This work studies which storage mechanisms in automata permit decidability of the emptiness problem. The question is formalized using valence automata, an abstract model of automata in which the storage mechanism is given by a monoid. For each of a variety of storage mechanisms, one can choose a (typically infinite) monoid $M$ such that valence automata over $M$ are equivalent to (one-way) automata with this type of storage. In fact, many important storage mechanisms can be realized by monoids defined by finite graphs, called graph monoids. Examples include pushdown stacks, partially blind counters (which behave like Petri net places), blind counters (which may attain negative values), and combinations thereof.
Hence, we study for which graph monoids the emptiness problem for valence automata is decidable. A particular model realized by graph monoids is that of Petri nets with a pushdown stack. For these, decidability is a long-standing open question and we do not answer it here.
However, if one excludes subgraphs corresponding to this model, a characterization can be achieved. Moreover, we provide a description of those storage mechanisms for which decidability remains open. This leads to a model that naturally generalizes both pushdown Petri nets and the priority multicounter machines introduced by Reinhardt.
The cases that are proven decidable constitute a natural and apparently new extension of Petri nets with decidable reachability. It is finally shown that this model can be combined with another such extension by Atig and Ganty: We present a further decidability result that subsumes both of these Petri net extensions.
Subzero automata is a class of tree automata whose acceptance condition can express probabilistic constraints. Our main result is that the problem of determining if a subzero automaton accepts some regular tree is decidable.
This paper proves the decidability of the emptiness problem for two models which recognize graphs: graph-walking automata, and tilings of graphs by star subgraphs (star automata). Furthermore, it is proved that the non-emptiness problem for graph-walking automata (that is, whether a given automaton accepts at least one graph) is NEXP-complete. For star automata, which generalize nondeterministic tree automata to the case of graphs, it is proved that their non-emptiness problem is NP-complete.
Zero automata are a probabilistic extension of parity automata on infinite trees. The satisfiability of a certain probabilistic variant of mso, called tmso + zero, reduces to the emptiness problem for zero automata. We introduce a variant of zero automata called nonzero automata. We prove that for every zero automaton there is an equivalent nonzero automaton of quadratic size and the emptiness problem of nonzero automata is decidable and both in NP and in coNP. These results imply that tmso + zero has decidable satisfiability.
The model of tree automata with equality and disequality constraints was introduced in 2007 by Filiot, Talbot and Tison. In this paper we show that if there is at least one disequality constraint, the emptiness problem is NP-hard.
Model checkers for timed automata are widely used to verify safety-critical, real-time systems. State-of-the-art tools achieve scalability by intricate abstractions. We aim at further increasing the trust in their verification results, in particular for checking liveness properties. To this end, we develop an approach for extracting certificates for the emptiness of timed Büchi automata from model checking runs. These certificates can be double-checked by a certifier that we formally verify in Isabelle/HOL. We study liveness certificates in an abstract setting and show that our approach is sou…
We study alternating automata with qualitative semantics over infinite binary trees: alternation means that two opposing players construct a decoration of the input tree called a run, and the qualitative semantics says that a run of the automaton is accepting if almost all branches of the run are accepting. In this paper we prove a positive and a negative result for the emptiness problem of alternating automata with qualitative semantics. The positive result is the decidability of the emptiness problem for the case of Büchi acceptance condition. An interesting aspect of our approach is that we…
The languages of infinite timed words accepted by timed automata are traditionally defined using Buchi-like conditions. These acceptance conditions focus on the set of locations visited infinitely often along a run, but completely ignore quantitative timing aspects. In this paper we propose a natural quantitative semantics for timed automata based on the so-called frequency, which measures the proportion of time spent in the accepting locations. We study various properties of timed languages accepted with positive frequency, and in particular the emptiness and universality problems.
A saturation algorithm for collapsible pushdown systems was published in ICALP 2012. This work introduced a class of stack automata used to recognised regular sets of collapsible pushdown configurations. It was shown that these automata form an effective boolean algebra, have a linear time membership problem, and are equivalent to an alternative automata representation appearing in LICS 2010. It was also claimed that the emptiness problem for stack automata is PSPACE-complete. Unfortunately, this claim is not true. We show that the problem is in fact NEXPTIME-complete when the stacks being acc…
We investigate the decidability of the emptiness problem for three classes of distributed automata. These devices operate on finite directed graphs, acting as networks of identical finite-state machines that communicate in an infinite sequence of synchronous rounds. The problem is shown to be decidable in LogSpace for a class of forgetful automata, where the nodes see the messages received from their neighbors but cannot remember their own state. When restricted to the appropriate families of graphs, these forgetful automata are equivalent to classical finite word automata, but strictly more e…
Parametric timed automata extend the standard timed automata with the possibility to use parameters in the clock guards. In general, if the parameters are real-valued, the problem of language emptiness of such automata is undecidable even for various restricted subclasses. We thus focus on the case where parameters are assumed to be integer-valued, while the time still remains continuous. On the one hand, we show that the problem remains undecidable for parametric timed automata with three clocks and one parameter. On the other hand, for the case with arbitrary many clocks where only one of th…
The Emptiness Problem for Quantum Finite Automata with Classical States
National Taipei University of Technology（国立台北科技大学）；School of Computer Science and Informatics, University of Liverpool（利物浦大学计算机科学与信息学院）；National Institute of Informatics and SOKENDAI（日本国立情报学研究所与综合研究大学院大学）
In this paper we study a subclass of pebble automata (PA) for data languages for which the emptiness problem is decidable. Namely, we introduce the so-called top view weak PA. Roughly speaking, top view weak PA are weak PA where the equality test is performed only between the data values seen by the two most recently placed pebbles. The emptiness problem for this model is decidable. We also show that it is robust: alternating, nondeterministic and deterministic top view weak PA have the same recognition power. Moreover, this model is strong enough to accept all data languages expressible in Li…
Automata over infinite alphabets have recently come to be studied extensively as potentially useful tools for solving problems in verification and database theory. One popular model of automata studied is the Class Memory Automata (CMA), for which the emptiness problem is equivalent to Petri Net Reachability. We identify a restriction - which we call weakness - of CMA, and show that their emptiness problem is equivalent to Petri Net Coverability. Further, we show that in the deterministic case they are closed under all Boolean operations. We clarify the connections between weak CMA and existin…
The emptiness and containment problems for probabilistic automata are natural quantitative generalisations of the classical language emptiness and inclusion problems for Boolean automata. It is well known that both problems are undecidable. In this paper we provide a more refined view of these problems in terms of the degree of ambiguity of probabilistic automata. We show that a gap version of the emptiness problem (that is known be undecidable in general) becomes decidable for automata of polynomial ambiguity. We complement this positive result by showing that the emptiness problem remains un…
The aim of this work is to thoroughly investigate Buchi automata augmented with spatial constraints. The input trees of such an automaton are infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest. The constraints, from an RCC8-like spatial Relation Algebra (RA) x, are used to impose spatial constraints on objects of the spatial scene, eventually at different nodes of the input trees. We show that a Buchi alternating automaton augm…
In [1], we introduced the weakly synchronizing languages for probabilistic automata. In this report, we show that the emptiness problem of weakly synchronizing languages for probabilistic automata is undecidable. This implies that the decidability result of [1-3] for the emptiness problem of weakly synchronizing language is incorrect.
The problem DFA-Intersection-Nonemptiness asks if a given number of deterministic automata accept a common word. In general, this problem is PSPACE-complete. Here, we investigate this problem for the subclasses of commutative automata and automata recognizing sparse languages. We show that in both cases DFA-Intersection-Nonemptiness is complete for NP and for the parameterized class $W[1]$, where the number of input automata is the parameter, when the alphabet is fixed. Additionally, we establish the same result for Tables Non-Empty Join, a problem that asks if the join of several tables (poss…
It is well known that the emptiness problem for binary probabilistic automata and so for quantum automata is undecidable. We present the current status of the emptiness problems for unary probabilistic and quantum automata with connections with Skolem's and positivity problems. We also introduce the concept of linear recurrence automata in order to show the connection naturally. Then, we also give possible generalizations of linear recurrence relations and automata on vectors.
In this paper, we study the extension of 1-clock Alternating Timed Automata (1-ATA) with the ability to read in both forward and backward direction, the 2-Way 1-clock Alternating Timed Automata (2-Way 1-ATA). We show that subclass of 2-Way 1-ATA with reset free loops (2-Way 1-ATA-rfl) is expressively equivalent to MSO[<] extended with Guarded Metric Quantifiers (GQMSO). Emptiness Checking problem for 2-Way 1-ATA-rfl (and hence GQMSO) is undecidable, in general. We propose a "non-punctuality" like restriction, called non-adjacency, for 2-Way 1-ATA-rfl, and also for GQMSO, for which the emptines…
We introduce a model of one-way language acceptors (a variant of a checking stack automaton) and show the following decidability properties: (1) The deterministic version has a decidable membership problem but has an undecidable emptiness problem. (2) The nondeterministic version has an undecidable membership problem and emptiness problem. There are many models of accepting devices for which there is no difference with these problems between deterministic and nondeterministic versions, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first on…
We introduce a new class of automata on infinite trees called \emph{alternating nonzero automata}, which extends the class of non-deterministic nonzero automata. We reduce the emptiness problem for alternating nonzero automata to the same problem for non-deterministic ones, which implies decidability. We obtain as a corollary algorithms for the satisfiability of a probabilistic temporal logic extending both CTL* and the qualitative fragment of pCTL*.
In this paper we introduce a weighted LTL over product $ω$-valuation monoids that satisfy specific properties. We also introduce weighted generalized Büchi automata with $\varepsilon$-transitions, as well as weighted Büchi automata with $\varepsilon$-transitions over product $ω$-valuation monoids and prove that these two models are expressively equivalent and also equivalent to weighted Büchi automata already introduced in the literature. We prove that every formula of a syntactic fragment of our logic can be effectively translated to a weighted generalized Büchi automaton with $\varepsilon$-t…
The Int_reg-problem of a combinatorial problem P asks, given a nondeterministic automaton M as input, whether the language L(M) accepted by M contains any positive instance of the problem P. We consider the Int_reg-problem for a number of different graph problems and give general criteria that give decision procedures for these Int_reg-problems. To achieve this goal, we consider a natural graph encoding so that the language of all graph encodings is regular. Then, we draw the connection between classical pumping- and interchange-arguments from the field of formal language theory with the graph…
Register pushdown automata (RPDA) is an extension of classical pushdown automata to handle data values in a restricted way. RPDA attracts attention as a model of a query language for structured documents with data values. The membership and emptiness problems for RPDA are known to be EXPTIME-complete. This paper shows the membership problem becomes PSPACE-complete and NP-complete for nondecreasing and growing RPDA, respectively, while the emptiness problem remains EXPTIME-complete for these subclasses.
During the last decades, classical models in language theory have been extended by control mechanisms defined by monoids. We study which monoids cause the extensions of context-free grammars, finite automata, or finite state transducers to exceed the capacity of the original model. Furthermore, we investigate when, in the extended automata model, the nondeterministic variant differs from the deterministic one in capacity. We show that all these conditions are in fact equivalent and present an algebraic characterization. In particular, the open question of whether every language generated by a …
In the Intersection Non-Emptiness problem, we are given a list of finite automata $A_1,A_2,\dots,A_m$ over a common alphabet $Σ$ as input, and the goal is to determine whether some string $w\in Σ^*$ lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-Emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the conte…
Motivated by the success of bounded model checking framework for finite state machines, Ouaknine and Worrell proposed a time-bounded theory of real-time verification by claiming that restriction to bounded-time recovers decidability for several key decision problem related to real-time verification. In support of this theory, the list of undecidable problems recently shown decidable under time-bounded restriction is rather impressive: language inclusion for timed automata, emptiness problem for alternating timed automata, and emptiness problem for rectangular hybrid automata. The objective of …
Parikh automata extend automata with counters whose values can only be tested at the end of the computation, with respect to membership into a semi-linear set. Parikh automata have found several applications, for instance in transducer theory, as they enjoy decidable emptiness problem. In this paper, we study two-way Parikh automata. We show that emptiness becomes undecidable in the non-deterministic case. However, it is PSpace-C when the number of visits to any input position is bounded and the semi-linear set is given as an existential Presburger formula. We also give tight complexity bounds…
Valence automata are a generalization of various models of automata with storage. Here, each edge carries, in addition to an input word, an element of a monoid. A computation is considered valid if multiplying the monoid elements on the visited edges yields the identity element. By choosing suitable monoids, a variety of automata models can be obtained as special valence automata. This work is concerned with the accepting power of valence automata. Specifically, we ask for which monoids valence automata can accept only context-free languages or only languages with semilinear Parikh image, resp…
Hamburg University of Technology（汉堡工业大学）；Department of Computer Engineering（计算机工程系）
Stochastic automata over monoids as input sets are studied. The well-definedness of these automata requires an extension postulate that replaces the inherent universal property of free monoids. As a generalization of Turakainen's result, it will be shown that the generalized automata over monoids have the same acceptance power as their stochastic counterparts. The key to homomorphisms is a commuting property between the monoid homomorphism of input states and the monoid homomorphism of transition matrices. Closure properties of the languages accepted by stochastic automata over monoids are inv…
The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature. Lifting this result to a weighted setting has been widely studied. Moreover, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalisin…
We consider the value 1 problem for probabilistic automata over finite words: it asks whether a given probabilistic automaton accepts words with probability arbitrarily close to 1. This problem is known to be undecidable. However, different algorithms have been proposed to partially solve it; it has been recently shown that the Markov Monoid algorithm, based on algebra, is the most correct algorithm so far. The first contribution of this paper is to give a characterisation of the Markov Monoid algorithm. The second contribution is to develop a profinite theory for probabilistic automata, calle…
The focus of this paper is the analysis of real-time systems with recursion, through the development of good theoretical techniques which are implementable. Time is modeled using clock variables, and recursion using stacks. Our technique consists of modeling the behaviours of the timed system as graphs, and interpreting these graphs on tree terms by showing a bound on their tree-width. We then build a tree automaton that accepts exactly those tree terms that describe realizable runs of the timed system. The emptiness of the timed system thus boils down to emptiness of a finite tree automaton t…
Automata for unordered unranked trees are relevant for defining schemas and queries for data trees in Json or Xml format. While the existing notions are well-investigated concerning expressiveness, they all lack a proper notion of determinism, which makes it difficult to distinguish subclasses of automata for which problems such as inclusion, equivalence, and minimization can be solved efficiently. In this paper, we propose and investigate different notions of "horizontal determinism", starting from automata for unranked trees in which the horizontal evaluation is performed by finite state aut…
We show that some results from the theory of group automata and monoid automata still hold for more general classes of monoids and models. Extending previous work for finite automata over commutative groups, we demonstrate a context-free language that can not be recognized by any rational monoid automaton over a finitely generated permutable monoid. We show that the class of languages recognized by rational monoid automata over finitely generated completely simple or completely 0-simple permutable monoids is a semi-linear full trio. Furthermore, we investigate valence pushdown automata, and pr…
We study \emph{multiplicity equivalence} testing of automata over partially commutative monoids (pc monoids) and show efficient algorithms in special cases, exploiting the structure of the underlying non-commutation graph of the monoid. Specifically, if the clique cover number of the non-commutation graph (the minimum number of cliques covering the graph) of the pc monoid is a constant, we obtain a deterministic quasi-polynomial time algorithm. As a consequence, we also obtain the first deterministic quasi-polynomial time algorithms for multiplicity equivalence testing of $k$-tape automata and…
Inspired by distributed algorithms, we introduce a new class of finite graph automata that recognize precisely the graph languages definable in monadic second-order logic. For the cases of words and trees, it has been long known that the regular languages are precisely those definable in monadic second-order logic. In this regard, the automata proposed in the present work can be seen, to some extent, as a generalization of finite automata to graphs. Furthermore, we show that, unlike for finite automata on words and trees, the deterministic, nondeterministic and alternating variants of our auto…
In 2007 Kambites presented an algebraic interpretation of Chomsky-Schutzenberger theorem for context-free languages. We give an interpretation of the corresponding theorem for the class of displacement context-free languages which are equivalent to well-nested multiple context-free languages. We also obtain a characterization of k-displacement context-free languages in terms of monoid automata and show how such automata can be simulated on two stacks. We introduce the simultaneous two-stack automata and compare different variants of its definition. All the definitions considered are shown to …
We introduce first order alternating automata, a generalization of boolean alternating automata, in which transition rules are described by multisorted first order formulae, with states and internal variables given by uninterpreted predicate terms. The model is closed under union, intersection and complement, and its emptiness problem is undecidable, even for the simplest data theory of equality. To cope with this limitation, we develop an abstraction refinement semi-algorithm based on lazy annotation of the symbolic execution paths with interpolants, obtained by applying (i) quantifier elimin…
We define a class of languages of infinite words over infinite alphabets, and the corresponding automata. The automata used for recognition are a generalisation of deterministic Muller automata to the setting of nominal sets. Remarkably, the obtained languages are determined by their ultimately periodic fragments, as in the classical case. Closure under complement, union and intersection, and decidability of emptiness and equivalence are preserved by the generalisation. This is shown by using finite representations of the (otherwise infinite-state) defined class of automata.
Probabilistic automata are an extension of nondeterministic finite automata in which transitions are annotated with probabilities. Despite its simplicity, this model is very expressive and many of the associated algorithmic questions are undecidable. In this work we focus on the emptiness problem (and its variant the value problem), which asks whether a given probabilistic automaton accepts some word with probability greater than a given threshold. We consider a natural and well-studied structural restriction on automata, namely the degree of ambiguity, which is defined as the maximum number o…
Extensions to finite-state automata on strings, such as multi-head automata or multi-counter automata, have been successfully used to encode many infinite-state non-regular verification problems. In this paper, we consider a generalization of automata-theoretic infinite-state verification from strings to labeled series-parallel graphs. We define a model of non-deterministic, 2-way, concurrent automata working on series-parallel graphs and communicating through shared registers on the nodes of the graph. We consider the following verification problem: given a family of series-parallel graphs de…
For a group $G$ and a finite set $A$, denote by $\text{CA}(G;A)$ the monoid of all cellular automata over $A^G$ and by $\text{ICA}(G;A)$ its group of units. We study the minimal cardinality of a generating set, known as the rank, of $\text{ICA}(G;A)$. In the first part, when $G$ is a finite group, we give upper bounds for the rank in terms of the number of conjugacy classes of subgroups of $G$. The case when $G$ is a finite cyclic group has been studied before, so here we focus on the cases when $G$ is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lowe…
Combining ideas from distributed algorithms and alternating automata, we introduce a new class of finite graph automata that recognize precisely the languages of finite graphs definable in monadic second-order logic. By restricting transitions to be nondeterministic or deterministic, we also obtain two strictly weaker variants of our automata for which the emptiness problem is decidable. As an application, we suggest how suitable graph automata might be useful in formal verification of distributed algorithms, using Floyd-Hoare logic.
Linear temporal logic was introduced in order to reason about reactive systems. It is often considered with respect to infinite words, to specify the behaviour of long-running systems. One can consider more general models for linear time, using words indexed by arbitrary linear orderings. We investigate the connections between temporal logic and automata on linear orderings, as introduced by Bruyère and Carton. We provide a doubly exponential procedure to compute from any LTL formula with Until, Since, and the Stavi connectives an automaton that decides whether that formula holds on the input …
We make a connection between the subgroup membership and identity problems for matrix groups and extended finite automata. We provide an alternative proof for the decidability of the subgroup membership problem for $ 2 \times 2 $ integer matrices. We show that the emptiness problem for extended finite automata over $ 4 \times 4 $ integer matrix semigroups is undecidable. We prove that the decidability of the universe problem for extended finite automata is a sufficient condition for the decidability of the subgroup membership and identity problems.
