Succinct Population Protocols for Presburger Arithmetic

International audienceIn [5], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula $ϕ$ of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with $2 O(poly(|ϕ|))$ states that computes $ϕ$. More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula $ϕ$ of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with $O(poly(|ϕ|))$ states. Our proof is based on several new constructions, which may be of independent interest. Given a formula $ϕ$ of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with $O(|ϕ| 3)$ leaders) that computes ϕ; this completes the work initiated in [8], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes $ϕ$. Our last construction gets rid of this leader for small inputs

The Well Structured Problem for Presburger Counter Machines

International audienceWe introduce the well structured problem as the question of whether a model (here a counter machine) is well structured (here for the usual ordering on integers). We show that it is undecidable for most of the (Presburger-defined) counter machines except for Affine VASS of dimension one. However, the strong well structured problem is decidable for all Presburger counter machines. While Affine VASS of dimension one are not, in general, well structured, we give an algorithm that computes the set of predecessors of a configuration; as a consequence this allows to decide the well structured problem for 1-Affine VASS

Fast and Succinct Population Protocols for Presburger Arithmetic

In their 2006 seminal paper in Distributed Computing, Angluin et al. present a construction that, given any Presburger predicate as input, outputs a leaderless population protocol that decides the predicate. The protocol for a predicate of size $m$ (when expressed as a Boolean combination of threshold and remainder predicates with coefficients in binary) runs in $\mathcal{O}(m \cdot n^2 \log n)$ expected number of interactions, which is almost optimal in $n$. However, the number of states of the protocol is exponential in $m$. Blondin et al. described in STACS 2020 another construction that produces protocols with a polynomial number of states, but exponential expected number of interactions. We present a construction that produces protocols with $\mathcal{O}(m)$ states that run in expected $\mathcal{O}(m^{7} \cdot n^2)$ interactions, optimal in $n$, for all inputs of size $\Omega(m)$. For this we introduce population computers, a carefully crafted generalization of population protocols easier to program, and show that our computers for Presburger predicates can be translated into fast and succinct population protocols.Comment: 52 pages, 4 figures, to be published in SAND 202

Decidability of Difference Logic over the Reals with Uninterpreted Unary Predicates

First-order logic fragments mixing quantifiers, arithmetic, and uninterpreted predicates are often undecidable, as is, for instance, Presburger arithmetic extended with a single uninterpreted unary predicate. In the SMT world, difference logic is a quite popular fragment of linear arithmetic which is less expressive than Presburger arithmetic. Difference logic on integers with uninterpreted unary predicates is known to be decidable, even in the presence of quantifiers. We here show that (quantified) difference logic on real numbers with a single uninterpreted unary predicate is undecidable, quite surprisingly. Moreover, we prove that difference logic on integers, together with order on reals, combined with uninterpreted unary predicates, remains decidable.Comment: This is the preprint for the submission published in CADE-29. It also includes an additional detailed proof in the appendix. The Version of Record of this contribution will be published in CADE-2

Revisiting Parameter Synthesis for One-Counter Automata

We study the synthesis problem for one-counter automata with parameters. One-counter automata are obtained by extending classical finite-state automata with a counter whose value can range over non-negative integers and be tested for zero. The updates and tests applicable to the counter can further be made parametric by introducing a set of integer-valued variables called parameters. The synthesis problem for such automata asks whether there exists a valuation of the parameters such that all infinite runs of the automaton satisfy some ?-regular property. Lechner showed that (the complement of) the problem can be encoded in a restricted one-alternation fragment of Presburger arithmetic with divisibility. In this work (i) we argue that said fragment, called ??_RPAD^+, is unfortunately undecidable. Nevertheless, by a careful re-encoding of the problem into a decidable restriction of ??_RPAD^+, (ii) we prove that the synthesis problem is decidable in general and in 2NEXP for several fixed ?-regular properties. Finally, (iii) we give polynomial-space algorithms for the special cases of the problem where parameters can only be used in counter tests

Semilinear Representations for Series-Parallel Atomic Congestion Games

We consider atomic congestion games on series-parallel networks, and study the structure of the sets of Nash equilibria and social local optima on a given network when the number of players varies. We establish that these sets are definable in Presburger arithmetic and that they admit semilinear representations whose all period vectors have a common direction. As an application, we prove that the prices of anarchy and stability converge to 1 as the number of players goes to infinity, and show how to exploit these semilinear representations to compute these ratios precisely for a given network and number of players

Fluted Logic with Counting

The fluted fragment is a fragment of first-order logic in which the order of quantification of variables coincides with the order in which those variables appear as arguments of predicates. It is known that the fluted fragment possesses the finite model property. In this paper, we extend the fluted fragment by the addition of counting quantifiers. We show that the resulting logic retains the finite model property, and that the satisfiability problem for its (m+1)-variable sub-fragment is in m-NExpTime for all positive m. We also consider the satisfiability and finite satisfiability problems for the extension of any of these fragments in which the fluting requirement applies only to sub-formulas having at least three free variables