Searcharxiv⌕ Search

arXiv subjects

Petr Jancar

Publications and source records attributed to Petr Jancar.

17 recordsLinked to original sources
arXiv2026

Well-Formed Free-Choice Petri Nets Revisited

The theory of free-choice Petri nets is an established field, initiated in the 1970s by F. Commoner and M. Hack. We revisit well-formed free-choice nets (those admitting markings that are both live and bounded) and provide a new characterisation by introducing semi-T-components. This notion is dual to that of semi-S-components, which in turn correspond to the well-known minimal siphons. By highlighting the symmetry between these dual concepts, we derive the classical coverability theorems for T- and S-components, as well as the duality theorem---stating that a free-choice net is well-formed if and only if its reverse-dual is also well-formed---using highly symmetric arguments.

cs.FL↗
arXiv2024

A concise proof of Commoner's theorem

The textbook proofs of Commoner's theorem characterizing liveness in free-choice Petri nets are given in contexts of technical notions and claims that make the proofs look a bit long. The aim of this note is to give a concise self-contained proof.

cs.LO↗
arXiv2023

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

We answer an open complexity question by Hofman, Lasota, Mayr, Totzke (LMCS 2016) for simulation preorder on the class of succinct one-counter nets (i.e., one-counter automata with no zero tests where counter increments and decrements are integers written in binary); the problem was known to be PSPACE-hard and in EXPSPACE. We show that all relations between bisimulation equivalence and simulation preorder are EXPSPACE-hard for these nets; simulation preorder is thus EXPSPACE-complete. The result is proven by 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 an alternative proof for the upper bound by Hofman et al. In particular, we give a new simplified proof of the belt theorem that yields a simple graphic presentation of simulation preorder on (non-succinct) one-counter nets and leads to a polynomial-space algorithm (which is trivially extended to an exponential-space algorithm for succinct one-counter nets).

cs.LO↗
arXiv2021

Structural Liveness of Immediate Observation Petri Nets

We look in detail at the structural liveness problem (SLP) for subclasses of Petri nets, namely immediate observation nets (IO nets) and their generalized variant called branching immediate multi-observation nets (BIMO nets), that were recently introduced by Esparza, Raskin, and Weil-Kennedy. We show that SLP is PSPACE-hard for IO nets and in PSPACE for BIMO nets. In particular, we discuss the (small) bounds on the token numbers in net places that are decisive for a marking to be (non)live.

cs.LO↗
arXiv2021

Simplest Non-Regular Deterministic Context-Free Language

We introduce a new notion of C-simple problems for a class C of decision problems (i.e. languages), w.r.t. a particular reduction. A problem is C-simple if it can be reduced to each problem in C. This can be viewed as a conceptual counterpart to C-hard problems to which all problems in C reduce. Our concrete example is the class of non-regular deterministic context-free languages (DCFL'), with a truth-table reduction by Mealy machines (which proves to be a preorder). The main technical result is a proof that the DCFL' language $L=\{0^n1^n; n\geq 1\}$ is DCFL'-simple, which can thus be viewed as the simplest problem in the class DCFL'. This result has already provided an application, to the computational model of neural networks 1ANN at the first level of analog neuron hierarchy. This model was proven not to recognize $L$, by using a specialized technical argument that can hardly be generalized to other languages in DCFL'. By the result that $L$ is DCFL'-simple, w.r.t. the reduction that can be implemented by 1ANN, we immediately obtain that 1ANN cannot accept any language in DCFL'. It thus seems worthwhile to explore if looking for C-simple problems in other classes C under suitable reductions could provide effective tools for expanding the lower-bound results known for single problems to the whole classes of problems.

cs.FL↗
arXiv2020

Equivalence of pushdown automata via first-order grammars

A decidability proof for bisimulation equivalence of first-order grammars is given. It is an alternative proof for a result by Sénizergues (1998, 2005) that subsumes his affirmative solution of the famous decidability question for deterministic pushdown automata. The presented proof is conceptually simpler, and a particular novelty is that it is not given as two semidecision procedures but it provides an explicit algorithm that might be amenable to a complexity analysis.

cs.LO↗
arXiv2019

Deciding semantic finiteness of pushdown processes and first-order grammars w.r.t. bisimulation equivalence

The problem if a given configuration of a pushdown automaton (PDA) is bisimilar with some (unspecified) finite-state process is shown to be decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic (i.e. alternative-free) epsilon-steps are allowed, i.e. to the model for which Sénizergues showed an involved procedure deciding bisimilarity (1998, 2005). Such a procedure is here used as a black-box part of the algorithm. The result extends the decidability of the regularity problem for deterministic PDA that was shown by Stearns (1967), and later improved by Valiant (1975) regarding the complexity. The decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Göller, 2012).

cs.LO↗
arXiv2018

EXPSPACE-hardness of behavioural equivalences of succinct one-counter nets

We note that the remarkable EXPSPACE-hardness result in [Göller, Haase, Ouaknine, Worrell, ICALP 2010] ([GHOW10] for short) allows us to answer an open complexity question 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). This problem, as well as bisimulation equivalence, turn out to be EXPSPACE-complete. The technique of [GHOW10] was referred to by Hunter [RP 2015] for deriving EXPSPACE-hardness of reachability games on succinct one-counter nets. We first give a direct self-contained EXPSPACE-hardness proof for such reachability games (by adjusting a known PSPACE-hardness proof for emptiness of alternating finite automata with one-letter alphabet); then we reduce reachability games to (bi)simulation games by using a standard "defender-choice" technique.

cs.LO↗
arXiv2017

Branching Bisimilarity of Normed BPA Processes as a Rational Monoid

The paper presents an elaborated and simplified version of the structural result for branching bisimilarity on normed BPA (Basic Process Algebra) processes that was the crux of a conference paper by Czerwinski and Jancar (arxiv 7/2014 and LiCS 2015). That paper focused on the computational complexity, and a NEXPTIME-upper bound has been derived; the authors built on the ideas by Fu (ICALP 2013), and strengthened his decidability result. Later He and Huang announced the EXPTIME-completeness of this problem (arxiv 1/2015, and LiCS 2015), giving a technical proof for the EXPTIME membership. He and Huang indirectly acknowledge the decomposition ideas by Czerwinski and Jancar on which they also built, but it is difficult to separate their starting point from their new ideas. One aim here is to present the previous decomposition result of Czerwinski and Jancar in a technically new framework, noting that branching bisimulation equivalence on normed BPA processes corresponds to a rational monoid (in the sense of [Sakarovitch, 1987]); in particular it is shown that the mentioned equivalence can be decided by normal-form computing deterministic finite transducers. Another aim is to provide a complete description, including an informal overview, that should also make clear how Fu's ideas were used, and to give all proofs in a form that should be readable and easily verifiable.

cs.LO↗
arXiv2016

Deciding structural liveness of Petri nets

Place/transition Petri nets are a standard model for a class of distributed systems whose reachability spaces might be infinite. One of well-studied topics is the verification of safety and liveness properties in this model; despite the extensive research effort, some basic problems remain open, which is exemplified by the open complexity status of the reachability problem. The liveness problems are known to be closely related to the reachability problem, and many structural properties of nets that are related to liveness have been studied. Somewhat surprisingly, the decidability status of the problem if a net is structurally live, i.e. if there is an initial marking for which it is live, has remained open, as also a recent paper (Best and Esparza, 2016) emphasizes. Here we show that the structural liveness problem for Petri nets is decidable. A crucial ingredient of the proof is the result by Leroux (LiCS 2013) showing that we can compute a finite (Presburger) description of the reachability set for a marked Petri net if this set is semilinear.

cs.LO↗
arXiv2014

Bisimulation Equivalence of First-Order Grammars

A decidability proof for bisimulation equivalence of first-order grammars (finite sets of labelled rules for rewriting roots of first-order terms) is presented. The equivalence generalizes the DPDA (deterministic pushdown automata) equivalence, and the result corresponds to the result achieved by Senizergues (1998, 2005) in the framework of equational graphs, or of PDA with restricted epsilon-steps. The framework of classical first-order terms seems particularly useful for providing a proof that should be understandable for a wider audience. We also discuss an extension to branching bisimilarity, announced by Fu and Yin (2014).

cs.LO↗
arXiv2013

Bisimulation equivalence of first-order grammars is Ackermann-hard

Bisimulation equivalence (or bisimilarity) of first-order grammars is decidable, as follows from the decidability result by Senizergues (1998, 2005) that has been given in an equivalent framework of equational graphs with finite out-degree, or of pushdown automata (PDA) with only deterministic and popping epsilon-transitions. Benedikt, Goeller, Kiefer, and Murawski (2013) have shown that the bisimilarity problem for PDA (even) without epsilon-transitions is nonelementary. Here we show Ackermann-hardness for bisimilarity of first-order grammars. The grammars do not use explicit epsilon-transitions, but they correspond to the above mentioned PDA with (deterministic and popping) epsilon-transitions, and this feature is substantial in the presented lower-bound proof. The proof is based on a (polynomial) reduction from the reachability problem of reset (or lossy) counter machines, for which the Ackermann-hardness has been shown by Schnoebelen (2010); in fact, this reachability problem is known to be Ackermann-complete in the hierarchy of fast-growing complexity classes defined by Schmitz (2013).

cs.LO↗
arXiv2013

Bisimilarity on Basic Process Algebra is in 2-ExpTime (an explicit proof)

Burkart, Caucal, Steffen (1995) showed a procedure deciding bisimulation equivalence of processes in Basic Process Algebra (BPA), i.e. of sequential processes generated by context-free grammars. They improved the previous decidability result of Christensen, Hüttel, Stirling (1992), since their procedure has obviously an elementary time complexity and the authors claim that a close analysis would reveal a double exponential upper bound. Here a self-contained direct proof of the membership in 2-ExpTime is given. This is done via a Prover-Refuter game which shows that there is an alternating Turing machine deciding the problem in exponential space. The proof uses similar ingredients (size-measures, decompositions, bases) as the previous proofs, but one new simplifying factor is an explicit addition of infinite regular strings to the state space. An auxiliary claim also shows an explicit exponential upper bound on the equivalence level of nonbisimilar normed BPA processes. The importance of clarifying the 2-ExpTime upper bound for BPA bisimilarity has recently increased due to the shift of the known lower bound from PSpace (Srba, 2002) to ExpTime (Kiefer, 2012).

cs.LO↗
arXiv2012

Bisimilarity of Probabilistic Pushdown Automata

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). Our first contribution is a general construction that reduces checking bisimilarity of probabilistic transition systems to checking bisimilarity of non-deterministic transition systems. This construction directly yields decidability of bisimilarity for pPDA, as well as an elementary upper bound for the bisimilarity problem on the subclass of probabilistic basic process algebras, i.e., single-state pPDA. We further show that, with careful analysis, the general reduction can be used to prove an EXPTIME upper bound for bisimilarity of probabilistic visibly pushdown automata. Here we also provide a matching lower bound, establishing EXPTIME-completeness. Finally we prove that deciding bisimilarity of probabilistic one-counter automata, another subclass of pPDA, is PSPACE-complete. Here we use a more specialised argument to obtain optimal complexity bounds.

cs.FL↗
arXiv2011

A Short Decidability Proof for DPDA Language Equivalence via First-Order Grammars

The main aim of the paper is to give a short self-contained proof of the decidability of language equivalence for deterministic pushdown automata, which is the famous problem solved by G. Senizergues, for which C. Stirling has derived a primitive recursive complexity upper bound. The proof here is given in the framework of first-order grammars, which seems to be particularly apt for the aim. An appendix presents a modification of Stirling's approach, yielding a complexity bound of the form tetr(2,g(n)) where tetr is the (nonelementary) operator of iterated exponentiation (tetration) and g is an elementary function of the input size.

cs.FL↗
arXiv2010

Reachability Games on Extended Vector Addition Systems with States

We consider two-player turn-based games with zero-reachability and zero-safety objectives generated by extended vector addition systems with states. Although the problem of deciding the winner in such games is undecidable in general, we identify several decidable and even tractable subcases of this problem obtained by restricting the number of counters and/or the sets of target configurations.

cs.GT↗