Undecidability of Consequence Relation in Full Non-Associative Lambek Calculus

Abstract

We prove that the consequence relation in the Full Non-associative Lambek Calculus is undecidable. An encoding of the halting problem for 2-tag systems using finitely many sequents in the language {⋅,∨} is presented. Therefore already the consequence relation in this fragment is undecidable. Moreover, the construction works even when the structural rules of exchange and contraction are added.

Decision Problems for Propositional Non-associative Linear Logic and Extensions

“…Immediately, it turns out that involutive full non-associative Lambek calculus (denoted by InFNL) is strongly conservative over full non-associative Lambek calculus (denoted by FNL). In conjunction with the undecidability result established by Chvalovský [6], it follows that the deducibility problem for InFNL is undecidable. Moreover, using the idea in [19], we show that the provability problem for NACCLL − is undecidable.…”

“…In view of this result, it is natural to ask how the lack of associativity of multiplication affects the decision problems for linear logic and related systems. So far, however, it has hardly been investigated whether the decision problems for non-associative versions of propositional linear logic are decidable or not, whereas several substructural logicians investigated the decision problems for various non-associative logics, see e.g., [2,4,5,6,8,9,14].…”

Complexity of the universal theory of bounded residuated distributive lattice-ordered groupoids

“…Next, we mention cases in which the lattice is not required to be distributive. This class of 'lattice-ordered residuated groupoids' has an undecidable universal theory [8], whereas its equational theory is in PSPACE [6]. If associativity of • is assumed, the undecidability of the universal theory is proved in [19].…”

Full Lambek Calculus With Contraction Is Undecidable

“…Finally, an encoding of atomic conditional SRSs in RL c is shown. Roughly speaking the conditionality in rules is expressed by join and an auxiliary rewriting system (inspired by [1]), the rewriting symbol is encoded by an implication and a set of rules by a meet of encoded rules. Although the constant 1 plays also an important role in this encoding, it can be shown that it is not necessary.…”

“…The idea of using the lattice disjunction for similar purposes comes from [14], where it is used for linear logic. In fact, our application of this idea was inspired by [3].…”