A Gentzen System and Decidability for Residuated Lattices

This note presents details of a result of Okada and Terui[2] which shows that the equational theory of residuated lattices is decidable and gives an effective algorithm based on a Gentzen system for propositional intuitionistic linear logic. The variety of residuated lattices is denoted by RL. An algebra (L,∨,∧, ∗, \, /, 1) is a member of this variety if (L,∨,∧) is a lattice, (L, ∗, 1) is a monoid, and x ∗ y ≤ z iff y ≤ x\z iff x ≤ z/y for all x, y, z ∈ L (these equivalences can be expressed by equations). We use s, t, u for terms in the language of RL, and γ, δ, ρ, σ for (finite) sequences of terms. Concatenation of sequences γ and δ is denoted by γδ, and terms are considered as sequences of length 1. (In this note, multiplication in RL is written explicitly as s ∗ t.) A pair (σ, t) is called a sequent and is written σ ` t. The symbol ` is read as yields, and the semantic interpretation of s1s2 . . . sn ` t is that the inclusion s1∗s2∗ · · · ∗sn ≤ t holds in RL. The empty sequence ε is interpreted as the multiplicative unit 1. Using this notation, we list below some quasi-inclusions s1 ≤ t1 & · · ·& sn ≤ tn ⇒ s ≤ t in the style of Gentzen rules: s1 ` t1 . . . sn ` tn s ` t name of rule. With the given semantic interpretation it is straight forward to check that all these quasi-inclusions hold in RL. The details in the proof of the completeness lemma (Lemma 3 below) justify the specific form of these rules. t ` t Id γδ ` u γ1δ ` u 1-left ε ` 1 1-right γstδ ` u γs∗tδ ` u ∗left γ ` s δ ` t γδ ` s∗t ∗right σ ` s γtδ ` u γσs\tδ ` u \left sγ ` t γ ` s\t \right Date: April 3, 2000.