Pushdown Automata Research Articles

A formal model of semantic computing

In the existing works of semantic computing (SC), the word “computing” in the phrase “semantic computing” means computational implementations of semantics reasoning (e.g., ontology reasoning, rule reasoning, semantic query, and semantic search) but is irrelevant to the formal theory of computation (e.g., computational models such as finite automaton, pushdown automaton, and Turing machine). In this paper, we propose a different understanding of “semantic computing” from a computation theory perspective. Concretely, we present a formal model of SC in terms of automata and discuss SC for the two most important and simplest types of automata, namely finite automata and pushdown automata. For each automaton, we first consider a simple case (equivalent concepts) and then we further investigate a general situation (semantically related concepts). That is, some new automata for SC are provided: finite (or pushdown) automaton for SC under equivalent concepts, finite (or pushdown) automaton for SC w.r.t. external words, nondeterministic finite automaton for SC under equivalent concepts (or w.r.t. external words), fuzzy finite (or pushdown) automaton for SC under semantically related concepts, and fuzzy finite (or pushdown) automaton for SC w.r.t. external words. Furthermore, we give some properties of these new automata for SC and prove that these new automata are extensions (or enlargements) of traditional (fuzzy) automata.

Visibly pushdown transducers

Visibly pushdown transducers (VPT ) extend visibly pushdown automata (VPA) with outputs. They read nested words, i.e. finite words on a structured alphabet partitioned into call, return and internal symbols, and produce output words, which are not necessarily nested. As for VPA, the behavior of the stack is synchronized with the types of input symbols: on reading a call symbol, exactly one stack symbol is pushed onto the stack; on reading a return symbol, exactly one stack symbol is popped from the stack; and the stack is untouched when reading internal symbols. Along their transitions VPT can append words on their output tape, whose concatenation define the resulting output word. Therefore VPT define transformations from nested words to words and can be seen as a subclass of pushdown transducers. In this paper, we study the algorithmic, closure and expressiveness properties of VPT.

On store languages of language acceptors

It is well known that the “store language” of every pushdown automaton — the set of store configurations (state and stack contents) that can appear as an intermediate step in accepting computations — is a regular language. Here many models of language acceptors with various store structures are examined, along with a study of their store languages. For each model, an attempt is made to find the simplest model that accepts their store languages. Some connections between store languages of one-way and two-way machines are demonstrated, as with connections between nondeterministic and deterministic machines. A nice application of these store language results is also presented, showing a general technique for proving families accepted by many deterministic models are closed under right quotient with regular languages, resolving some open questions (and significantly simplifying proofs for others that are known) in the literature. Lower bounds on the space complexity of Turing machines for having non-regular store languages are obtained.

Production of Supra-regular Spatial Sequences by Macaque Monkeys

Understanding and producing embedded sequences in language, music, or mathematics, is a central characteristic of our species. These domains are hypothesized to involve a human-specific competence for supra-regular grammars, which can generate embedded sequences that go beyond the regular sequences engendered by finite-state automata. However, is this capacity truly unique to humans? Using a production task, we show that macaque monkeys can be trained to produce time-symmetrical embedded spatial sequences whose formal description requires supra-regular grammars or, equivalently, a push-down stack automaton. Monkeys spontaneously generalized the learned grammar to novel sequences, including longer ones, and could generate hierarchical sequences formed by an embedding of two levels of abstract rules. Compared to monkeys, however, preschool children learned the grammars much faster using a chunking strategy. While supra-regular grammars are accessible to nonhuman primates through extensive training, human uniqueness may lie in the speed and learning strategy with which they are acquired.

Quantum Pushdown Automata with Garbage Tape

Several kinds of quantum pushdown automata models have been proposed, and their computational power has been investigated intensively. However, for some quantum pushdown automaton models, it is unknown whether quantum models are at least as powerful as their classical counterparts or not. This is due to the reversibility restriction. In this paper, we introduce a new quantum pushdown automaton model that has a garbage tape. This model can overcome the reversibility restriction by exploiting the garbage tape to store popped symbols. We show that the proposed model can simulate any quantum pushdown automaton with classical stack as well as any probabilistic pushdown automaton. We also show that our model can solve a certain promise problem exactly while deterministic pushdown automata cannot. These results imply that our model is strictly more powerful than its classical counterparts in the setting of exact, one-sided error and non-deterministic computation. Showing impossibility for a promise problem is a difficult task in general. However, by analyzing the behavior of a deterministic pushdown automaton carefully, we obtained the impossibility result. This is one of the main contributions of the paper.

Periodic Properties of Autonomous Automata with Pushdown Storage

The paper presents a short proof that pushdown transducers without input generate periodic sequences. We also provide lower and upper bounds for the maximal period of output sequence that can be generated by a pushdown transducer with fixed characteristics.

On the Density of Context-Free and Counter Languages

A language [Formula: see text] is said to be dense if every word in the universe is an infix of some word in [Formula: see text]. This notion has been generalized from the infix operation to arbitrary word operations [Formula: see text] in place of the infix operation ([Formula: see text]-dense, with infix-dense being the standard notion of dense). It is shown here that it is decidable, for a language [Formula: see text] accepted by a one-way nondeterministic reversal-bounded pushdown automaton, whether [Formula: see text] is infix-dense. However, it becomes undecidable for both deterministic pushdown automata (with no reversal-bound), and for nondeterministic one-counter automata. When examining suffix-density, it is undecidable for more restricted families such as deterministic one-counter automata that make three reversals on the counter, but it is decidable with less reversals. Other decidability results are also presented on dense languages, and contrasted with a marked version called [Formula: see text]-marked-density. Also, new languages are demonstrated to be outside various deterministic language families after applying different deletion operations from smaller families. Lastly, bounded-dense languages are defined and examined.

Diagnostic computational model for neuronal disorder through glycogen metabolism in astrocytes

An automaton called embedded automaton is introduced to monitor the glycogen metabolism with an end product as lactate in astrocytes. It is designed to observe the glycogen metabolism with the states of acceptance and rejection. It is a hybridisation of both non-deterministic finite automaton and push-down automaton. The acceptance state of the embedded automaton depicts the normal level of glycogen synthesis and degradation. The rejection state of this machine shows the inhibition of metabolic processes in the absence of active enzymes. The process of glycogen metabolism in astrocytes can be regulated with a group of regulatory enzymes. It assists in enhancing the energy supply to neurons by regulating the neuronal activity. The construction of computational model using automata theory can help in targeting research on the enzymatic activity in brain glycogen pathways. Such an interdisciplinary approach can be a promising therapeutic application for both diagnosis and therapy for neuronal injuries.

Watson-Crick pushdown automata

A multi-head 1-way pushdown automaton with $k$ heads is a pushdown automaton with $k$ 1-way read heads on the input tape and a stack. It was previously shown that the deterministic variant of the model cannot accept all the context free languages. In this paper, we introduce a 2-tape, 2-head model namely Watson-Crick pushdown automata where the content of the second tape is determined using a complementarity relation, similar to Watson-Crick automata. We show computational powers of nondeterministic two-head pushdown automata and nondeterministic Watson-Crick pushdown automata are same. Moreover, deterministic Watson-Crick pushdown automata can accept all the context free languages.

A generic framework for checking semantic equivalences between pushdown automata and finite-state automata

For a given process equivalence, we say that a process g is fully equivalent to a process f of a transition system T if g is equivalent to f and every reachable state of g is equivalent to some state of T. We propose a generic method for deciding full equivalence between pushdown processes and finite-state processes applicable to every process equivalence satisfying certain abstract conditions. Then, we show that these conditions are satisfied by bisimulation-like equivalences (including weak and branching bisimilarity), weak simulation equivalence, and weak trace equivalence, which are the main conceptual representatives of the linear/branching time spectrum. The list of particular results obtained by applying our method includes items which are first of their kind, and the associated upper complexity bounds are essentially optimal.

On characterization of fuzzy tree pushdown automata

This paper introduces the concepts of fuzzy tree pushdown automata and fuzzy context-free tree grammars. At first, we show that any fuzzy context-free tree grammar can be converted into a corresponding definition of fuzzy Chomsky normal form for trees. Then, the relation between a fuzzy context-free tree grammar and a fuzzy tree pushdown automaton is investigated. In fact, we show that the class of languages accepted by fuzzy tree pushdown automata is identical to the one generated by fuzzy context-free tree grammars. Some examples are given to clarify the results.

Descriptional complexity of limited automata

A k-limited automaton is a linear bounded automaton that may rewrite each tape cell only in the first k visits, where k≥0 is a fixed constant. It is known that these automata accept context-free languages only. We investigate the descriptional complexity of limited automata. Since the unary languages accepted are necessarily regular, we first study the cost in the number of states when finite automata simulate a unary k-limited automaton. For the conversion of a 4n-state deterministic 1-limited automaton into one-way or two-way deterministic or nondeterministic finite automata, we show a lower bound of n⋅F(n) states, where F denotes Landau's function. So, even the ability to deterministically rewrite any cell only once gives an enormous descriptional power. For the simulation cost for removing the ability to rewrite each cell k≥1 times, more precisely, the cost for the simulation of sweeping unary k-limited automata by deterministic finite automata, we obtain a lower bound of n⋅F(n)k. The upper bound of the cost for the simulation by two-way deterministic finite automata is a polynomial whose degree is quadratic in k. If the k-limited automaton is rotating, the upper bound reduces to O(nk+1) and the lower bound derived is Ω(nk+1) even for nondeterministic two-way finite automata. So, for rotating k-limited automata, the trade-off for the simulation is tight in the order of magnitude. Finally, we consider the simulation of k-limited automata over general alphabets by pushdown automata. It turns out that the cost is an exponential blow-up of the size. Furthermore, an exponential size is also necessary.

On the complexity and decidability of some problems involving shuffle

The complexity and decidability of various decision problems involving the shuffle operation (denoted by ⧢) are studied. The following three problems are all shown to be NP-complete: given a nondeterministic finite automaton (NFA) M, and two words u and v, is L(M)⊈u⧢v, is u⧢v⊈L(M), and is L(M)≠u⧢v? It is also shown that there is a polynomial-time algorithm to determine, for NFAs M1,M2, and a deterministic pushdown automaton M3, whether L(M1)⧢L(M2)⊆L(M3). The same is true when M1,M2,M3 are one-way nondeterministic l-reversal-bounded k-counter machines, with M3 being deterministic. Other decidability and complexity results are presented for testing whether given languages L1,L2, and R from various languages families satisfy L1⧢L2⊆R, and R⊆L1⧢L2. Several closure results on shuffle are also shown.

Genetic Algorithm for Grammar Induction and Rules Verification through a PDA Simulator

The focus of this paper is towards developing a grammatical inference system uses a genetic algorithm (GA), has a powerful global exploration capability that can exploit the optimum offspring. The implemented system runs in two phases: first, generation of grammar rules and verification and then applies the GA’s operation to optimize the rules. A pushdown automata simulator has been developed, which parse the training data over the grammar’s rules. An inverted mutation with random mask and then ‘XOR’ operator has been applied introduces diversity in the population, helps the GA not to get trapped at local optimum. Taguchi method has been incorporated to tune the parameters makes the proposed approach more robust, statistically sound and quickly convergent. The performance of the proposed system has been compared with: classical GA, random offspring GA and crowding algorithms. Overall, a grammatical inference system has been developed that employs a PDA simulator for verification.

Input-Driven Double-Head Pushdown Automata

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 and is called a signature. Since our automaton model works with two heads either both heads respect the same signature or each head owes its own signature. This results in two variants of input-driven double-head pushdown automata. The induced language families on input-driven double-head pushdown automata are studied from the perspectives of their language describing capability, their closure properties, and decision problems.

Algorithmic games for full ground references

We present a full classification of decidable and undecidable cases for contextual equivalence in a finitary ML-like language equipped with full ground storage (both integers and reference names can be stored). The simplest undecidable type is mathsf {unit}rightarrow mathsf {unit}rightarrow mathsf {unit}. At the technical level, our results marry game semantics with automata-theoretic techniques developed to handle infinite alphabets. On the automata-theoretic front, we show decidability of the emptiness problem for register pushdown automata extended with fresh-symbol generation.

Deterministic Stack Transducers

We introduce and investigate stack transducers, which are one-way stack automata with an output tape. A one-way stack automaton is a classical pushdown automaton with the additional ability to move the stack head inside the stack without altering the contents. For stack transducers, we distinguish between a digging and a non-digging mode. In digging mode, the stack transducer can write on the output tape when its stack head is inside the stack, whereas in non-digging mode, the stack transducer is only allowed to emit symbols when its stack head is at the top of the stack. These stack transducers have a motivation from natural-language interface applications, as they capture long-distance dependencies in syntactic, semantic, and discourse structures. We study the computational capacity for deterministic digging and non-digging stack transducers, as well as for their non-erasing and checking versions. We finally show that even for the strongest variant of stack transducers the stack languages are regular.

Realizability of concurrent recursive programs

We study the realizability problem for concurrent recursive programs: given a distributed system architecture and a sequential specification over words, find a distributed automata implementation that is equivalent to the specification. This problem is well-studied as far as finite-state processes are concerned, and it has a solution in terms of Zielonka’s Theorem. We lift Zielonka’s Theorem to the case where processes are recursive and modeled as visibly pushdown (or, equivalently, nested-word) automata. However, contrarily to the finite-state case, it is undecidable whether a specification is realizable or not. Therefore, we also consider suitable underapproximation techniques from the literature developed for multi-pushdown systems, and we show that they lead to a realizability framework with effective algorithms.

On the complexity of ω-pushdown automata

Finite automata over infinite words (called $\omega$-automata) play an important role in the automata-theoretic approach to system verification.Different types of $\omega$-automata differ in their succinctness and complexity of their emptiness problems, as a result, theory of $\omega$-automata hasreceived considerable research attention. Pushdown automata over infinite words (called $\omega$-PDAs), a generalization of $\omega$-automata, are a naturalmodel of recursive programs. Our goal in this paper is to conduct a relatively complete investigation on the complexity of the emptiness problems for variantsof $\omega$-PDAs. For this purpose, we consider $\omega$-PDAs of five standard acceptance types: Buchi, Parity, Rabin, Streett and Muller acceptances.Based on the transformation for $\omega$-automata and the efficient algorithm proposed by Esparza et al. in CAV00 for verifying the emptiness problem of $\omega$-PDAswith Buchi acceptance, it is trivial to check the emptiness problem of other $\omega$-PDAs. However, this naive approach is not optimal. In this paper,we propose novel algorithms for the emptiness problem of $\omega$-PDAs based on the observations of the structure of accepting runs. Our algorithms outperform algorithmsthat go through Buchi PDAs. In particular, the space complexity of the algorithm for Streett acceptance that goes through Buchi acceptance is exponential,while ours is polynomial. The algorithm for Parity acceptance that goes through Buchi acceptance is in ${\boldsymbol O}(k^3n^2m)$ time and ${\boldsymbol O}(k^2nm)$ space,while ours is in ${\boldsymbol O}(kn^2m)$ time and ${\boldsymbol O}(nm)$ space, where $n$ (resp. $m$ and $k$) is the number of control states (resp. transitions and index).Finally, we show that our algorithms yield a better solution for the pushdown model checking problem against linear temporal logic with fairness.

Deterministic Deep Pushdown Transducer and its Parallel Version

Deterministic Deep Pushdown Transducer and its Parallel Version