Empty Stack Research Articles

Rubik Tables and object rearrangement

A great number of robotics applications demand the rearrangement of many mobile objects, for example, organizing products on store shelves, shuffling containers at shipping ports, reconfiguring fleets of mobile robots, and so on. To boost the efficiency/throughput in systems designed for solving these rearrangement problems, it is essential to minimize the number of atomic operations that are involved, for example, the pick-n-places of individual objects. However, this optimization task poses a rather difficult challenge due to the complex inter-dependency between the objects, especially when they are tightly packed together. In this work, in tackling the aforementioned challenges, we have developed a novel algorithmic tool, called Rubik Tables, that provides a clean abstraction of object rearrangement problems as the proxy problem of shuffling items stored in a table or lattice. In its basic form, a Rubik Table is an n × n table containing n2 items. We show that the reconfiguration of items in such a Rubik Table can be achieved using at most n column and n row shuffles in the partially labeled setting, where each column (resp., row) shuffle may arbitrarily permute the items stored in a column (resp., row) of the table. When items are fully distinguishable, additional n shuffles are needed. Rubik Tables allow many generalizations, for example, adding an additional depth dimension or extending to higher dimensions. Using Rubik Table results, we have designed a first constant-factor optimal algorithm for stack rearrangement problems where items are stored in stacks, accessible only from the top. We show that, for nd items stored in n stacks of depth d each, using one empty stack as the swap space, O(nd) stack pop-push operations are sufficient for an arbitrary reconfiguration of the stacks where for arbitrary fixed m > 0. Rubik Table results also allow the development of constant-factor optimal solutions for solving multi-robot motion planning problems under extreme robot density. These algorithms based on Rubik Table results run in low-polynomial time.

Note on the dominance rules in the exact algorithm for the container pre-marshalling problem by Tanaka & Tierney (2018)

In this note, we identify an over-pruning issue that occurs in the iterative deepening branch-and-bound algorithm by Tanaka and Tierney (2018). We show that this over-pruning issue is caused by an inconsistency between two dominance rules, namely, the transitive move rule and the empty stack rule, and it will in turn cause the exact algorithm to produce incorrect optimal solutions. To address this issue, we propose a new concept termed the lexicographic dominance principle, and prove that all dominance rules under the lexicographic dominance principle are mutually consistent. Finally, we re-examine all the dominance rules described in the referred paper, and present a compatible set of dominance rules to guarantee mutual consistency.

Formula omitted] Watson-Crick pushdown automata

Watson-Crick automata work on two-stranded tape, and, consequently, they have two reading heads. In the case of sensing 5′→3′ Watson-Crick automata, the two heads start from different ends of the input and they move in opposite directions until they meet. In this paper, an extension of these automata is presented by adding a pushdown stack. This model can also be seen as a 2-head extension of the pushdown automata. Acceptance by empty stack and acceptance by final states are proven to have the same recognition power in this new model. The language family recognized by this model contains only semi-linear context-sensitive languages and it includes the class of context-free languages. Some well-known non-context-free languages are shown as examples. Normal form like results, a pumping lemma and various closure properties are also proven.

Fuzzy Pushdown Termination Games

The computational study on finite/infinite-state systems, probabilistic systems, and finite-state fuzzy systems, has received much attention recently. In contrast, there are very few results for algorithmic analysis of infinite-state fuzzy systems. In this paper, we introduce fuzzy pushdown termination games (FPDTGs), which are an extension of fuzzy pushdown automata with a game feature and can serve as a formal model of infinite-state fuzzy systems. We investigate some computational issues of the games under termination objectives for two players: the goal of player-1 is to maximize the truth value of eventually terminating at some given configurations with the empty stack, while player-2 aims at the opposite. Some interesting results are obtained. For example, we show that both players have optimal memoryless strategies and the same value. The problem of computing the value can be solved in exponential time when the triangular norm is chosen as the minimum one. Furthermore, we present efficient algorithms for computing the values of two special subclasses of FPDTGs. The potential for practical use of our model is demonstrated by a case study on a manufacturing system.

LR(0) conjunctive grammars and deterministic synchronized alternating pushdown automata

The paper introduces a subfamily of synchronized alternating pushdown automata, deterministic synchronized alternating pushdown automata, and a subfamily of conjunctive grammars, LR(0)conjunctive grammars. It is shown that deterministic synchronized alternating pushdown automata and LR(0) conjunctive grammars have the same recognition/generation power, analogously to the classical equivalence between acceptance by empty stack of deterministic pushdown automata and LR(0) grammars. These models form the theoretical basis for efficient linear time parsing of a subfamily of conjunctive languages which properly includes the classical LR(0) languages.

The algebraic characterizations for a formal power series over complete strong bimonoids.

On the basis of run semantics and breadth-first algebraic semantics, the algebraic characterizations for a classes of formal power series over complete strong bimonoids are investigated in this paper. As recognizers, weighted pushdown automata with final states (WPDAs for short) and empty stack (WPDAs^{emptyset }) are shown to be equivalent based on run semantics. Moreover, it is demonstrated that for every WPDA there is an equivalent crisp-simple weighted pushdown automaton with final states by run semantics if the underlying complete strong bimonoid satisfies multiplicatively local finiteness condition. As another type of generators, weighted context-free grammars over complete strong bimonoids are introduced, which are proven to be equivalent to WPDAs^{emptyset } based on each one of both run semantics and breadth-first algebraic semantics. Finally examples are presented to illuminate the proposed methods and results.

On Intuitionistic Fuzzy Context-Free Languages

Taking intuitionistic fuzzy sets as the structures of truth values, we propose the notions of intuitionistic fuzzy context-free grammars (IFCFGs, for short) and pushdown automata with final states (IFPDAs). Then we investigate algebraic characterization of intuitionistic fuzzy recognizable languages including decomposition form and representation theorem. By introducing the generalized subset construction method, we show that IFPDAs are equivalent to their simple form, called intuitionistic fuzzy simple pushdown automata (IF-SPDAs), and then prove that intuitionistic fuzzy recognizable step functions are the same as those accepted by IFPDAs. It follows that intuitionistic fuzzy pushdown automata with empty stack and IFPDAs are equivalent by classical automata theory. Additionally, we introduce the concepts of Chomsky normal form grammar (IFCNF) and Greibach normal form grammar (IFGNF) based on intuitionistic fuzzy sets. The results of our study indicate that intuitionistic fuzzy context-free languages generated by IFCFGs are equivalent to those generated by IFGNFs and IFCNFs, respectively, and they are also equivalent to intuitionistic fuzzy recognizable step functions. Then some operations on the family of intuitionistic fuzzy context-free languages are discussed. Finally, pumping lemma for intuitionistic fuzzy context-free languages is investigated.

Automata theory based on complete residuated lattice-valued logic: Pushdown automata

Automata theory based on complete residuated lattice-valued logic, called L -valued automata, has been proposed by Qiu [Automata theory based on complete residuated latticed-valued logic, Sci. China (Ser. F) 44 (6) (2001) 419–429; Automata theory based on complete residuated latticed-valued logic (II), Sci. China (Ser. F) 45 (6) (2002) 442–452]. In this paper, we discuss some properties of L -valued context-free grammars, languages, and pushdown automata. We show that, for such grammars, Chomsky and Greibach Normal Forms can be equivalently constructed, and we also prove that the languages accepted by final states and by empty stack in L -valued pushdown automata are equivalent. In particular, we prove the equivalence between L -valued context-free grammars and L -valued pushdown automata.

Polynomial Time Identification of Strict Deterministic Restricted One-Counter Automata in Some Class from Positive Data

A deterministic pushdown automaton (dpda) having just one stack symbol is called a deterministic restricted one-counter automaton (droca). When it accepts an input by empty stack, it is called strict. This paper is concerned with a subclass of real-time strict droca's, called Szilard strict droca's, and studies the problem of identifying the subclass in the limit from positive data. The class of languages accepted by Szilard strict droca's coincides with the class of Szilard languages (or, associated languages) of strict droca's and is incomparable to each of the class of regular languages and that of simple languages. After providing some properties of languages accepted by Szilard strict droca's, we show that the class of Szilard strict droca's is polynomial time identifiable in the limit from positive data in the sense of Yokomori. This identifiability is proved by giving an exact characteristic sample of polynomial size for a language accepted by a Szilard strict droca. The class of very simple languages, which is a proper subclass of simple languages, is also proved to be polynomial time identifiable in the limit from positive data by Yokomori, but it is yet unknown whether there exists a characteristic sample of polynomial size for any very simple language.

Automata theory based on quantum logic: Reversibilities and pushdown automata

Automata theory based on quantum logic, called l - valued finite automata ( l -VFAs), may be viewed as a logical approach to quantum computing. This work is mainly divided into two parts: one part deals with reversibility of l -VFAs, and the other establishes a basic framework of l - valued pushdown automata ( l -VPDAs). First we provide some preliminaries concerning quantum logic and l -VFAs, and we prove a useful property of l -valued successor and source operators. Then we clarify the relationships between various reversibilities closely related to quantum finite automata in the literature. In particular, we define a reversibility of l -VFAs which is termed as retrievability, and we clarify the relationships between a number of different fashions regarding retrievability of l -VFAs. We prove that some of them are equivalent, but for the others to be equivalent the truth-value set is required to satisfy a certain condition. This is an essential difference from the classical situation. Afterwards, we introduce l -VPDAs and show that the class of the languages accepted by l -VPDAs by empty stack coincides with that accepted by l -VPDAs by final state. Finally, we provide some examples of l -VFAs and conclude with some remarks.

Temporal Constraints for Concurrent Objects: Extended Abstract

to refer to such problems [3].Commonly,in object oriented code,the set of messages accepted by anobject is not uniform in time. Depending on the object’s state,some of itsmethods will be unavailable,as e.g.,pop from a empty stack,orput on afull buffer. To avoid inconsistencies in a concurrent setting,programmersuse

Quantum Pushdown Automata

Quantum automata are mathematical models for quantum computing. We analyze the existing quantum pushdown automata, propose a q quantum pushdown automata (qQPDA), and partially clarify their connections. We emphasize some advantages of our qQPDA over others. We demonstrate the equivalence between qQPDA and another QPDA. We indicate that qQPDA are at least as powerful as the QPDA of Moore and Crutchfield with accepting words by empty stack. We introduce the quantum languages accepted by qQPDA and prove that every η-q quantum context-free language is also an η′-q quantum context-free language for any η ∈ (0, 1) and η′ ∈ (0, 1).

A NEW TYPE OF PUSHDOWN AUTOMATA ON INFINITE TREES

In this paper we consider pushdown automata on infinite trees with empty stack as the accepting condition (ω-EPDTA). We provide the following regarding ω-EPDTA: (a) its relationship to other Pushdown automata on infinite trees, (b) a Kleene-Closure theorem and (c) a single exponential time algorithm for checking emptiness. We demonstrate the usefulness of ω-EPDTA through two example applications: defining the temporal uniform inevitability property and specifying a context-free process with unbounded state space, both of which cannot be defined and/or specified by the classical finite state automata on infinite trees. We also discuss the relevance of the results presented here to model-checking.

Probabilistic analysis of some distributed algorithms

In this paper we analyze: (i) a storage allocation algorithm (D.E. Knuth, The Art of Computer Programming-Vol. I, Addison-Wesley, Reading, MA, 1969, Ex. 2.2.2.13) which permits one to maintain two stacks inside a shared (continuous) memory area of a fixed size, and (ii) the well-known banker algorithm which plays a fundamental role in parallel processing (J. Francon, Combinatoire et Parallelisme; Journees Informatique et Mathematique; Lumigny, October 15-17, 1987; A. N. Habermann, in Current Trends in Programming Methodology, Vol. 3, K. M. Chandy and R. T. Yeh, Eds., Prentice-Hall, Englewood Cliffs, NJ, 1987; J. Peterson and A. Silberschatz, Operating Systems Concepts, Addison-Wesley, Reading, MA, 1983). The natural formulation of the problems to be solved here is in terms of random walks. For (i) the random walk Ym(-) takes place in a triangle in a two-dimensional lattice space with two reflecting barriers along the axes (a deletion has no effect on an empty stack) and one absorbing barrier along the diagonal (the algorithm stops when the combined sizes of the stacks exhaust the available storage). For (ii) the random walk takes place in a rectangle with broken corner and has four reflecting barriers and one absorbing barrier (see Fig. 10). With the help of diffusion techniques, we obtain, asymptotically as m: ∞. the distributions of the hitting place (Zm) and hitting time (Tm) on the absorbing boundary. the joint distribution of Zm and Tm. the distribution P[Ym(n) ≦ ym, n < Tm].

A direct branching algorithm for checking equivalence of some classes of deterministic pushdown automata

A new direct algorithm is presented for checking equivalence of some classes of deterministic pushdown automata (dpda's), after Korenjak and Hopcroft's branching algorithm. It is not only powerful enough to be applicable to two dpda's accepting by empty stack, one of which is real-time, but also simple even for dpda's in lower subclasses. This is the first time the branching algorithm has been used to give such a general decision procedure without ever “mixing” the two languages in question. In other words, it deals with only the equivalence equation whose left-hand side consists of a pure reachable configuration of one dpda and whose right-hand side that of the other.

A simplicity test for deterministic pushdown automata

A context-free language is said to be simple if it is accepted by a single-state deterministic pushdown automaton with empty stack acceptance. This paper proves that it is decidable whether a deterministic pushdown automaton (dpda) accepts a simple language. For this purpose, we prove that it is decidable whether a real-time dpda with empty stack acceptance accepts a simple language. To prove this, we present two conditions called nonsingularity and separability conditions and show that for a real-time dpda M with empty stack acceptance, (i) the language accepted by M is simple if and only if M satisfies the two conditions, and (ii) it is decidable whether M satisfies the two conditions.

A real-time strictness test for deterministic pushdown automata

A context-free language is said to be real-time strict if it is accepted by a real-time deterministic pushdown automaton with empty stack acceptance. This paper proves that there exists an effective procedure for deciding whether the language accepted by a deterministic pushdown automaton (dpda) is real-time strict. The result implies the following: For any subclass ℰ of real-time dpda's with empty stack acceptance, the problem for deciding whether a dpda is equivalent to some machine in ℰ reduces to that for deciding whether a real-time dpda with empty stack acceptance is equivalent to some machine in ℰ.

The equivalence problem for real-time strict deterministic languages

The equivalence problem is decidable for deterministic real-time pushdown store automata accepting by empty stack. Further, the equivalence problem is decidable for two deterministic pushdown store automata, one of which is real-time accepting by empty stack.

One-way weak-stack-counter automata

Stack-counter automata are defined as one-way stack automata with a storage alphabet of just one symbol and the ability to recognize the left (bottom) and right (top) ends of the stack. Weak-stack-counter automata considered in this paper are stack-counter automata which do not have the ability to recognize the left end of the stack. We show that stack-counter automata are more powerful than weak-stack-counter automata. This answers the open question given by S. Ginsburg and G. F. Rose (J. Comput. System Sci. 8 (1974), 243–269). We also show that nondeterministic weak-stack-counter automata that accept by simultaneous terminal (final) state and empty stack are more powerful than those that accept by terminal state alone. This contrasts with the case of stack-counter automata, for which both methods are equally powerful.

A result on the equivalence problem for deterministicpushdown automata

This paper shows that it is decidable whether or not two deterministic pushdownautomata, one of which is nonsingular (introduced by L. G. Valiant), are equivalent (in the sense that they accept the same languages by empty stack). This is an extension of Valiant's result that the equivalence of two nonsingular deterministic pushdown automata is decidable.