Presburger Arithmetic Research Articles

Presburger Arithmetic with algebraic scalar multiplications

We consider Presburger arithmetic (PA) extended by scalar multiplication by an algebraic irrational number $\alpha$, and call this extension $\alpha$-Presburger arithmetic ($\alpha$-PA). We show that the complexity of deciding sentences in $\alpha$-PA is substantially harder than in PA. Indeed, when $\alpha$ is quadratic and $r\geq 4$, deciding $\alpha$-PA sentences with $r$ alternating quantifier blocks and at most $c\ r$ variables and inequalities requires space at least $K 2^{\cdot^{\cdot^{\cdot^{2^{C\ell(S)}}}}}$ (tower of height $r-3$), where the constants $c, K, C>0$ only depend on $\alpha$, and $\ell(S)$ is the length of the given $\alpha$-PA sentence $S$. Furthermore deciding $\exists^{6}\forall^{4}\exists^{11}$ $\alpha$-PA sentences with at most $k$ inequalities is PSPACE-hard, where $k$ is another constant depending only on~$\alpha$. When $\alpha$ is non-quadratic, already four alternating quantifier blocks suffice for undecidability of $\alpha$-PA sentences.

Multi-Party Campaigning

We study a social choice setting of manipulation in elections and extend the usual model in two major ways: first, instead of considering a single manipulating agent, in our setting there are several, possibly competing ones; second, instead of evaluating an election after the first manipulative action, we allow several back-and-forth rounds to take place. We show that in certain situations, such as in elections with only a few candidates, optimal strategies for each of the manipulating agents can be computed efficiently. Our algorithmic results rely on formulating the problem of finding an optimal strategy as sentences of Presburger arithmetic that are short and only involve small coefficients, which we show is fixed-parameter tractable -- indeed, one of our contributions is a general result regarding fixed-parameter tractability of Presburger arithmetic that might be useful in other settings. Following our general theorem, we design quite general algorithms; in particular, we describe how to design efficient algorithms for various settings, including settings in which we model diffusion of opinions in a social network, complex budgeting schemes available to the manipulating agents, and various realistic restrictions on adversary actions.

Interpolating bit-vector formulas using uninterpreted predicates and Presburger arithmetic

The inference of program invariants over machine arithmetic, commonly called bit-vector arithmetic, is an important problem in verification. Techniques that have been successful for unbounded arithmetic, in particular Craig interpolation, have turned out to be difficult to generalise to machine arithmetic: existing bit-vector interpolation approaches are based either on eager translation from bit-vectors to unbounded arithmetic, resulting in complicated constraints that are hard to solve and interpolate, or on bit-blasting to propositional logic, in the process losing all arithmetic structure. We present a new approach to bit-vector interpolation, as well as bit-vector quantifier elimination (QE), that works by lazy translation of bit-vector constraints to unbounded arithmetic. Laziness enables us to fully utilise the information available during proof search (implied by decisions and propagation) in the encoding, and this way produce constraints that can be handled relatively easily by existing interpolation and QE procedures for Presburger arithmetic. The lazy encoding is complemented with a set of native proof rules for bit-vector equations and non-linear (polynomial) constraints, this way minimising the number of cases a solver has to consider. We also incorporate a method for handling concatenations and extractions of bit-vector efficiently.

Truncations of ordered abelian groups

We give axioms for a class of ordered structures, called truncated ordered abelian groups (TOAG’s) carrying an addition. TOAG’s come naturally from ordered abelian groups with a 0 and a +, but the addition of a TOAG is not necessarily even a cancellative semigroup. The main examples are initial segments [0, tau ] of an ordered abelian group, with a truncation of the addition. We prove that any model of these axioms (i.e. a truncated ordered abelian group) is an initial segment of an ordered abelian group. We define Presburger TOAG’s, and give a criterion for a TOAG to be a Presburger TOAG, and for two Presburger TOAG’s to be elementarily equivalent, proving analogues of classical results on Presburger arithmetic. Their main interest for us comes from the model theory of certain local rings which are quotients of valuation rings valued in a truncation [0, a] of the ordered group {mathbb {Z}} or more general ordered abelian groups, via a study of these truncations without reference to the ambient ordered abelian group. The results are used essentially in a forthcoming paper (D’Aquino and Macintyre, The model theory of residue rings of models of Peano Arithmetic: The prime power case, 2021, arXiv:2102.00295) in the solution of a problem of Zilber about the logical complexity of quotient rings, by principal ideals, of nonstandard models of Peano arithmetic.

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Let \(f_1(n), \ldots , f_k(n)\) be polynomial functions of n. For fixed \(n\in \mathbb {N}\), let \(S_n\subseteq \mathbb {N}\) be the numerical semigroup generated by \(f_1(n),\ldots ,f_k(n)\). As n varies, we show that many invariants of \(S_n\) are eventually quasi-polynomial in n, most notably the Betti numbers, but also the type, the genus, and the size of the \(\Delta \)-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups \(S_n\subseteq \mathbb {N}^m\) generated by vectors whose coordinates are polynomial functions of n, and we prove that in this case the Betti numbers are also eventually quasi-polynomial functions of n.

Reachability relations of timed pushdown automata

Timed pushdown automata (tpda) are an expressive formalism combining recursion with a rich logic of timing constraints. We prove that reachability relations of tpda are expressible in linear arithmetic, a rich logic generalising Presburger arithmetic and rational arithmetic. The main technical ingredients are a novel quantifier elimination result for clock constraints (used to simplify the syntax of tpda transitions), the use of clock difference relations to express reachability relations of the fractional clock values, and an application of Parikh's theorem to reconstruct the integral clock values.

Multi-dimensional Interpretations of Presburger Arithmetic in Itself

Abstract Presburger arithmetic is the true theory of natural numbers with addition. We study interpretations of Presburger arithmetic in itself. The main result of this paper is that all self-interpretations are definably isomorphic to the trivial one. Here we consider interpretations that might be multi-dimensional. We note that this resolves a conjecture by Visser (1998, An overview of interpretability logic. Advances in Modal Logic, pp. 307–359). In order to prove the result, we show that all linear orderings that are interpretable in $({\mathbb{N}},+)$ are scattered orderings with the finite Hausdorff rank and that the ranks are bounded in the terms of the dimensions of the respective interpretations.

Logic-based specification and verification of homogeneous dynamic multi-agent systems

We develop a logic-based framework for formal specification and algorithmic verification of homogeneous and dynamic concurrent multi-agent transition systems. Homogeneity means that all agents have the same available actions at any given state and the actions have the same effects regardless of which agents perform them. The state transitions are therefore determined only by the vector of numbers of agents performing each action and are specified symbolically, by means of conditions on these numbers definable in Presburger arithmetic. The agents are divided into controllable (by the system supervisor/controller) and uncontrollable, representing the environment or adversary. Dynamicity means that the numbers of controllable and uncontrollable agents may vary throughout the system evolution, possibly at every transition. As a language for formal specification we use a suitably extended version of Alternating-time Temporal Logic, where one can specify properties of the type “a coalition of (at least) n controllable agents can ensure against (at most) m uncontrollable agents that any possible evolution of the system satisfies a given objective gamma″, where gamma is specified again as a formula of that language and each of n and m is either a fixed number or a variable that can be quantified over. We provide formal semantics to our logic {mathcal {L}}_{textsc {hdmas}} and define normal form of its formulae. We then prove that every formula in {mathcal {L}}_{textsc {hdmas}} is equivalent in the finite to one in a normal form and develop an algorithm for global model checking of formulae in normal form in finite HDMAS models, which invokes model checking truth of Presburger formulae. We establish worst case complexity estimates for the model checking algorithm and illustrate it on a running example.

Logics for Sizes with Union or Intersection

This paper presents the most basic logics for reasoning about the sizes of sets that admit either the union of terms or the intersection of terms. That is, our logics handle assertions All x y and AtLeast x y, where x and y are built up from basic terms by either unions or intersections. We present a sound, complete, and polynomial-time decidable proof system for these logics. An immediate consequence of our work is the completeness of the logic additionally permitting More x y. The logics considered here may be viewed as efficient fragments of two logics which appear in the literature: Boolean Algebra with Presburger Arithmetic and the Logic of Comparative Cardinality.

Definable groups in models of Presburger Arithmetic

This paper is devoted to understand groups definable in Presburger Arithmetic. We prove the following theorems:Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite.Theorem 2. Every bounded abelian group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z,+)n mod out by a lattice.

Logic and rational languages of scattered and countable series-parallel posets

Let A be an alphabet and SP⋄(A) denote the class of all countable N-free partially ordered sets labeled by A, in which chains are scattered linear orderings and antichains are finite. We characterize the rational languages of SP⋄(A) by means of logic. We define an extension of monadic second-order logic by Presburger arithmetic, named P-MSO, such that a language L of SP⋄(A) is rational if and only if L is the language of a sentence of P-MSO, with effective constructions from one formalism to the other. As a corollary, the P-MSO theory of SP⋄(A) is decidable.

Expressive cardinality restrictions on concepts in a description logic with expressive number restrictions

In two previous publications we have, on the one hand, extended the description logic (DL) ALCQ by more expressive number restrictions using numerical and set constraints expressed in the quantifier-free fragment of Boolean Algebra with Presburger Arithmetic (QFBAPA). The resulting DL was called ALCSCC . On the other hand, we have extended the terminological formalism of the well-known description logic ALC from concept inclusions (CIs) to more general cardinality constraints expressed in QFBAPA, which we called extended cardinality constraints. Here, we combine the two extensions, i.e., we consider extended cardinality constraints on ALCSCC concepts. We show that this does not increase the complexity of reasoning, which is NExpTime-complete both for extended cardinality constraints in the DL ALC and in its extension ALCSCC . The same is true for a restricted version of such cardinality constraints, where the complexity of reasoning decreases to ExpTime, not just for ALC , but also for ALCSCC.

Short Presburger Arithmetic Is Hard

We study the computational complexity of short sentences in Presburger arithmetic (Short-PA). Here by “short” we mean sentences with a bounded number of variables, quantifiers, inequalities, and Bo...

One-variable logic meets Presburger arithmetic

We consider the one-variable fragment of first-order logic extended with Presburger constraints. The logic is designed in such a way that it subsumes the previously-known fragments extended with counting, modulo counting or cardinality comparison and combines their expressive powers. We prove ▪-completeness of the logic by presenting an optimal algorithm for solving its finite satisfiability problem.

Parametric Presburger arithmetic: complexity of counting and quantifier elimination

AbstractWe consider an expansion of Presburger arithmetic which allows multiplication by k parameters . A formula in this language defines a parametric set as varies in , and we examine the counting function as a function of t. For a single parameter, it is known that can be expressed as an eventual quasi‐polynomial (there is a period m such that, for sufficiently large t, the function is polynomial on each of the residue classes mod m). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming ) we construct a parametric set such that is not even polynomial‐time computable on input . In contrast, for parametric sets with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that is always polynomial‐time computable in the size of t, and in fact can be represented using the gcd and similar functions.

Parameterized verification of algorithms for oblivious robots on a ring

We study verification problems for autonomous swarms of mobile robots that self-organize and cooperate to solve global objectives. In particular, we focus in this paper on the model proposed by Suzuki and Yamashita of anonymous robots evolving in a discrete space with a finite number of locations (here, a ring). A large number of algorithms have been proposed working for rings whose size is not a priori fixed and can be hence considered as a parameter. Handmade correctness proofs of these algorithms have been shown to be error-prone, and recent attention had been given to the application of formal methods to automatically prove those. Our work is the first to study the verification problem of such algorithms in the parameterized case. We show that safety and reachability problems are undecidable for robots evolving asynchronously. On the positive side, we show that safety properties are decidable in the synchronous case, as well as in the asynchronous case for a particular class of algorithms. Several other properties of the protocol can be decided as well. Decision procedures rely on an encoding in Presburger arithmetics formulae that can be verified by an SMT-solver. Feasibility of our approach is demonstrated by the encoding of several case studies.

Decidable weighted expressions with Presburger combinators

In this paper, we investigate the expressive power and the algorithmic properties of weighted expressions, which define functions from finite words to integers. First, we consider a slight extension of an expression formalism, introduced by Chatterjee et al. in the context of infinite words, in order to combine values given by unambiguous sum-automata, using Presburger arithmetic. We show that important decision problems such as emptiness, universality, inclusion and equivalence are PSpace-C for these expressions. We then investigate the extension of these expressions with Kleene star. This allows to iterate an expression over smaller fragments of the input word, and to combine the results by taking their iterated sum. The decision problems turn out to be undecidable, but we introduce a decidable and still expressive class of synchronised expressions.

Rigid models of Presburger arithmetic

AbstractWe present a description of rigid models of Presburger arithmetic (i.e., ‐groups). In particular, we show that Presburger arithmetic has rigid models of all infinite cardinalities up to the continuum, but no larger.

VC-Dimensions of Short Presburger Formulas

We study VC-dimensions of short formulas in Presburger Arithmetic, defined to have a bounded number of variables, quantifiers and atoms. We give both lower and upper bounds, which are tight up to a polynomial factor in the bit length of the formula.

A survival guide to presburger arithmetic

The first-order theory of the integers with addition and order, commonly known as Presburger arithmetic, has been a central topic in mathematical logic and computer science for almost 90 years. Presburger arithmetic has been the starting point for numerous lines of research in automata theory, model theory and discrete geometry. In formal verification, Presburger arithmetic is the first-choice logic to represent and reason about systems with infinitely many states. This article provides a broad yet concise overview over the history, decision procedures, extensions and geometric properties of Presburger arithmetic.