A deterministic finite automaton is synchronizing if there exists a word that sends all states of the automaton to the same state. v{C}erny conjectured in 1964 that a synchronizing automaton with $n$ states has a synchronizing word of length at most $(n-1)^2$. We introduce the notion of aperiodically $1-$contracting automata and prove that in these automata all subsets of the state set are reachable, so that in particular they are synchronizing. Furthermore, we give a sufficient condition under which the v{C}erny conjecture holds for aperiodically $1-$contracting automata. As a special case, we prove some results for circular automata.
Priced timed games are optimal-cost reachability games played between two players---the controller and the environment---by moving a token along the edges of infinite graphs of configurations of priced timed automata. The goal of the controller is to reach a given set of target locations as cheaply as possible, while the goal of the environment is the opposite. Priced timed games are known to be undecidable for timed automata with $3$ or more clocks, while they are known to be decidable for automata with $1$ clock. In an attempt to recover decidability for priced timed games Bouyer, Markey, and Sankur studied robust priced timed games where the environment has the power to slightly perturb delays proposed by the controller. Unfortunately, however, they showed that the natural problem of deciding the existence of optimal limit-strategy---optimal strategy of the controller where the perturbations tend to vanish in the limit---is undecidable with $10$ or more clocks. In this paper we revisit this problem and improve our understanding of the decidability of these games. We show that the limit-strategy problem is already undecidable for a subclass of robust priced timed games with $5$ or more clocks. On a positive side, we show the decidability of the existence of almost optimal strategies for the same subclass of one-clock robust priced timed games by adapting a classical construction by Bouyer at al. for one-clock priced timed games.
We present a new aperiodic tileset containing 11 Wang tiles on 4 colors, and we show that this tileset is minimal, in the sense that no Wang set with either fewer than 11 tiles or fewer than 4 colors is aperiodic. This gives a definitive answer to the problem raised by Wang in 1961.
A distributed protocol is typically modeled as a set of communicating processes, where each process is described as an extended state machine along with fairness assumptions, and its correctness is specified using safety and liveness requirements. Designing correct distributed protocols is a challenging task. Aimed at simplifying this task, we allow the designer to leave some of the guards and updates to state variables in the description of extended state machines as unknown functions. The protocol completion problem then is to find interpretations for these unknown functions while guaranteeing correctness. In many distributed protocols, process behaviors are naturally symmetric, and thus, synthesized expressions are further required to obey symmetry constraints. Our counterexample-guided synthesis algorithm consists of repeatedly invoking two phases. In the first phase, candidates for unknown expressions are generated using the SMT solver Z3. This phase requires carefully orchestrating constraints to enforce the desired symmetry in read/write accesses. In the second phase, the resulting completed protocol is checked for correctness using a custom-built model checker that handles fairness assumptions, safety and liveness requirements, and exploits symmetry. When model checking fails, our tool examines a set of counterexamples to safety/liveness properties to generate constraints on unknown functions that must be satisfied by subsequent completions. For evaluation, we show that our prototype is able to automatically discover interesting missing details in distributed protocols for mutual exclusion, self stabilization, and cache coherence.
The third author noticed in his 1992 PhD Thesis [Sim92] that every regular tree language of infinite trees is in a class $Game (D_n({bfSigma}^0_2))$ for some natural number $ngeq 1$, where $Game$ is the game quantifier. We first give a detailed exposition of this result. Next, using an embedding of the Wadge hierarchy of non self-dual Borel subsets of the Cantor space $2^omega$ into the class ${bfDelta}^1_2$, and the notions of Wadge degree and Veblen function, we argue that this upper bound on the topological complexity of regular tree languages is much better than the usual ${bfDelta}^1_2$.
This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and motivate the fundamental concepts. We do this for the most part without proofs, and often without precise definitions, leaving these to the formal development of the theory that begins in Section 2. Our presentation of the theory draws heavily on the work of Gehrke, Grigorieff and Pin (2008) on the equational theory of lattices of regular languages. In the subsequent sections we consider in more detail aspects of varieties that were only briefly evoked in the introduction: Decidability, operations on languages, and characterizations in formal logic.
It is not known whether Thompsons group F is automatic. With the recent extensions of the notion of an automatic group to graph automatic by Kharlampovich, Khoussainov and Miasnikov and then to C-graph automatic by the authors, a compelling question is whether F is graph automatic or C-graph automatic for an appropriate language class C. The extended definitions allow the use of a symbol alphabet for the normal form language, replacing the dependence on generating set. In this paper we construct a 1-counter graph automatic structure for F based on the standard infinite normal form for group elements.
Syntax-directed translation tools require the specification of a language by means of a formal grammar. This grammar must conform to the specific requirements of the parser generator to be used. This grammar is then annotated with semantic actions for the resulting system to perform its desired function. In this paper, we introduce ModelCC, a model-based parser generator that decouples language specification from language processing, avoiding some of the problems caused by grammar-driven parser generators. ModelCC receives a conceptual model as input, along with constraints that annotate it. It is then able to create a parser for the desired textual syntax and the generated parser fully automates the instantiation of the language conceptual model. ModelCC also includes a reference resolution mechanism so that ModelCC is able to instantiate abstract syntax graphs, rather than mere abstract syntax trees.
We consider the cyclic closure of a language, and its generalisation to the operators $C^k$ introduced by Brandstadt. We prove that the cyclic closure of an indexed language is indexed, and that if $L$ is a context-free language then $C^k(L)$ is indexed.
The concept of promise problems was introduced and started to be systematically explored by Even, Selman, Yacobi, Goldreich, and other scholars. It has been argued that promise problems should be seen as partial decision problems and as such that they are more fundamental than decision problems and formal languages that used to be considered as the basic ones for complexity theory. The main purpose of this paper is to explore the promise problems accepted by classical, quantum and also semi-quantum finite automata. More specifically, we first introduce two acceptance modes of promise problems, recognizability and solvability, and explore their basic properties. Afterwards, we show several results concerning descriptional complexity on promise problems. In particular, we prove: (1) there is a promise problem that can be recognized exactly by measure-once one-way quantum finite automata (MO-1QFA), but no deterministic finite automata (DFA) can recognize it; (2) there is a promise problem that can be solved with error probability $epsilonleq 1/3$ by one-way finite automaton with quantum and classical states (1QCFA), but no one-way probability finite automaton (PFA) can solve it with error probability $epsilonleq 1/3$; and especially, (3) there are promise problems $A(p)$ with prime $p$ that can be solved {em with any error probability} by MO-1QFA with only two quantum basis states, but they can not be solved exactly by any MO-1QFA with two quantum basis states; in contrast, the minimal PFA solving $A(p)$ with any error probability (usually smaller than $1/2$) has $p$ states. Finally, we mention a number of problems related to promise for further study.
