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

MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44642

Horbach,  Matthias
Automation of Logic, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons101767

Voigt,  Marco
Automation of Logic, MPI for Informatics, Max Planck Society;

http://pubman.mpdl.mpg.de/cone/persons/resource/persons45719

Weidenbach,  Christoph
Automation of Logic, MPI for Informatics, Max Planck Society;

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Fulltext (public)

arXiv:1703.01212.pdf
(Preprint), 298KB

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Citation

Horbach, M., Voigt, M., & Weidenbach, C. (2017). The Universal Fragment of Presburger Arithmetic with Unary Uninterpreted Predicates is Undecidable. Retrieved from http://arxiv.org/abs/1703.01212.


Cite as: http://hdl.handle.net/11858/00-001M-0000-002C-A5E7-D
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.