Beyond Language Equivalence on Visibly Pushdown Automata

We study (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods for proving complexity upper and lower bounds for a number of studied preorders and equivalences like simulation, completed simulation, ready simulation, 2-nested simulation preorders/equivalences and bisimulation equivalence. Our main results are that all the mentioned equivalences and preorders are EXPTIME-complete on visibly pushdown automata, PSPACE-complete on visibly one-counter automata and P-complete on visibly BPA. Our PSPACE lower bound for visibly one-counter automata improves also the previously known DP-hardness results for ordinary one-counter automata and one-counter nets. Finally, we study regularity checking problems for visibly pushdown automata and show that they can be decided in polynomial time.Comment: Final version of paper, accepted by LMC

Об эффективном моделировании неограниченного ресурса при помощи односчетчиковых контуров

A class of infinite-state automata with a simple periodic behaviour and a convenient graphical representation is studied. A positive one-counter circuit is defined as a strongly connected one-counter net (one-counter nondeterministic finite automata without zero-testing) with at least one positive cycle. It is shown that in a positive circuit an infinite part of a reachability set is an arithmetic progression; numerical properties of this progression are estimated. A specific graphical representation of circuits is presented. General one-counter nets are equivalent to Petri Nets with at most one unbounded place and to pushdown automata with a single-symbol stack alphabet. It is shown that an arbitrary one-counter net can be represented by a finite tree of circuits. A one-counter net is called sound, if a counter is used only for “infinite-state” (periodic) behaviour. It is shown that for an arbitrary one-counter net an equivalent sound net can be effectively constructed from the corresponding tree of circuits.Вводится и исследуется специфический формализм счетчиковых автоматов с одним неограниченным счетчиком без проверки на ноль, обладающий достаточно простым почти периодическим поведением и удобным графическим представлением.Положительным односчетчиковым контуром называется сильно связная односчетчиковая сеть (недетерминированный конечный автомат с одним счетчиком без проверки на ноль), содержащая по крайней мере один цикл, увеличивающий значение счетчика. Показано, что в положительном контуре бесконечная часть множества достижимости описывается арифметической прогрессией; получены оценки параметров этой прогрессии через структурные свойства диаграммы переходов. Представлен компактный и наглядный способ графического представления контура.В общем случае односчетчиковые сети обладают такой же выразительной мощностью, как сети Петри с одной неограниченной позицией и магазинные автоматы с односимвольным стековым алфавитом. Показано, что произвольная односчетчиковая сеть может быть представлена в виде конечного дерева односчетчиковых контуров.Вводится понятие правильно сформированной односчетчиковой сети. Односчетчиковая сеть называется правильно сформированной, если счетчик используется только при порождении бесконечных периодических подмножеств множества достижимых состояний. Показано, что для любой односчетчиковой сети существует эквивалентная (в смысле совпадения множеств достижимости) правильно сформированная сеть, которая может быть эффективно построена из соответствующего дерева контуров

Verification for Timed Automata extended with Unbounded Discrete Data Structures

We study decidability of verification problems for timed automata extended with unbounded discrete data structures. More detailed, we extend timed automata with a pushdown stack. In this way, we obtain a strong model that may for instance be used to model real-time programs with procedure calls. It is long known that the reachability problem for this model is decidable. The goal of this paper is to identify subclasses of timed pushdown automata for which the language inclusion problem and related problems are decidable

Verification of qualitative Z\mathbb{Z} constraints

International audienceWe introduce an LTL-like logic with atomic formulae built over a constraint language interpreting variables in $\mathbb{Z}$. The constraint language includes periodicity constraints, comparison constraints of the form $x = y$ and $x < y$, it is closed under Boolean operations and it admits a restricted form of existential quantification. This is the largest set of qualitative constraints over $\mathbb{Z}$ known so far, shown to admit a decidable LTL extension. Such constraints are those used for instance in calendar formalisms or in abstractions of counter automata by using congruences modulo some power of two. Indeed, various programming languages perform arithmetic operators modulo some integer. We show that the satisfiability and model-checking problems (with respect to an appropriate class of constraint automata) for this logic are decidable in polynomial space improving significantly known results about its strict fragments. As a by-product, LTL model-checking over integral relational automata is proved complete for polynomial space which contrasts with the known undecidability of its CTL counterpart

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

DP lower bounds for equivalence-checking and model-checking of one-counter automata

We present a general method for proving DP-hardness of problems related to formal verification of one-counter automata. For this we show a reduction of the S-U problem to the truth problem for a fragment of (Presburger) arithmetic. The fragment contains only special formulas with one free variable, and is particularly apt for transforming to simulation-like equivalences on one-counter automata. In this way we show that the membership problem for any relation subsuming bisimilarity and subsumed by simulation preorder is DP-hard (even) for one-counter nets (where the counter cannot be tested for zero). We also show DP-hardness for deciding simulation between one-counter automata and finite-state systems (in both directions), and for the model-checking problem with one-counter nets and the branching-time temporal logic EF

Countdown games, and simulation on (succinct) one-counter nets

We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS 2016) [HLMT16] for simulation preorder of succinct one-counter nets (i.e., one-counter automata with no zero tests where counter increments and decrements are integers written in binary), by showing that all relations between bisimulation equivalence and simulation preorder are EXPSPACE-hard for these nets. We describe a reduction from reachability games whose EXPSPACE-completeness in the case of succinct one-counter nets was shown by Hunter [RP 2015], by using other results. We also provide a direct self-contained EXPSPACE-completeness proof for a special case of such reachability games, namely for a modification of countdown games that were shown EXPTIME-complete by Jurdzinski, Sproston, Laroussinie [LMCS 2008]; in our modification the initial counter value is not given but is freely chosen by the first player. We also present a new simplified proof of the belt theorem that gives a simple graphic presentation of simulation preorder on one-counter nets and leads to a polynomial-space algorithm; it is an alternative to the proof from [HLMT16].Comment: A part of this paper elaborates arxiv-paper 1801.01073 and the related paper presented at Reachability Problems 201