Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

Vectorial Languages and Linear Temporal Logic

International audienceDetermining for a given deterministic complete automaton the sequence of visited states while reading a given word is the core of important problems with automata-based solutions, such as approximate string matching. The main difficulty is to do this computation efficiently, especially when dealing with very large texts. Considering words as vectors and working on them using vectorial (parallel) operations allows to solve the problem faster than in linear time using sequential computations. In this paper, we show first that the set of vectorial operations needed by an algorithm representing a given automaton depends only on the language accepted by the automaton. We give precise characterizations of vectorial algorithms for star-free, solvable and regular languages in terms of the vectorial operations allowed. We also consider classes of languages associated with restricted sets of vectorial operations and relate them with languages defined by fragments of linear temporal logic. Finally, we consider the converse problem of constructing an automaton from a given vectorial algorithm. As a byproduct, we show that the satisfiability problem for some extensions of linear-time temporal logic characterizing solvable and regular languages is PSPACE-complete

Languages of Dot-depth One over Infinite Words

Over finite words, languages of dot-depth one are expressively complete for alternation-free first-order logic. This fragment is also known as the Boolean closure of existential first-order logic. Here, the atomic formulas comprise order, successor, minimum, and maximum predicates. Knast (1983) has shown that it is decidable whether a language has dot-depth one. We extend Knast's result to infinite words. In particular, we describe the class of languages definable in alternation-free first-order logic over infinite words, and we give an effective characterization of this fragment. This characterization has two components. The first component is identical to Knast's algebraic property for finite words and the second component is a topological property, namely being a Boolean combination of Cantor sets. As an intermediate step we consider finite and infinite words simultaneously. We then obtain the results for infinite words as well as for finite words as special cases. In particular, we give a new proof of Knast's Theorem on languages of dot-depth one over finite words.Comment: Presented at LICS 201

Runtime Enforcement for Component-Based Systems

Runtime enforcement is an increasingly popular and effective dynamic validation technique aiming to ensure the correct runtime behavior (w.r.t. a formal specification) of systems using a so-called enforcement monitor. In this paper we introduce runtime enforcement of specifications on component-based systems (CBS) modeled in the BIP (Behavior, Interaction and Priority) framework. BIP is a powerful and expressive component-based framework for formal construction of heterogeneous systems. However, because of BIP expressiveness, it remains difficult to enforce at design-time complex behavioral properties. First we propose a theoretical runtime enforcement framework for CBS where we delineate a hierarchy of sets of enforceable properties (i.e., properties that can be enforced) according to the number of observational steps a system is allowed to deviate from the property (i.e., the notion of k-step enforceability). To ensure the observational equivalence between the correct executions of the initial system and the monitored system, we show that i) only stutter-invariant properties should be enforced on CBS with our monitors, ii) safety properties are 1-step enforceable. Given an abstract enforcement monitor (as a finite-state machine) for some 1-step enforceable specification, we formally instrument (at relevant locations) a given BIP system to integrate the monitor. At runtime, the monitor observes and automatically avoids any error in the behavior of the system w.r.t. the specification. Our approach is fully implemented in an available tool that we used to i) avoid deadlock occurrences on a dining philosophers benchmark, and ii) ensure the correct placement of robots on a map.Comment: arXiv admin note: text overlap with arXiv:1109.5505 by other author

Tree Languages Defined in First-Order Logic with One Quantifier Alternation

We study tree languages that can be defined in \Delta_2 . These are tree languages definable by a first-order formula whose quantifier prefix is forall exists, and simultaneously by a first-order formula whose quantifier prefix is . For the quantifier free part we consider two signatures, either the descendant relation alone or together with the lexicographical order relation on nodes. We provide an effective characterization of tree and forest languages definable in \Delta_2 . This characterization is in terms of algebraic equations. Over words, the class of word languages definable in \Delta_2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil

Linear temporal logic for regular cost functions

Regular cost functions have been introduced recently as an extension to the notion of regular languages with counting capabilities, which retains strong closure, equivalence, and decidability properties. The specificity of cost functions is that exact values are not considered, but only estimated. In this paper, we define an extension of Linear Temporal Logic (LTL) over finite words to describe cost functions. We give an explicit translation from this new logic to automata. We then algebraically characterize the expressive power of this logic, using a new syntactic congruence for cost functions introduced in this paper

Fragments of first-order logic over infinite words

We give topological and algebraic characterizations as well as language theoretic descriptions of the following subclasses of first-order logic FO[<] for omega-languages: Sigma_2, FO^2, the intersection of FO^2 and Sigma_2, and Delta_2 (and by duality Pi_2 and the intersection of FO^2 and Pi_2). These descriptions extend the respective results for finite words. In particular, we relate the above fragments to language classes of certain (unambiguous) polynomials. An immediate consequence is the decidability of the membership problem of these classes, but this was shown before by Wilke and Bojanczyk and is therefore not our main focus. The paper is about the interplay of algebraic, topological, and language theoretic properties.Comment: Conference version presented at 26th International Symposium on Theoretical Aspects of Computer Science, STACS 200