The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable

TL;DR

This paper proves that adding an uninterpreted unary predicate to Presburger arithmetic makes its purely universal fragment undecidable, revealing a sharp boundary between decidability and undecidability with implications for verification.

Contribution

It establishes the undecidability of the purely universal fragment of the extended Presburger arithmetic with unary predicates, and analyzes the complexity of sentences with quantifier alternations.

Findings

Purely universal fragment is undecidable with unary predicates.

Quantifier alternation leads to $ ext{Sigma}_1^1$-completeness.

Results impact verification and undecidability of related logical theories.

Abstract

The first-order theory of addition over the natural numbers, known as Presburger arithmetic, is decidable in double exponential time. Adding an uninterpreted unary predicate to the language leads to an undecidable theory. We sharpen the known boundary between decidable and undecidable in that we show that the purely universal fragment of the extended theory is already undecidable. Our proof is based on a reduction of the halting problem for two-counter machines to unsatisfiability of sentences in the extended language of Presburger arithmetic that does not use existential quantification. On the other hand, we argue that a single $\forall\exists$ quantifier alternation turns the set of satisfiable sentences of the extended language into a $\Sigma^1_1$-complete set. Some of the mentioned results can be transfered to the realm of linear arithmetic over the ordered real numbers. This concerns the undecidability of the purely universal fragment and the $\Sigma^1_1$-hardness for sentences with at least one quantifier alternation. Finally, we discuss the relevance of our results to verification. In particular, we derive undecidability results for quantified fragments of separation logic, the theory of arrays, and combinations of the theory of equality over uninterpreted functions with restricted forms of integer arithmetic. In certain cases our results even imply the absence of sound and complete deductive calculi.