n-contraction (BLn -alg.) (WD 2006)

Corollary (WD, 2010) If a logic has projective unification, then it is almost structurally complete. Assume: L has projective unification and r : α/β an admissible rule in L, non-passive: α - unifiable, let σ be its projective unifier, i.e. `L σα and (?) α `L σx ←→ x, but r is admissible, hence `L σβ and, by (?) α `L σβ ←→ β, thus α `L β, i.e. r is derivable in L.

Corollary (WD, P.Wojtylak 2011). Every modal logic L containing S4.3 is almost structurally complete. For a modal logic L ⊇ S4.3: L is hereditarily structurally complete iff McKinsey ax. M : ♦α → ♦α is in L iff L 6⊆ L(•, •)

L(•, •)

S4.3 + M

S4.3

Unification versus structural completeness in Ext S4

L(f2 )

S4.3 S4.2

S4 Unitary unification and (almost) structural completeness independent for logics L 6⊇ S4.3. • L(f2 ) is structurally complete but its unification - not unitary • S4.2 has unitary unification but is not - even almost structurally complete.

S. Ghilardi : unification depends on finitely presented and projective objects in a category. Finitely presented projective algebras are retracts of n-generated free algebras (Lindenbaum-Tarski algebras) FL (n) of a logic L. Let [A) denote the principal filter, generated by a formula A, in FL (n), where Var (A) = {x1 , . . . , xn }. Corollary A finitely presented S4.3 algebra A is projective iff A u FL (n)/[A), for some unifiable formula A. In particular: for every formula A such that A 6` (♦x1 ∧ ♦ ∼ x1 ) ∨ · · · ∨ (♦xn ∧ ♦ ∼ xn ), xi ∈ Var (A), (equivalently, for every unifiable formula A), the quotient F(n)/[A) of the n-generated free algebra F(n) is a projective S4.3-algebra.

Applications to Admissible Rules r : α/β is admissible in L if `L τ α ⇒ `L τ β , for every τ . Let U(α) = maximal unifiers for α in L. r : α/β is admissible in L if `L τ β, for τ ∈ U(α). L - unitary: card U(α) = 1, (mgu), L - finitary: card U(α) < ω, Example:

the Scott rule

((♦x → x) → (x ∨ ¬x)) / (♦x ∨ ¬x)) - is admissible in S4.

References [1] Baader,F., Ghilardi, S., Unification in Modal and Discription logic, Logic Journal of the IGPL, doi:10.1093/jigpal/jzq008 (2010), [2] Baader,F. and Snyder, W., Unification Theory. In: Robinson, A. and Voronkov, A. (eds.) Handbook of Automated Reasoning, Ch.8, Elsevier Science Publ., MIT, 1 (2001), 445–533. [3] Burris S., Discriminator varieties and symbolic computation, J. Symbolic Computation 13 (1992), 175-207. [4] Dzik, W., Splittings of Lattices of Theories and Unification Types, Contributions to General Algebra, 17 (2006), 71–81. [5] Dzik, W., Unification Types in Logic, USl, Katowice 2007. [6] Dzik, W., Wojtylak, P. Projective uUnification in Modal Logic, Logic Journal of the IGPL, accepted (2011), [7] Ghilardi, S., Unification throught Projectivity , J. of Symb. Computation, 7(1997), 733-752. [8] Ghilardi, S., Unification in Intuitionistic Logic, J. Symb. Log. 64(2): 859-880 (1999) [8] Ghilardi, S., Sacchetti, L., Filtering Unification and Most General Unifiers in Modal Logic, J. of Symbolic Logic, 69 (2004), 879–906. [9] Wro´ nski, A., Transparent Unification Problem, Reports on Mathematical Logic, 29 (1995), 105–107. [10] Wro´ nski, A., Transparent Verifiers for Intermediate Logics, Abstracts of 54 KHL, Krak´ ow 2008.