Pushdown Automata Research Articles

Superiority of one-way and realtime quantum machines

In automata theory, quantum computation has been widely examined for finite state machines, known as quantum finite automata (QFAs), and less attention has been given to QFAs augmented with counters or stacks. In this paper, we focus on such generalizations of QFAs where the input head operates in one-way or realtime mode, and present some new results regarding their superiority over their classical counterparts. Our first result is about the nondeterministic acceptance mode: Each quantum model architecturally intermediate between realtime finite state automaton and one-way pushdown automaton (one-way finite automaton, realtime and one-way finite automata with one-counter, and realtime pushdown automaton) is superior to its classical counterpart. The second and third results are about bounded error language recognition: for any k > 0, QFAs with k blind counters outperform their deterministic counterparts; and, a one-way QFA with a single head recognizes an infinite family of languages, which can be recognized by one-way probabilistic finite automata with at least two heads. Lastly, we compare the nondeterminictic and deterministic acceptance modes for classical finite automata with k blind counter(s), and we show that for any k > 0, the nondeterministic models outperform the deterministic ones.

Transductions Computed by One-Dimensional Cellular Automata

Cellular automata are investigated towards their ability to compute transductions, that is, to transform inputs into outputs. The families of transductions computed are classified with regard to the time allowed to process the input and to compute the output. Since there is a particular interest in fast transductions, we mainly focus on the time complexities real time and linear time. We first investigate the computational capabilities of cellular automaton transducers by comparing them to iterative array transducers, that is, we compare parallel input/output mode to sequential input/output mode of massively parallel machines. By direct simulations, it turns out that the parallel mode is not weaker than the sequential one. Moreover, with regard to certain time complexities cellular automaton transducers are even more powerful than iterative arrays. In the second part of the paper, the model in question is compared with the sequential devices single-valued finite state transducers and deterministic pushdown transducers. It turns out that both models can be simulated by cellular automaton transducers faster than by iterative array transducers.

The family of languages accepted by pushdown automata with 5-components.

The family of languages accepted by pushdown automata with 5-components.

On the Existence of Universal Finite or Pushdown Automata

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.

Analyzing probabilistic pushdown automata

The paper gives a summary of the existing results about algorithmic analysis of probabilistic pushdown automata and their subclasses.

Descriptional complexity of two-way pushdown automata with restricted head reversals

Two-way nondeterministic pushdown automata (2PDA) are classical nondeterministic pushdown automata (PDA) enhanced with two-way motion of the input head. In this paper, the subclass of 2PDA accepting bounded languages and making at most a constant number of input head turns is studied with respect to descriptional complexity aspects. In particular, the effect of reducing the number of pushdown reversals to a constant number is of interest. It turns out that this reduction leads to an exponential blow-up in case of nondeterministic devices, and to a doubly-exponential blow-up in case of deterministic devices. If the restriction on boundedness of the languages considered and on the finiteness of the number of head and pushdown turns is dropped, the resulting trade-offs are no longer bounded by recursive functions, and so-called non-recursive trade-offs are shown.

On synchronized multi-tape and multi-head automata

Motivated by applications to verification problems in string manipulating programs, we look at the problem of whether the heads in a multi-tape automaton are synchronized. Given an n-tape pushdown automaton M with a one-way read-only head per tape and a right end marker $ on each tape, and an integer k≥0, we say that M is k-synchronized if at any time during any computation of M on any input n-tuple (x1,…,xn) (whether or not it is accepted), no pair of input heads that are not on $ are more than k cells apart. This requirement is automatically satisfied if one of the heads has reached $. Note that an n-tuple (x1,…,xn) is accepted if M reaches the configuration where all n heads are on $ and M is in an accepting state. The automaton can be deterministic (DPDA) or nondeterministic (NPDA) and, in the special case, may not have a pushdown stack (DFA, NFA). We obtain decidability and undecidability results for these devices for both one-way and two-way versions. We also consider the notion of k-synchronized one-way and two-way multi-head automata and investigate similar problems.

Guest Editorial: Computing and Combinatorics

“Computing and Combinatorics” is an interesting and fundamental research area, which involves several fields, ranging from algorithms, theory of computation, and computational complexity to combinatorics related to computing. This issue consists of six papers, which are briefly discussed as follows. The “Shorthand Universal Cycles for Permutations” paper investigates SP-cycles with maximum and minimum ’weight’. The authors show that periodic min-weight SP-cycles correspond to spanning trees of the (n− 1) permutohedron and provides two constructions by using ’half-hunts’ from bell-ringing and the cool-lex order. In “Zero-Knowledge Argument for Simultaneous Discrete Logarithms,” the authors present an EQDL (the equality of two discrete logarithms) protocol in the ROM (random oracle model) which saves approximately 40 % of the computational cost and approximately 33 % of the prover’s outgoing message size. This improvement benefits a variety of interesting cryptosystems, ranging from signatures and anonymous credential systems, to verifiable secret sharing and threshold cryptosystems. The issue also includes the study of computational complexity. Namely, the paper “Tile-Packing Tomography is NP-hard” shows that for a fixed tile, it is NP-hard to reconstruct its tilings from their projections for all types of tiles. The #NC1 upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton is considered in “Counting paths in VPA is complete for #NC1”. The paper also shows that the difference between #BWBP and #NC1 is captured exactly by the addition of a visible stack to a nondeterministic finite-state automaton. Due to the NP-hardness of the problems, approximation algorithms are needed to provide a near-optimal solution. In “Exact and Approximation Algorithms for Geometric and General Capacitated Set Cover Problems”, the first known polynomial-

Relating automata to other fields

Automata classes including Turing Machines, Pushdown Automata and Finite Automata define the most elegant models of computation in terms of set processors. This study elaborates pedagogically motivated intuitive and formal relations between mathematical models and other computer science areas, so that students can relate different areas of computer science in a meaningful way. Cases that promote learning about theoretical models are presented with other related areas, such as programming languages, compilers and software design. Dynamic aspects of software can be appropriately modeled by certain automata based models. Visualizations are developed to help students in their initial stages of understanding of these relations. The visualizations demonstrate that mathematical models such as Pushdown Automata are reasonable processors of some programming language features such as balanced {'s and }'s which are evidently helpful in learning this kind of relation.

An Application of Iterative Pushdown Automata to Contour Words of Balls and Truncated Balls in Hyperbolic Tessellations

We give an application of iterated pushdown automata to contour words of balls and two other domains in infinitely many tilings of the hyperbolic plane. We also give a similar application for the tiling of the hyperbolic 3D space and for the tiling of the hyperbolic 4D space as well.

A theory of computation based on unsharp quantum logic: Finite state automata and pushdown automata

When generalizing the projection-valued measurements to the positive operator-valued measurements, the notion of the quantum logic generalizes from the sharp quantum logic to the unsharp quantum logic. It is known that: (i) the distributive law is one of the main differences between the sharp quantum logic and the boolean logic, and the block or the center of the sharp quantum structures are boolean algebras; (ii) the unsharp quantum logic does not satisfy the non-contradiction law, which forces the block or the center of unsharp quantum structures to be multiple valued algebras, rather than boolean algebras. Multiple valued algebras, as special quantum structures, are the algebraic semantics of multiple valued logic. Interestingly, we recently discovered that the difference between some unsharp quantum structures and multiple valued algebras is also some kind of distributive law.Choosing an orthomodular lattice (an algebraic model of a sharp quantum logic) to be the truth valued lattice, Ying et al. have systematically developed automata theory based on sharp quantum logic. In this paper, choosing a lattice ordered quantum multiple valued algebra E (an extended lattice ordered effect algebra E, respectively) to be the truth valued lattice, we also systematically develop an automata theory based on unsharp quantum logic. We introduce E-valued finite-state automata and E-valued pushdown automata in the framework of unsharp quantum logic. We study the classes of languages accepted by these automata and re-examine their various properties in the framework of unsharp quantum logic. The study includes the equivalence between finite-state automata and regular expressions, as well as the equivalence between pushdown automata and context-free grammars. It is also demonstrated that the universal validity of some important properties (such as some closure properties of languages and Kleene theorem etc.) depends heavily on the aforementioned distributive law. More precisely, when the underlying model degenerates into an MV algebra, then all the counterparts of properties in classical automata are valid. This is the main difference between automata theory based on unsharp quantum logic and automata theory based on sharp quantum logic.

P automata revisited

We continue here the investigation of P automata, in their non-extended case, a class of devices which characterize non-universal family of languages. First, a recent conjecture is confirmed: any recursively enumerable language is obtained from a language recognized by a P automaton, to which an initial (arbitrarily large) string is added. Then, we discuss possibilities of extending P automata, following suggestions from string finite automata. For instance, automata with a memory (corresponding to push-down automata) are considered and their power is briefly investigated, as well as some closure properties of the family of languages recognized by P automata. In the context, a brief survey of results about P and dP automata (a distributed version of P automata) is provided, and several further research topics are formulated.

The expressive power of analog recurrent neural networks on infinite input streams

We consider analog recurrent neural networks working on infinite input streams, provide a complete topological characterization of their expressive power, and compare it to the expressive power of classical infinite word reading abstract machines. More precisely, we consider analog recurrent neural networks as language recognizers over the Cantor space, and prove that the classes of ω-languages recognized by deterministic and non-deterministic analog networks correspond precisely to the respective classes of Π20-sets and Σ11-sets of the Cantor space. Furthermore, we show that the result can be generalized to more expressive analog networks equipped with any kind of Borel accepting condition. Therefore, in the deterministic case, the expressive power of analog neural nets turns out to be comparable to the expressive power of any kind of Büchi abstract machine, whereas in the non-deterministic case, analog recurrent networks turn out to be strictly more expressive than any other kind of Büchi or Muller abstract machine, including the main cases of classical automata, 1-counter automata, k-counter automata, pushdown automata, and Turing machines.

Model checking probabilistic systems against pushdown specifications

Model checking is a fully automatic verification technique traditionally used to verify finite-state systems against regular specifications. Although regular specifications have been proven to be feasible in practice, many desirable specifications are non-regular. For instance, requirements which involve counting cannot be formalized by regular specifications but using pushdown specifications, i.e., context-free properties represented by pushdown automata. Research on model-checking techniques for pushdown specifications is, however, rare and limited to the verification of non-probabilistic systems.In this paper, we address the probabilistic model-checking problem for systems modeled by discrete-time Markov chains and specifications that are provided by deterministic pushdown automata over infinite words. We first consider finite-state Markov chains and show that the quantitative and qualitative model-checking problem is solvable via a product construction and techniques that are known for the verification of probabilistic pushdown automata. Then, we consider recursive systems modeled by probabilistic pushdown automata with an infinite-state Markov chain semantics. We first show that imposing appropriate compatibility (visibility) restrictions on the synchronizations between the pushdown automaton for the system and the specification, decidability of the probabilistic model-checking problem can be established. Finally we prove that slightly departing from this compatibility assumption leads to the undecidability of the probabilistic model-checking problem, even for qualitative properties specified by deterministic context-free specifications.

Indexing ordered trees for (nonlinear) tree pattern matching by pushdown automata

Trees are one of the fundamental data structures used in Computer Science. We present a new kind of acyclic pushdown automata, the tree pattern pushdown automaton and the nonlinear tree pattern pushdown automaton, constructed for an ordered tree. These automata accept all tree patterns and nonlinear tree patterns, respectively, which match the tree and represent a full index of the tree for such patterns. Given a tree with n nodes, the numbers of these distinct tree patterns and nonlinear tree patterns can be at most 2n?1 +n and at most (2+v)n?1+2, respectively, where v is the maximal number of nonlinear variables allowed in nonlinear tree patterns. The total sizes of nondeterministic versions of the two pushdown automata are O(n) and O(n2), respectively. We discuss the time complexities and show timings of our implementations using the bit-parallelism technique. The timings show that for a given tree the running time is linear to the size of the input pattern.

Reversible pushdown automata

Reversible pushdown automata are deterministic pushdown automata that are also backward deterministic. Therefore, they have the property that any configuration occurring in any computation has exactly one predecessor. In this paper, the computational capacity of reversible computations in pushdown automata is investigated and turns out to lie properly in between the regular and deterministic context-free languages. Furthermore, it is shown that a deterministic context-free language cannot be accepted reversibly if more than realtime is necessary for acceptance. Closure properties as well as decidability questions for reversible pushdown automata are studied. Finally, we show that the problem to decide whether a given nondeterministic or deterministic pushdown automaton is reversible is P-complete, whereas it is undecidable whether the language accepted by a given nondeterministic pushdown automaton is reversible.

TREE-BASED 2D INDEXING

A new approach to the 2D pattern matching and specifically to 2D text indexing is proposed. A transformation of a 2D text into the form of a tree is presented. It preserves the context of each element of the 2D text. The tree can be linearised using the prefix notation into the form of a string (a linear text) and the pattern matching is performed in this text. Pushdown automata indexing the 2D text are constructed over the tree representation. They allow to search for 2D prefixes, 2D suffixes, and 2D factors of the 2D text in time proportional to the size of the representation of a 2D pattern. This result achieves the properties analogous to the results obtained in tree pattern matching and string indexing.

Concise guide to computation theory

This textbook presents a thorough foundation to the theory of computation. Combining intuitive descriptions and illustrations with rigorous arguments and detailed proofs for key topics, the logically structured discussion guides the reader through the core concepts of automata and languages, computability, and complexity of computation. Topics and features: presents a detailed introduction to the theory of computation, complete with concise explanations of the mathematical prerequisites; provides end-of-chapter problems with solutions, in addition to chapter-opening summaries and numerous examples and definitions throughout the text; draws upon the authors extensive teaching experience and broad research interests; discusses finite automata, context-free languages, and pushdown automata; examines the concept, universality and limitations of the Turing machine; investigates computational complexity based on Turing machines and Boolean circuits, as well as the notion of NP-completeness.

A comment on “Automata theory based on complete residuated lattice-valued logic: Pushdown automata”

Automata theory based on complete residuated lattice-valued logic has been first established by Qiu, and then has been systematically studied by Qiu and others. The definition of L-valued Chomsky Normal Form in Xing and Qiu [Automata theory based on complete residuated lattice-valued logic: pushdown automata, Fuzzy Sets and Systems 160 (2009) 1125–1140] is somewhat different from that in Xing and Qiu [Pumping lemma in context-free grammar theory based on complete residuated lattice-valued logic, Fuzzy Sets and Systems 160 (2009) 1141–1151]. In this note, we give a more general L-valued Chomsky Normal Form to unify the two definitions. We mainly show that, for an L-valued context-free grammar, an L-valued Greibach Normal Form can be equivalently constructed.

Quasi-rocking real-time pushdown automata

A pushdown automaton (PDA) is quasi-rocking if it preserves the stack height for no more than a bounded number of consecutive moves. Every PDA can be transformed into an equivalent one that is quasi-rocking and real-time and every finite-turn (one-turn) PDA can be transformed into an equivalent one that is quasi-rocking or real-time. The quasi-rocking [quasi-rocking in the increasing mode, and quasi-rocking in the decreasing mode] real-time restriction in finite-turn (one-turn) PDAs coincides with the double Greibach [reverse Greibach, and Greibach] form in nonterminal-bounded (linear) context-free grammars. This provides complete grammatical characterizations of quasi-rocking and/or real-time (finite-turn and one-turn) PDAs and, together with known relations and other relations proved in the present paper, yields an extended hierarchy of PDA languages. Basic decision properties for PDAs can be stated in stronger forms by using the quasi-rocking and real-time restrictions and their undecidability/decidability status rests on the way PDAs quasi-rock.