Presburger arithmetic is the first-order theory of the integers with addition and ordering, but without multiplication. This theory is decidable and the sets it defines admit several different representations, including formulas, generators, and finite automata, the latter being the focus of this thesis. Finite-automata representations of Presburger sets work by encoding numbers as words and sets by automata-defined languages. With this representation, set operations are easily computable as automata operations, and minimized deterministic automata are a canonical representation of Presburger sets. However, automata-based representations are somewhat opaque and do not allow all operations to be performed efficiently. An ideal situation would be to be able to move easily between formula-based and automata-based representations but, while building an automaton from a formula is a well understood process, moving the other way is a much more difcult problem that has only attracted attention fairly recently. The main results of this thesis are new algorithms for extracting information about Presburger-definable sets represented by finite automata. More precisely, we present algorithms that take as input a finite-automaton representing a Presburger definable set S and compute in polynomial time the affine hull over Q or over Z of the set S, i.e., the smallest set defined by a conjunction of linear equations (and congruence relations in Z) which includes S. Also, we present an algorithm that takes as input a deterministic finite-automaton representing the integer elements of a polyhedron P and computes a quantifier-free formula corresponding to this set. The algorithms rely on a very detailed analysis of the scheme used for encoding integer vectors and this analysis sheds light on some structural properties of finite-automata representing Presburger definable sets. The algorithms presented have been implemented and the results are encouraging : automata with more than 100000 states are handled in seconds.

This thesis concerns decision procedures for fragments of linear arithmetic and their application to model-checking one-counter automata. The first part of this thesis covers the complexity of decision problems for different types of linear arithmetic, namely the existential subset of the first-order linear theory over the p-adic numbers and the existential subset of Presburger arithmetic with divisibility, with all integer constants and coefficients represented in binary. The most important result of this part is a new upper complexity bound of <b>NEXPTIME</b> for existential Presburger arithmetic with divisibility. The best bound that was known previously was <b>2NEXPTIME</b>, which followed directly from the original proof of decidability of this theory by Lipshitz in 1976. Lipshitz also gave a proof of <b>NP</b>-hardness of the problem in 1981. Our result is the first improvement of the bound since this original description of a decision procedure. Another new result, which is both an important building block in establishing our new upper complexity bound for existential Presburger arithmetic with divisibility and an interesting result in its own right, is that the decision problem for the existential linear theory of the p-adic numbers is in the Counting Hierarchy <b>CH</b>, and thus within <b>PSPACE</b>. The precise complexity of this problem was stated as open by Weispfenning in 1988, who showed that it is in <b>EXPTIME</b> and <b>NP</b>-hard. The second part of this thesis covers two problems concerning one-counter automata. The first problem is an LTL synthesis problem on one-counter automata with integer-valued and parameterised updates and with equality tests. The decidability of this problem was stated as open by G&ouml;ller et al. in 2010. We give a reduction of this problem to the decision problem of a subset of Presburger arithmetic with divisibility with one quantifier alternation and a restriction on existentially quantified variables. A proof of decidability of this theory is currently under review. The final result of this thesis concerns a type of one-counter automata that differs from the previous one in that it allows parameters only on tests, not actions, and it includes both equality and disequality tests on counter values. The decidability of the basic reachability problem on such one-counter automata was stated as an open problem by Demri and Sangnier in 2010. We show that this problem is decidable by a reduction to the decision problem for Presburger arithmetic.

of doctoral thesis Study of Arithmetical Structures and Theories with Regard to Representative and Descriptive Analysis Petr Glivický We are motivated by a problem of understanding relations between local and global properties of an operation o in a structure of the form B, o , with regard to an application for the study of models B, · of Peano arithmetic, where B is a model of Presburger arithmetic. We are particularly interested in a dependency problem, which we formulate as the problem of describing the dependency closure iclO (E) = {d ∈ Bn ; (∀o, o ∈ O)(o E = o E ⇒ o(d) = o (d))}, where B is a structure, O a set of n-ary operations on B, and E ⊆ Bn. We show, that this problem can be reduced to a definability question in certain expansion of B. In particular, if B is a saturated model of Presburger arithmetic, and O is the set of all (saturated) Peano products on B, we prove that, for a ∈ B, iclO ({a}×B) is the smallest possible, i.e. it contains just those pairs (d0, d1) ∈ B2 for which at least one of di equals p(a), for some polynomial p ∈ Q[x]. We show that the presented problematics is closely connected to the descriptive analysis of linear theories. That are theories, models of which are - up to a change of the language - certain discretely ordered modules over specific discretely ordered...

Model checking is a powerful technique for analyzing reach ability and temporal properties of finite state systems. Model-checking finite state systems has been well-studied and there are well known efficient algorithms for this problem. However these algorithms may not terminate when applied directly to in finite state systems. Counter systems are a class of in fininite state systems where the domain of counter values is possibly in finite. Many practical systems like cache coherence protocols, broadcast protocols etc, can naturally be modeled as counter systems. In this thesis we identify a class of counter systems, and propose a new technique to check whether a system from this class satires’ a given CTL formula. The key novelty of our approach is a way to use existing reach ability analysis techniques to answer both \until" and \global" properties; also our technique for \global" properties is different from previous techniques that work on other classes of counter systems, as well as other classes of in finite state systems. We also provide some results by applying our approach to several natural examples, which illustrates the scope of our approach.

Simple counting quantifiers that can be used to compare the number of role successors of an individual or the cardinality of a concept with a fixed natural number have been employed in Description Logics (DLs) for more than two decades under the respective names of number restrictions and cardinality restrictions on concepts. Recently, we have considerably extended the expressivity of such quantifiers by allowing to impose set and cardinality constraints formulated in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA) on sets of role successors and concepts, respectively. We were able to prove that this extension does not increase the complexity of reasoning. In the present paper, we investigate the expressive power of the DLs obtained in this way, using appropriate bisimulation characterizations and 0–1 laws as tools to differentiate between the expressiveness of different logics. In particular, we show that, in contrast to most classical DLs, these logics are no longer expressible in first-order predicate logic (FOL), and we characterize their first-order fragments. In most of our previous work on DLs with QFBAPA-based set and cardinality constraints we have employed finiteness restrictions on interpretations to ensure that the obtained sets are finite, as required by the standard semantics for QFBAPA. Here we dispense with these restrictions to ease the comparison with classical DLs, where one usually considers arbitrary models rather than finite ones, easier. It turns out that doing so does not change the complexity of reasoning.

We introduce a new description logic that extends the well-known logic ALCQ by allowing the statement of constraints on role successors that are more general than the qualified number restrictions of ALCQ. To formulate these constraints, we use the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA), in which one can express Boolean combinations of set constraints and numerical constraints on the cardinalities of sets. Though our new logic is considerably more expressive than ALCQ, we are able to show that the complexity of reasoning in it is the same as in ALCQ, both without and with TBoxes. / The first version of this report was put online on April 6, 2017. The current version, containing more information on related work, was put online on July 13, 2017. This is an extended version of a paper published in the proceedings of FroCoS 2017.