Subclasses of Presburger Arithmetic and the Weak EXP Hierarchy

It is shown that for any fixed $i>0$, the $\Sigma_{i+1}$-fragment of Presburger arithmetic, i.e., its restriction to $i+1$ quantifier alternations beginning with an existential quantifier, is complete for $\mathsf{\Sigma}^{\mathsf{EXP}}_{i}$, the $i$-th level of the weak EXP hierarchy, an analogue to the polynomial-time hierarchy residing between $\mathsf{NEXP}$ and $\mathsf{EXPSPACE}$. This result completes the computational complexity landscape for Presburger arithmetic, a line of research which dates back to the seminal work by Fischer & Rabin in 1974. Moreover, we apply some of the techniques developed in the proof of the lower bound in order to establish bounds on sets of naturals definable in the $\Sigma_1$-fragment of Presburger arithmetic: given a $\Sigma_1$-formula $\Phi(x)$, it is shown that the set of non-negative solutions is an ultimately periodic set whose period is at most doubly-exponential and that this bound is tight.Comment: 10 pages, 2 figure

Recompression: a simple and powerful technique for word equations

In this paper we present an application of a simple technique of local recompression, previously developed by the author in the context of compressed membership problems and compressed pattern matching, to word equations. The technique is based on local modification of variables (replacing X by aX or Xa) and iterative replacement of pairs of letters appearing in the equation by a `fresh' letter, which can be seen as a bottom-up compression of the solution of the given word equation, to be more specific, building an SLP (Straight-Line Programme) for the solution of the word equation. Using this technique we give a new, independent and self-contained proofs of most of the known results for word equations. To be more specific, the presented (nondeterministic) algorithm runs in O(n log n) space and in time polynomial in log N, where N is the size of the length-minimal solution of the word equation. The presented algorithm can be easily generalised to a generator of all solutions of the given word equation (without increasing the space usage). Furthermore, a further analysis of the algorithm yields a doubly exponential upper bound on the size of the length-minimal solution. The presented algorithm does not use exponential bound on the exponent of periodicity. Conversely, the analysis of the algorithm yields an independent proof of the exponential bound on exponent of periodicity. We believe that the presented algorithm, its idea and analysis are far simpler than all previously applied. Furthermore, thanks to it we can obtain a unified and simple approach to most of known results for word equations. As a small additional result we show that for O(1) variables (with arbitrary many appearances in the equation) word equations can be solved in linear space, i.e. they are context-sensitive.Comment: Submitted to a journal. Since previous version the proofs were simplified, overall presentation improve

Approaching Arithmetic Theories with Finite-State Automata

The automata-theoretic approach provides an elegant method for deciding linear arithmetic theories. This approach has recently been instrumental for settling long-standing open problems about the complexity of deciding the existential fragments of Büchi arithmetic and linear arithmetic over p-adic fields. In this article, which accompanies an invited talk, we give a high-level exposition of the NP upper bound for existential Büchi arithmetic, obtain some derived results, and further discuss some open problems

Some new results on decidability for elementary algebra and geometry

We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the AE fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilbert's 10th problem show that the EA fragments for metric and normed spaces and the AE fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3

Hilbert's tenth problem for weak theories of arithmetic

AbstractHilbert's tenth problem for a theory T asks if there is an algorithm which decides for a given polynomial p(x̄) from Z[x̄] whether p(x̄) has a root in some model of T. We examine some of the model-theoretic consequences that an affirmative answer would have in cases such as T = Open Induction and others, and apply these methods by providing a negative answer in the cases when T is some particular finite fragment of the weak theories IE1 (bounded existential induction) or IU-1 (parameter-free bounded universal induction)