CUT-ELIMINATION FOR DISTRIBUTIVE SUBSTRUCTURAL LOGICS NIKOLAOS GALATOS AND PETER JIPSEN
A residuated lattice is an algebra of the form A = (A, ∧, ∨, ·, \, /, 1) where (A, ∧, ∨) is a lattice, (A, ·, 1) is a monoid and the following residuation property holds for all x, y, z ∈ A (res)
xy ≤ z
iff
x ≤ z/y
iff
y ≤ x\z.
An FL-algebra is an expansion of a residuated lattice with a constant 0. A residuated lattice/FL-algebra is called distributive, if its underlying lattice is distributive. The variety of distributive residuated lattices/FL-algebras is denoted by DRL/DFL. Giambrone and Brady considered a Gentzen-style sequent calculus for commutative distributive residuated lattices/FL-algebras and Kozak extended it to the non-commutative case (DFL). The system has two punctuation symbols, comma and semicolon, corresponding to the connectives of fusion and conjunction. Extending our previous work on FL and residuated frames, we define distributive residuated frames, as structures of the form W = (W, W 0 , N, ◦, , , ε, f, i, h), where W and W 0 are sets, N ⊆ W × W 0 , ◦ and f are binary operations on W and ε ∈ W , while , i : W 0 × W → P(W 0 ), and , h : W × W 0 → P(W 0 ), such that • N is ◦-nuclear: for all x, y ∈ W and z ∈ W 0 – x ◦ y N z iff x N z y iff y N x z. • N is f-nuclear: for all x, y ∈ W and z ∈ W 0 – x f y N z iff x N z i y iff y N x h z. • N is f-distributive: for all x, y, z ∈ W and w ∈ W 0 , – (x f y) f z N w iff x f (y f z) N w; – x f y N w implies y f x N w; – x N w implies x f y N w; and – x f x N w implies x N w. For X ⊆ W and Y ⊆ W 0 , we define X B = {z ∈ W 0 : X N z} and Y C = {w ∈ W : w N Y }; here X N z means x N z for all x ∈ X. Given a distributive residuated frame W, we define its Galois algebra W+ = (W + , ∩, ∪N , ◦N , \, /, {ε}BC ), where W + = {X ⊆ W : X BC = X} X ∪N Y = (X ∪ Y )BC [X ◦ Y = {x ◦ y : x ∈ X, y ∈ Y }] X ◦N Y = (X ◦ Y )BC X\Y = {z : X ◦ {z} ⊆ Y } Y /X = {z : {z} ◦ X ⊆ Y }. 1
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NIKOLAOS GALATOS AND PETER JIPSEN
Lemma 1. The algebra W+ is a distributive residuated lattice. Distributive Gentzen frames are pairs (W, B), where • W is a distributive residuated frame • B is a subset of W that {◦, f, ε}-generates W , • W 0 contains a copy of B, • B is an algebra in the type of residuated lattices, and • the following conditions (read the fraction line as implication) hold for all a, b ∈ B, x, y ∈ W and z ∈ W 0 x N a a N z (CUT) xN z
aN a
(Id)
x N a b N z (\L) x ◦ (a\b) N z
a ◦ x N b (\R) x N a\b
x N a b N z (/L) (b/a) ◦ x N z
x ◦ a N b (/R) x N b/a
a ◦ b N z (·L) a·bN z a f b N z (∧L) a∧bN z a N z b N z (∨L) a∨bN z
xN a yN b (·R) x◦y N a·b x N a x N b (∧R) xN a∧b
x N a (∨R`) xN a∨b
x N b (∨Rr) xN a∨b
ε N z (1L) (1R) 1N z εN 1 A cut-free distributive Gentzen frame is one that may not satisfy the cut property. We construct the structure WDFL by selecting as W the set of left-hand sides of sequents in DFL; as W 0 = SW × F m, where the elements of SW are left-hand sides of sequents in DFL with an empty slot (unary linear polynomials) and F m is the set of all propositional formulas on the language of residuated lattices; as ◦ the comma; as f the semicolon; and where x N (u, b) means the sequent u[x] ⇒ b is provable in DFL. Lemma 2. The structure (WDFL , Fm) is a distributive Gentzen frame. Likewise the structure associated with cut-free DFL is a cut-free distributive Gentzen frame. Theorem 3. Let (W, B) be a cut-free distributive Gentzen frame. For all a, b ∈ B, X, Y ∈ W+ and for every connective •, if a ∈ X ⊆ {a}C and b ∈ Y ⊆ {b}C , then + a •B b ∈ X •W Y ⊆ {a •B b}C . [If (W, B) is a distributive Gentzen frame, the map x 7→ {x}C from B to W+ is a homomorphism from the algebra B into the distributive residuated lattice W+ .] We define validity of a sequent in a Gentzen frame by interpreting ⇒ as N , comma as ◦, semicolon as f and the empty sequence by ε.
CUT-ELIMINATION FOR DISTRIBUTIVE SUBSTRUCTURAL LOGICS
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Theorem 4. If (W, B) is a cut free distributive Gentzen frame, then every sequent that is valid in W+ is also valid in (W, B). Theorem 5. The system DFL enjoys the cut elimination property. A simple equation is one of the form t0 ≤ t1 ∨ · · · ∨ tn , where ti are terms over {∧, ·, 1}, and t0 is linear. Lemma 6. Every equation over {∧, ∨, ·, 1} is equivalent over DRL to a conjunction of simple equations. Lemma 7. Simple equations ε correspond to structural rules R(ε), namely rules that do not involve logical connectives. We call the corresponding rules simple. Theorem 8. Let (W, B) be a cut free distributive Gentzen frame and let ε be an equation over {∧, ∨, ·, 1}. Then (W, B) satisfies R(ε) iff W+ satisfies ε. Corollary 9. The system DFL + R enjoys the cut elimination property, for every set R of simple rules, and in particular for sets of rules corresponding to equations over {∧, ∨, ·, 1}. References [1] S. Giambrone, PTW+ and RW+ are decidable, J. Philosophical Logic 14 (1985), 235–254. [2] R. T. Brady, The Gentzenization and decidability of RW, J. Philosophical Logic 19 (1990), 35–73. [3] M Kozak, Distributive full Lambek calculus has the finite model property. manuscript. Department of Mathematics, University of Denver, 2360 S. Gaylord St., Denver, CO 80208, USA E-mail address:
[email protected] Chapman University, Department of Mathematics and Computer Science, One University Drive, Orange, CA 92866, USA E-mail address:
[email protected] 