Input-driven pushdown automata (also known as visibly pushdown automata and as nested word automata) are a subclass of deterministic pushdown automata and a superclass of the parenthesis languages. Nguyen and Ogawa ("Event-clock visibly pushdown automata", SOFSEM 2009) defined a timed extension of these automata under the event-clock model, and showed that this model can be determinized using the method of region construction. This paper defines a further extension of this model with the event clock on the call-return operations, and proposes a new, direct determinization procedure for these automata: an $n$-state nondeterministic automaton with $k$ different clock constraints is transformed to a deterministic automaton with $2^{n^2}$ states, $2^{n^2+k}$ stack symbols and the same clock constraints as in the original automaton. The construction is shown to be asymptotically optimal with respect to both the number of states and the number of stack symbols.
We study the problem of determinizing $ω$-automata whose acceptance condition is defined on the transitions using Boolean formulas, also known as transition-based Emerson-Lei automata (TELA). The standard approach to determinize TELA first constructs an equivalent generalized Büchi automaton (GBA), which is later determinized. We introduce three new ways of translating TELA to GBA. Furthermore, we give a new determinization construction which determinizes several GBA separately and combines them using a product construction. An experimental evaluation shows that the product approach is competi…
Visibly pushdown automata (VPA), introduced by Alur and Madhusuan in 2004, is a subclass of pushdown automata whose stack behavior is completely determined by the input symbol according to a fixed partition of the input alphabet. Since its introduce, VPAs have been shown to be useful in various context, e.g., as specification formalism for verification and as automaton model for processing XML streams. Due to high complexity, however, implementation of formal verification based on VPA framework is a challenge. In this paper we consider the problem of implementing VPA-based model checking algor…
In this paper we adapt the well-known Brzozowski determinization method to fuzzy automata. This method gives better results than all previously known methods for determinization of fuzzy automata developed by Bělohlávek [Inform Sciences 143 (2002) 205--209], Li and Pedrycz [Fuzzy Set Syst 156 (2005) 68--92], Ignjatović et al. [Inform Sciences 178 (2008) 164--180], and Jančić et al. [Inform Sciences 181 (2011) 1358--1368]. Namely, as in the case of ordinary nondeterministic automata, Brzozowski type determinization of a fuzzy automaton results in a minimal crisp-deterministic fuzzy automaton eq…
In this work, we present multiple new optimizations and heuristics for the determinization of Büchi automata that exploit a number of semantic and structural properties, most of which may be applied together with any determinization procedure. We built a prototype implementation where all the presented heuristics can be freely combined and evaluated them, comparing our implementation with the state-of-the-art tool spot on multiple data sets with different characteristics. Our results show that the proposed optimizations and heuristics can in some cases significantly decrease the size of the re…
In this paper we combine determinization and state reduction methods into two-in-one algorithms that simultaneously perform determinization and state reduction. These algorithms perform better than all previous determinization algorithms for fuzzy finite automata, developed by Belohlavek [Inform Sciences 143 (2002) 205-209], Li and Pedrycz [Fuzzy Set Syst 156 (2005) 68-92], Ignjatović et al. [Inform Sciences 178 (2008) 164-180], and Jančić et al. [Inform Sciences 181 (2011) 1358-1368], in the sense that they produce smaller automata, while require the same computation time. The only exception …
Deterministic timed automata are strictly less expressive than their non-deterministic counterparts, which are again less expressive than those with silent transitions. As a consequence, timed automata are in general non-determinizable. This is unfortunate since deterministic automata play a major role in model-based testing, observability and implementability. However, by bounding the length of the traces in the automaton, effective determinization becomes possible. We propose a novel procedure for bounded determinization of timed automata. The procedure unfolds the automata to bounded trees,…
Determinization of Büchi automata is a long-known difficult problem and after the seminal result of Safra, who developed the first asymptotically optimal construction from Büchi into Rabin automata, much work went into improving, simplifying or avoiding Safra's construction. A different, less known determinization construction was derived by Muller and Schupp and appears to be unrelated to Safra's construction on the first sight. In this paper we propose a new meta-construction from nondeterministic Büchi to deterministic parity automata which strictly subsumes both the construction of Safra a…
Determinization of fuzzy finite automata is understood here as a procedure of their conversion into equivalent crisp-deterministic fuzzy automata, which can be viewed as being deterministic with possibly infinitely many states, but with fuzzy sets of terminal states. Particularly significant determinization methods are those that provide a minimal crisp-deterministic fuzzy automaton equivalent to the original fuzzy finite automaton, called canonization methods. One canonization method for fuzzy finite automata, the Brzozowski type determinization, has been developed recently by Jančić and Ćiri…
Recently, an infinite hierarchy of languages accepted by stateless deterministic pushdown automata has been established based on the number of pushdown symbols. However, the witness language for the n-th level of the hierarchy is over an input alphabet with 2(n-1) elements. In this paper, we improve this result by showing that a binary alphabet is sufficient to establish this hierarchy. As a consequence of our construction, we solve the open problem formulated by Meduna et al. Then we extend these results to m-state realtime deterministic pushdown automata, for all m at least 1. The existence …
We propose an algorithm for schema-based determinization of finite automata on words and of step-wise hedge automata on nested words. The idea is to integrate schema-based cleaning directly into automata determinization. We prove the correctness of our new algorithm and show that it is alway smore efficient than standard determinization followed by schema-based cleaning. Our implementation permits to obtain a small deterministic automaton for an example of an XPath query, where standard determinization yields a huge stepwise hedge automaton for which schema-based cleaning runs out of memory.
Timed pushdown automata are pushdown automata extended with a finite set of real-valued clocks. Additionaly, each symbol in the stack is equipped with a value representing its age. The enabledness of a transition may depend on the values of the clocks and the age of the topmost symbol. Therefore, dense-timed pushdown automata subsume both pushdown automata and timed automata. We have previously shown that the reachability problem for this model is decidable. In this paper, we study the zenoness problem and show that it is EXPTIME-complete.
The determinization of a nondeterministic Büchi automaton (NBA) is a fundamental construction of automata theory, with applications to probabilistic verification and reactive synthesis. The standard determinization constructions, such as the ones based on the Safra-Piterman's approach, work on the whole NBA. In this work we propose a divide-and-conquer determinization approach. To this end, we first classify the strongly connected components (SCCs) of the given NBA as inherently weak, deterministic accepting, and nondeterministic accepting. We then present how to determinize each type of SCC i…
Optimal determinization construction of Streett automata is an important research problem because it is indispensable in numerous applications such as decision problems for tree temporal logics, logic games and system synthesis. This paper presents a transformation from nondeterministic Streett automata (NSA) with $n$ states and $k$ Streett pairs to equivalent deterministic Rabin transition automata (DRTA) with $n^{5n}(n!)^{n}$ states, $O(n^{n^2})$ Rabin pairs for $k=ω(n)$ and $n^{5n}k^{nk}$ states, $O(k^{nk})$ Rabin pairs for $k=O(n)$. This improves the state of the art Streett determinizatio…
We investigate the (non)-existence of universal automata for some classes of automata, such as finite automata and pushdown automata, and in particular the influence of the representation and encoding function. An alternative approach, using transition systems, is presented too.
The pumping lemma for context-free languages is a result about pushdown automata which is strikingly similar to the well-known pumping lemma for regular languages. However, though the lemma for regular languages is simply proved by using the pigeonhole principle on deterministic automata, the lemma for pushdown automata is proven through an equivalence with context-free languages and through the more powerful Ogden's lemma. We present here a proof of the pumping lemma for context-free languages which relies on pushdown automata instead of context-free grammars.
This papers presents a general framework for the uniform random generation of deterministic real-time accessible pushdown automata. A polynomial time algorithm to randomly generate a pushdown automaton having a fixed stack operations total size is proposed. The influence of the accepting condition (empty stack, final state) on the reachability of the generated automata is investigated.
An automaton is universal if it accepts every possible input. We study the notion of u-universality, which asserts that the automaton accepts every input starting with u. Universality and u-universality are both EXPTIME-hard for non-deterministic tree automata. We propose efficient antichain-based techniques to address these problems for visibly pushdown automata operating on trees. One of our approaches yields algorithms for the universality and u-universality of hedge automata.
Pushdown automata may contain transitions that are never used in any accepting run of the automaton. We present an algorithm for detecting such useless transitions. A finite automaton that captures the possible stack content during runs of the pushdown automaton, is first constructed in a forward procedure to determine which transitions are reachable, and then employed in a backward procedure to determine which of these transitions can lead to a final stat
In (Csuhaj-Varju et. al. 2000) Parallel Communicating Systems of Pushdown Automata (PCPA) were introduced and shown to be able to simulate nondeterministic one-way multi-head pushdown automata in returning mode, even if communication is restricted to be one-way having a single target component. A simulation of such centralized PCPA by one-way multi-head pushdown automata (Balan 2009) turned out to be incomplete (Otto 2012). Subsequently it was shown that centralized returning PCPA are universal even if the number of components is two (Petersen 2013) and thus are separated from one-way multi-he…
Parallel communicating systems of pushdown automata (PCPA) were introduced in (Csuhaj-Varj{ú} et. al. 2000) and in their centralized variants shown to be able to simulate nondeterministic one-way multi-head pushdown automata. A claimed converse simulation for returning mode (Balan 2009) turned out to be incomplete (Otto 2012) and a language was suggested for separating these PCPA of degree two (number of pushdown automata) from nondeterministic one-way two-head pushdown automata. We show that the suggested language can be accepted by the latter computational model. We present a different examp…
We introduce and study input-driven deterministic and nondeterministic double-head pushdown automata. A double-head pushdown automaton is a slight generalization of an ordinary pushdown automaton working with two input heads that move in opposite directions on the common input tape. In every step one head is moved and the automaton decides on acceptance if the heads meet. Demanding the automaton to work input-driven it is required that every input symbol uniquely defines the action on the pushdown store (push, pop, state change). Normally this is modeled by a partition of the input alphabet an…
We extend Antimirov's partial derivatives from regular expressions to $μ$-regular expressions that describe context-free languages. We prove the correctness of partial derivatives as well as the finiteness of the set of iterated partial derivatives. The latter are used as pushdown symbols in our construction of a nondeterministic pushdown automaton, which generalizes Antimirov's NFA construction.
The simulation of deterministic pushdown automata defined over a one-letter alphabet by finite state automata is investigated from a descriptional complexity point of view. We show that each unary deterministic pushdown automaton of size s can be simulated by a deterministic finite automaton with a number of states that is exponential in s. We prove that this simulation is tight. Furthermore, its cost cannot be reduced even if it is performed by a two-way nondeterministic automaton. We also prove that there are unary languages for which deterministic pushdown automata cannot be exponentially m…
Hankel matrices (aka connection matrices) of word functions and graph parameters have wide applications in automata theory, graph theory, and machine learning. We give a characterization of real-valued functions on nested words recognized by weighted visibly pushdown automata in terms of Hankel matrices on nested words. This complements C. Mathissen's characterization in terms of weighted monadic second order logic.
We show the diagonal problem for higher-order pushdown automata (HOPDA), and hence the simultaneous unboundedness problem, is decidable. From recent work by Zetzsche this means that we can construct the downward closure of the set of words accepted by a given HOPDA. This also means we can construct the downward closure of the Parikh image of a HOPDA. Both of these consequences play an important role in verifying concurrent higher-order programs expressed as HOPDA or safe higher-order recursion schemes.
We model collapsible and ordered pushdown systems with term rewriting, by encoding higher-order stacks and multiple stacks into trees. We show a uniform inverse preservation of recognizability result for the resulting class of term rewriting systems, which is obtained by extending the classic saturation-based approach. This result subsumes and unifies similar analyses on collapsible and ordered pushdown systems. Despite the rich literature on inverse preservation of recognizability for term rewrite systems, our result does not seem to follow from any previous study.
We introduce the notion of adaptive synchronisation for pushdown automata, in which there is an external observer who has no knowledge about the current state of the pushdown automaton, but can observe the contents of the stack. The observer would then like to decide if it is possible to bring the automaton from any state into some predetermined state by giving inputs to it in an \emph{adaptive} manner, i.e., the next input letter to be given can depend on how the contents of the stack changed after the current input letter. We show that for non-deterministic pushdown automata, this problem is…
We propose a new extension of higher-order pushdown automata, which allows to use an infinite alphabet. The new automata recognize languages of data words (instead of normal words), which beside each its letter from a finite alphabet have a data value from an infinite alphabet. Those data values can be loaded to the stack of the automaton, and later compared with some farther data values on the input. Our main purpose for introducing these automata is that they may help in analyzing normal automata (without data). As an example, we give a proof that deterministic automata with collapse can rec…
We present an algorithm which modifies a deterministic pushdown automaton (DPDA) such that (i) the marked language is preserved, (ii) lifelocks are removed, (iii) deadlocks are removed, (iv) all states and edges are accessible, and (v) operational blockfreeness is established (i.e., coaccessibility in the sense that every initial derivation can be continued to a marking configuration). This problem can be trivially solved for deterministic finite automata (DFA) but is not solvable for standard petri net classes. The algorithm is required for an operational extension of the supervisory control …
We study the average stack cost of Buechi pushdown automata (Buechi PDA). We associate a non-negative price with each stack symbol and define the cost of a stack as the sum of costs of all its elements. We introduce and study the average stack cost problem (ASC), which asks whether there exists an accepting run of a given Buechi PDA such that the long-run average of stack costs is below some given threshold. The ASC problem generalizes mean-payoff objective and can be used to express quantitative properties of pushdown systems. In particular, we can compute the average response time using the …
In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting. Here, we investigate weighted context-free languages of infinite words, a generalization of $ω$-context-free languages (Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Schützenberger 1963). As in the theory of formal grammars, these weighted context-free languages, or $ω$-algebraic series, can be represented as solutions of mixed $ω$-algebraic systems of equations and by weighted $ω$-pushdown automata. In our first main result, …
One-Counter Nets (OCNs) are finite-state automata equipped with a counter that is not allowed to become negative, but does not have zero tests. Their simplicity and close connection to various other models (e.g., VASS, Counter Machines and Pushdown Automata) make them an attractive model for studying the border of decidability for the classical decision problems. The deterministic fragment of OCNs (DOCNs) typically admits more tractable decision problems, and while these problems and the expressive power of DOCNs have been studied, the determinization problem, namely deciding whether an OCN ad…
Automata operating on strings of nested brackets, known as input-driven pushdown automata, and as visibly pushdown automata, have been studied since the 1980s. They were extended to the case of infinite strings by Alur and Madhusudan ("Visibly pushdown languages", STOC 2004). This paper investigates the properties of these automata under the assumption that a given infinite string is always well-nested. This restriction enables a complete characterization of the corresponding $ω$-languages in terms of classical $ω$-regular languages and input-driven automata on finite strings. This characteriz…
We consider the problem of state-space reduction for nondeterministic weakly-hierarchical visibly pushdown automata (VPA). VPA recognize a robust and algorithmically tractable fragment of context-free languages that is natural for modeling programs. We define an equivalence relation that is sufficient for language-preserving quotienting of VPA. Our definition allows to merge states that have different behavior, as long as they show the same behavior for reachable equivalent stacks. We encode the existence of such a relation as a Boolean partial maximum satisfiability (PMax-SAT) problem and pre…
Complementation and determinization are two fundamental notions in automata theory. The close relationship between the two has been well observed in the literature. In the case of nondeterministic finite automata on finite words (NFA), complementation and determinization have the same state complexity, namely Theta(2^n) where n is the state size. The same similarity between determinization and complementation was found for Buchi automata, where both operations were shown to have 2^Θ(n lg n) state complexity. An intriguing question is whether there exists a type of omega-automata whose determin…
Probabilistic pushdown automata (pPDA) are a standard model for discrete probabilistic programs with procedures and recursion. In pPDA, many quantitative properties are characterized as least fixpoints of polynomial equation systems. In this paper, we study the problem of certifying that these quantities lie within certain bounds. To this end, we first characterize the polynomial systems that admit easy-to-check certificates for validating bounds on their least fixpoint. Second, we present a sound and complete Optimistic Value Iteration algorithm for computing such certificates. Third, we show…
This article proposes novel off-line test generation techniques from non-deterministic timed automata with inputs and outputs (TAIOs) in the formal framework of the tioco conformance theory. In this context, a first problem is the determinization of TAIOs, which is necessary to foresee next enabled actions after an observable trace, but is in general impossible because not all timed automata are determinizable. This problem is solved thanks to an approximate determinization using a game approach. The algorithm performs an io-abstraction which preserves the tioco conformance relation and thus g…
Let S be a complete star-omega semiring and Sigma be an alphabet. For a weighted omega-pushdown automaton P with stateset 1...n, n greater or equal to 1, we show that there exists a mixed algebraic system over a complete semiring-semimodule pair ((S<<Sigma*>>)^nxn, (S<<Sigma^omega>>)^n) such that the behavior ||P|| of P is a component of a solution of this system. In case the basic semiring is the Boolean semiring or the semiring of natural numbers (augmented with infinity), we show that there exists a mixed context-free grammar that generates ||P||. The construction of the mixed context-free …
In this tutorial, we illustrate through examples how we can combine two classical models, namely those of pushdown automata (PDA) and timed automata, in order to obtain timed pushdown automata (TPDA). Furthermore, we describe how the reachability problem for TPDAs can be reduced to the reachability problem for PDAs.
The edit distance between two words $w_1, w_2$ is the minimal number of word operations (letter insertions, deletions, and substitutions) necessary to transform $w_1$ to $w_2$. The edit distance generalizes to languages $\mathcal{L}_1, \mathcal{L}_2$, where the edit distance from $\mathcal{L}_1$ to $\mathcal{L}_2$ is the minimal number $k$ such that for every word from $\mathcal{L}_1$ there exists a word in $\mathcal{L}_2$ with edit distance at most $k$. We study the edit distance computation problem between pushdown automata and their subclasses. The problem of computing edit distance to a pu…
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 present a disambiguation algorithm for weighted automata. The algorithm admits two main stages: a pre-disambiguation stage followed by a transition removal stage. We give a detailed description of the algorithm and the proof of its correctness. The algorithm is not applicable to all weighted automata but we prove sufficient conditions for its applicability in the case of the tropical semiring by introducing the *weak twins property*. In particular, the algorithm can be used with all acyclic weighted automata, relevant to applications. While disambiguation can sometimes be achieved using det…
By separating the principal acceptance mechanism from the concrete acceptance condition of a given Büchi automaton with $n$ states,Schewe presented the construction of an equivalent deterministic Rabin transition automaton with $o((1.65n)^n)$ states via \emph{history trees}, which can be simply translated to a standard Rabin automaton with $o((2.26n)^n)$ states. Apart from the inherent simplicity, Schewe's construction improved Safra's construction (which requires $12^nn^{2n}$ states). However, the price that is paid is the use of $2^{n-1}$ Rabin pairs (instead of $n$ in Safra's construction).…
We describe a uniform construction for converting $ω$-automata with arbitrary acceptance conditions (based on the notion of infinity sets i.e. the set of states visited infinitely often in a run of the automaton) to equivalent deterministic parity automata (DPW). Given a non-deterministic automaton with $n$ states, our construction gives a DPW with at most $2^{O(n^2 \log n)}$ states and $O(n^2)$ parity indices. The corresponding bounds when the original automaton is deterministic are O(n!) and O(n), respectively. Our algorithm gives better asymptotic bounds on the number of states and parity i…
We study an expressive model of timed pushdown automata extended with modular and fractional clock constraints. We show that the binary reachability relation is effectively expressible in hybrid linear arithmetic with a rational and an integer sort. This subsumes analogous expressibility results previously known for finite and pushdown timed automata with untimed stack. As key technical tools, we use quantifier elimination for a fragment of hybrid linear arithmetic and for cyclic order atoms, and a reduction to register pushdown automata over cyclic order atoms.
We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with $n$ states and $p \geq 2n + 4$ stack symbols that accepts one single long word for which every equivalent context-free grammar needs $Ω(n^2(p-2n-4))$ variables. This family shows that the classical algorithm for converting a PDA to an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family…
We address the verification problem of ordered multi-pushdown automata: A multi-stack extension of pushdown automata that comes with a constraint on stack transitions such that a pop can only be performed on the first non-empty stack. First, we show that the emptiness problem for ordered multi-pushdown automata is in 2ETIME. Then, we prove that, for an ordered multi-pushdown automata, the set of all predecessors of a regular set of configurations is an effectively constructible regular set. We exploit this result to solve the global model-checking which consists in computing the set of all con…
We consider parallel communicating pushdown automata systems (PCPA) and define a property called known communication for it. We use this property to prove that the power of a variant of PCPA, called returning centralized parallel communicating pushdown automata (RCPCPA), is equivalent to that of multi-head pushdown automata. The above result presents a new sub-class of returning parallel communicating pushdown automata systems (RPCPA) called simple-RPCPA and we show that it can be written as a finite intersection of multi-head pushdown automata systems.
