Danièle Beauquier

On translating one polyomino to tile the plane

Given a polyomino, we prove that we can decide whether translated copies of the polyomino can tile the plane. Copies that are rotated, for example, are not allowed in the tilings we consider. If such a tiling exists the polyomino is called anexact polyomino. Further, every such tiling of the plane by translated copies of the polyomino is half-periodic. Moreover, all the possible surroundings of an exact polyomino are described in a simple way.

Factors of Words

Let A be an alphabet of cardinality m, n k be a sequence of positive integers and ( ) n w A w k ∈ = . In this paper it is shown that if lim supn→ ∞ kn/ ln n < 1/ lnm then almost all words of length n over A contain the factor w, but if lim supn→ ∞ kn/ln n> 1/ lnm then this property is not true. Also, if lim infn→ ∞ kn/ lnn >1/ lnm then almost all words of length n over A do not contain the factor w. Moreover, if lim (ln ln ) n n n k m α → ∞ − = ∈ ¡ then limsup ( , , , / 1 exp( exp( )) n n n W n k w A m α → ∞ ≤ − − and liminf ( , , , / 1 exp( (1 1/ ) exp( )) n n n W n k w A m m α → ∞ ≥ − − − , where W(n, kn,w,A) denotes the set of words of length n over A containing the factor w of length n k .

A first order logic for specification of timed algorithms: basic properties and a decidable class

Abstract We consider one aspect of the problem of specification and verification of reactive real-time systems which involve operations and constraints concerning time. Time is continuous what is motivated by specifications of hybrid systems. Our goal is to try to find a framework that is based on applied first order logic that permits to represent the verification problem directly, completely and conservatively (as explained in Introduction), and that is apt to describe interesting decidable classes, maybe showing way to feasible algorithms. To achieve this goal we use a first order timed logic that is an extension of a decidable theory of reals with timed functions. This logic permits, on the one hand, to rewrite directly and completely requirements and, on the other hand, to describe executions of various timed algorithms—here we consider block Gurevich abstract state machines because of their theoretical clarity and sufficient expressive power. Then we describe one decidable class of the verification problem that is based on notions reflecting finiteness properties of systems of control. These notions may be of independent interest, as, in particular, they give a way to describe a limited usage of arithmetics preserving decidability that is not covered by existing model-theoretic approaches. As an example we consider the generalized railroad crossing problem that we analyze in its entirety.

Polyomino tilings, cellular automata and codicity

Abstract As usual, a 4-connex finite part of Z2 is called a polyomino. Recognizing whether a given polyomino can be tiled by translated copies of tiles taken from a given family of polyominoes is obviously decidable. On the contrary, deciding whether a given set of polyominoes is a code has been shown to be undecidable (Beauquier and Nivat, 1993). In this paper, we define various classes of codes and study the complexity of tiling recognition for these classes and their mutual relations. Specially, we study the class of polyominoe families, which we call neighbourhood codes, which generate tilings which are recognizable by cellular automata using only neighbourhood relations. We prove that there exist codes which are not neighbourhood codes, and we give an example of such a code.

A codicity undecidable problem in the plane

In this paper we give a new undecidability result about tiling problems. Given a finite set of polyomino types, the problem whether this set is a code, is undecidable. The same result holds for dominoes.

A Logic of Probability with Decidable Model-Checking

A predicate logic of probability, close to logics of probability of Halpern and al., is introduced. Our main result concerns the following model-checking problem: deciding whether a given formula holds on the structure defined by a given Finite Probabilistic Process. We show that this model-checking problem is decidable for a rather large subclass of formulas of a second-order monadic logic of probability. We discuss also the decidability of satisfiability and compare our logic of probability with the probabilistic temporal logic pCTL*.

Tiling figures of the plane with two bars

Abstract Given two “bars”, a horizontal one, and a vertical one (both of length at least two), we are interested in the following decision problem: is a finite figure drawn on a plane grid tilable with these bars. It turns out that if one of the bars has length at least three, the problem is NP- complete . If bars are dominoes, the problem is in P, and even linear (in the size of the figure) for certain classes of figures. Given a general pair of bars, we give two results: (1) a necessary condition to have a unique tiling for finite figures without holes, (2) a linear algorithm (in the size of the figure) deciding whether a unique tiling exists, and computing this one if it does exist. Finally, given a tiling of a figure (not necessarily finite), this tiling is the unique one for the figure if and only if there exists no subtiling covering a “canonical” rectangle .

The Railroad Crossing Problem: Towards Semantics of Timed Algorithms and Their Model Checking in High Level Languages

The goal of this paper is to analyse semantics of algorithms with explicit continuous time with further aim to find approaches to automatize model checking in high level, easily understandable languages. We give here a general notion of timed transition system and its formula representation that are sufficient to deal with some known examples of timed algorithms. We prove that the general semantics gives the same executions as direct, more intuitive interpretations of executions of algorithms. In a way, we try to give a general treatment of considerations of Yu.Gurevich and his co-authors concerning concrete Gurevich machines (called evolving algebras in [Gur95]), in particular, related to Railroad Crossing Problem [GH96]. Besides that we formalize specifications of this problem in a high level language which permits to rewrite directly natural language formulations, and to give a formal proof of correctness of the railroad crossing algorithm using rather a small amount of logical means, and this leads to hypotheses how automatize inference search.

Security Policies Enforcement Using Finite Edit Automata

Edit automata have been introduced by J. Ligatti et al. as a model for security enforcement mechanisms which work at run time. In a distributed interacting system, they play a role of monitor that runs in parallel with a target program and transforms its execution sequence into a sequence that obeys the security property. In this paper we characterize security properties which are enforceable by finite edit automata, i.e. edit automata with a finite set of states. We prove that these properties are a sub-class of ~-regular sets. Moreover given an ~-regular set P, one can decide in time O(n^2) whether P is enforceable by a finite edit automaton (where n is the number of states of the finite automaton recognizing P) and we give an algorithm to synthesize the controller.

Pumping Lemmas for Timed Automata

We remark that languages recognized by timed automata in the general case do not satisfy classical Pumping Lemma (PL) well known in the theory of finite automata. In this paper we prove two weaker versions of Pumping Lemma for timed words : a general one (DPL) where iterations preserve the duration of timed word, and another more restricted one, (LPL) when iterations preserve the length of timed word.