The Logic of Action Lattices is Undecidable

Abstract

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an ω-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved Π 0 1 -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by D. Kozen in 1994. In this article, we show that it is undecidable, more precisely, Σ 0 1 -complete. We also prove the same complexity results for all recursively enumerable logics between action logic and infinitary action logic; for fragments of those only one of the two lattice (additive) connectives; for action logic extended with the law of distributivity.Here the lower bound is due to Buszkowski and the upper one is due to Palka; recently A. Das and D. Pous [11] gave another proof of the Π 0 1 upper bound for ACT ω , based on non-well-founded proofs.

Commutative Action Logic

“…The former is a commutative counterpart of results by Buszkowski and Palka [1,18]. The latter is a commutative counterpart of an earlier result by the author [9,11].…”

“…A similar idea was used to prove undeciability in the non-commutative case, for ACT [9]. In the commutative situation it is even more straightforward, since Minsky computation here is represented directly, without a detour through totality of context-free grammars [1,9].…”

“…In contrast, ACT ω is Π 0 1 -complete, as shown by Buszkowski and Palka [1,18]. For the general case, ACT is Σ 0 1 -complete [9,11], which is already the maximal possible complexity: iteration in action lattices in general allows a finite axiomatization, unlike the *-continuous situation, which requires infinitary mechanisms.…”

“…In Section 4 we prove Σ 0 1 -completeness for CommACT by encoding circular behaviour of Minsky machines and the technique of effective inseparability (Myhill's theorem). This argument is even more straightforward than the one from [9,11], since we do not need intermediate context-free grammars.…”

Iterative Division in the Distributive Full Non-associative Lambek Calculus

“…Firstly, it has been shown that associativity often leads to undecidability. Buszkowski [5] and Palka [20] have shown that the logic of all * -continuous action lattices (the ones where, roughly, Kleene star is equivalent to an infinite disjunction) is undecidable; Kuznetsov [15] established recently that the logic of all action lattices is undecidable as well. His proof applies also to Pratt's Action Logic.…”