The logic of bunched implications is undecidable

TL;DR

This paper proves that the logic of bunched implications (BI) and related logics are undecidable by encoding Wang tilings, challenging previous beliefs of their decidability and extending to various fragments.

Contribution

It demonstrates the undecidability of BI and related logics through tiling encodings, covering a broad class of non-commutative and disjointive negation logics.

Findings

Proves BI is undecidable via Wang tilings.

Extends undecidability to non-commutative and disjointive negation fragments.

Contradicts previous assumptions of BI's decidability.

Abstract

The logic of bunched implications (BI), introduced by O'Hearn and Pym (1999), has attracted significant attention due to its elegant proof calculus, varied semantics, and close connections to the propositional fragment of separation logic. We show here that provability in BI is undecidable by encoding Wang tilings into its ternary relational semantics. Equivalently, this yields the undecidability of the equational theory of BI-algebras. Our result is much more general, applying to the {and, or, not, --*}-fragment of stronger and weaker logics: the negation simply needs to be disjointive, and the multiplicative conjunction need not be commutative (then --* splits into two divisions \, /). Consequently, our result covers an interval that includes BI, the non-commutative logic GBI, and Boolean BI (BBI), the latter already known to be undecidable. This result contrasts with a long-standing expectation that BI might be decidable. We also identify the gaps in the publications claiming decidability.