Stack Automata Research Articles

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.

Variations of checking stack automata: Obtaining unexpected decidability properties

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, i.e., the membership problem for both versions are either decidable or undecidable, and the same holds for the emptiness problem. As far as we know, the model we introduce above is the first one-way model to exhibit properties (1) and (2). We define another family of one-way acceptors where the nondeterministic version has an undecidable emptiness problem, but the deterministic version has a decidable emptiness problem. We also know of no other model with this property in the literature. We also investigate decidability properties of other variations of checking stack automata (e.g., allowing multiple stacks, two-way input, etc.). Surprisingly, two-way deterministic machines with multiple checking stacks and multiple reversal-bounded counters are shown to have a decidable membership problem, a very general model with this property.

Tinput-Driven Pushdown, Counter, and Stack Automata

Tinput-Driven Pushdown, Counter, and Stack Automata

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.

Prediction of Infinite Words with Automata

In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the predictor’s guesses are correct. In this paper we consider the case in which the predictor is an automaton and the emitted values are drawn from a finite set; i.e., the emitted sequence is an infinite word. We examine the predictive capabilities of finite automata, pushdown automata, stack automata (a generalization of pushdown automata), and multihead finite automata. We relate our predicting automata to purely periodic words, ultimately periodic words, and multilinear words, describing novel prediction algorithms for mastering these sequences.

GROUPS WITH INDEXED CO-WORD PROBLEM

We investigate co-indexed groups, that is groups whose co-word problem (all words defining nontrivial elements) is an indexed language. We show that all Higman–Thompson groups and a large class of tree automorphism groups defined by finite automata are co-indexed groups. The latter class is closely related to dynamical systems and includes the Grigorchuk 2-group and the Gupta–Sidki 3-group. The co-word problems of all these examples are in fact accepted by nested stack automata with certain additional properties, and we establish various closure properties of this restricted class of co-indexed groups, including closure under free products.

Tree-stack automata

We introduce a new model of stack automata, the “tree-stack automata,” extending the linear stack to a tree-stack. A main subject of our investigations is to explore the relationship between tree-stack automata and stack automata. The main result of this paper is that tree-stack have the same recognition power as stack-pushdown automata, another (well-known) extension of stack automata. Therefore we obtain that the class of languages accepted by the one-way (linear) stack automata is a proper subset of the class of languages accepted by the one-way tree-stack automata and that two-way tree-stack automata have the same recognition power as two-way (linear) stack automata. As a special case of tree-stack automata we consider tree-pushdown automata. As opposed to stack automata the tree-pushdown storage does not extend the recognition power of one-way (resp. two-way) pushdown automata.

Computational complexity of symbolic dynamics at the onset of chaos.

In a variety of studies of dynamical systems, the edge of order and chaos has been singled out as a region of complexity. It was suggested by Wolfram, on the basis of qualitative behavior of cellular automata, that the computational basis for modeling this region is the universal Turing machine. In this paper, following a suggestion of Crutchfield, we try to show that the Turing machine model may often be too powerful as a computational model to describe the boundary of order and chaos. In particular we study the region of the first accumulation of period doubling in unimodal and bimodal maps of the interval, from the point of view of language theory. We show that in relation to the "extended" Chomsky hierarchy, the relevant computational model in the unimodal case is the nested stack automaton or the related indexed languages, while the bimodal case is modeled by the linear bounded automaton or the related context-sensitive languages.

The parser generating system PGS

AbstractPGS is a parser generating system accepting LALR(1) and related grammars in extended BNF notation and producing parsers based on table‐driven stack automata. To enable syntax‐directed translation, semantic actions can be attached to rules of the input grammar. An attribution mechanism allows the transfer of information between rules. The generated parsers have an automatic error recovery which can be tailored to satisfy specific needs of the language to be accepted. PGS generates parsers written in Pascal, Modula‐2, C or Ada. Compared with existing systems, e.g. YACC1, a parser generated by PGS is twice as fast and the parse tables require 25 per cent less storage. This paper gives a survey of algorithms involved in the generator and the generated parsers, and compares them with algorithms used in other systems. In detail, it compares several parse‐table representations and their implications for space and time efficiency of the generated parsers.

Some characterizations of multihead finite automata

We give characterizations of multihead two-way finite automata in terms of multihead reversal-bounded pushdown automata and restricted checking stack automata. In particular, we show that a language is accepted by a 2 k -head two-way finite automaton if and only if it is accepted by a k -head two-way pushdown automaton which makes one reversal on its stack. We also show that a 2-head two-way deterministic finite automaton is equivalent to a simple type of two-way deterministic checking stack automaton. This is in contrast to a previously known result which shows that simple two-way nondeterministic checking stack automata are equivalent to nondeterministic linear bounded automata.

Real‐time shuffle stack automaton

AbstractAs models to describe concurrent systems, Petri net, shuffle stack automaton (SSA), flow expression, event expression, and shuffle grammar have been proposed. SSA is an automaton which is obtained by adding to the finite automaton a stack with shuffle function. Its descriptive power has already been discussed. This paper discusses the real‐time shuffle stack automaton (RSSA, which is an SSA without λ‐transition in regard to the input), which is a subclass of SSA. It is shown that RSSA has the same descriptive power, independently of the accepting mode (empty‐stack acceptance or final‐state acceptance), and of whether or not a terminating symbol is provided at the end of the input string. It is also shown that the descriptive power of RSSA is incomparable to that of the push‐down automaton, but is properly contained in that of the linear bounded automaton. If a language is bounded and if the set obtained by applying Parikh mapping to that language is semi‐linear, the language is accepted by RSSA. Another property of the language accepted by RSSA is that if it is a bounded context‐free language, it is accepted by RSSA.

Extended macro grammars and stack controlled machines

K-extended basic macro grammars are introduced, where K is any class of languages. The class B(K) of languages generated by such grammars is investigated, together with the class LB(K) of languages generated by the corresponding linear basic grammars. For any full semi-AFL K, B(K) is a full AFL closed under iterated LB(K)-substitution, but not necessarily under substitution. For any machine type D, the stack controlled machine type corresponding to D is introduced, denoted S(D), and the checking-stack controlled machine type CS(D). The data structure of this machine is a stack which controls a pushdown of data structures from D. If D accepts K, then S(D) accepts B(K) and CS(D) accepts LB(K). Thus the classes B(K) are characterized by stack controlled machines and the classes LB(K), i.e., the full hyper-AFLs, by checking-stack controlled machines. A full basic-AFL is a full AFL K such that B( K) ⊆ K. Every full basic-AFL is a full hyper-AFL, but not vice versa. The class of OI macro languages (i.e., indexed languages, i.e., nested stack automaton languages) is a full basic-AFL, properly containing the smallest full basic-AFL. The latter is generated by the ultrabasic macro grammars and accepted by the nested stack automata with bounded depth of nesting (and properly contains the stack languages, the ETOL languages, i.e., the smallest full hyper-AFL, and the basic macro languages). The full basic-AFLs are characterized by bounded nested stack controlled machines.

(Semi)alternating stack automata

New results concerning the space complexity of languages accepted by stack automata, alternating stack automata, and alternating pushdown automata are derived. Some of the results generalize previously known results.

A note on dpda transductions of {0,1}∗and inverse dpda transductions of the dyck set

It is shown that the simple deterministic pushdown transductions of {0,1}∗ equal the class of context free languages (by means of an elementary proof). Several new results are presented concerning the class of source languages which can be compiled into Dyck languages by a DPDA transducer, i.e., the class DPDT-1 (D 2). It is shown that this language family does not 2 have the Parikh property and is incomparable with the unambiguous context free languages, the bounded context free languages, the linear context free languages, the one-way deterministic stack automaton languages and the one-way non-deterministic stack automaton languages.

Complexity results for classes of quantificational formulas

We analyze the computational complexity of determining whether F is satisfiable when F is a formula of the classical predicate calculus obeying certain syntactic restrictions. For example, for the monadic predicate calculus and the Gödel or 3 … ∃∀∀∃ … 3 prefix class we obtain lower and upper nondeterministic time bounds of the form c n/log n . The lower bounds are established by using acceptance problems for time-bounded Turing machines and alternating pushdown and stack automata.

Tree transducers, L systems, and two-way machines

A relationship between parallel rewriting systems and two-way machines is investigated. Restrictions on the “copying power” of these devices endow them with rich structuring and give insight into the issues of determinism, parallelism, and copying. Among the parallel rewriting systems considered are the top-down tree transducer; the generalized syntax-directed translation scheme and the ETOL system, and among the two-way machines are the tree-walking automaton, the two-way finite-state transducer, and (generalizations of) the one-way checking stack automaton. The. relationship of these devices to macro grammars is also considered. An effort is made .to provide a systematic survey of a number of existing results.

Space-bounded complexity classes and iterated deterministic substitution

We investigate the effect on the space complexity when a language family K is extended by means of iterated λ-free deterministic substitution to the family η(K). If each language in K is accepted by a one-way nondeterministic multi-tape Turing machine within space S(n) for some monotonic space bound S(n) ⩾ log n, then η(K) is included in NSPACE(S(n)). Thus for each monotonic space bound S(n) ⩾ n, the inclusion K ⊆ NSPACE(S(n)) implies that η(K) is also included in NSPACE(S(n)). An implication similar to the latter one also holds for DSPACE(S(n)). Consequently, some well-known space-bounded complexity classes such as the families of (non)deterministic context-sensitive languages, of two-way (non)deterministic nonerasing stack automaton languages, and PLPACE are AFL's closed under intersection and iterated λ-free deterministic substitution. On the other hand no complexity class which includes DSPACE(log n) is closed under controlled iterated λ-free (non) deterministic substitution.

Bounded nesting in macro grammars

Macro grammars and context-free tree grammars are discussed in which a bound is put (dynamically or statically) on the nesting-depth of nested calls of nonterminals. The dynamic and static restrictions are closely related (and both are related to the nested stack automaton with bounded nesting of stacks). The corresponding classes of tree languages have derivation bounded contextfree languages as path-languages. Macro languages exist which cannot be generated by nesting-depth bounded macro grammars.

Stack languages and log n space

We consider the space complexity of stack languages. The main result is: if a language is accepted by a deterministic (nondeterministic) one-way stack automaton then it is the image under a nonerasing homomorphism of a language accepted by a deterministic (nondeterministic) Turing machine that operates within space log n.

On the complexity of finite, pushdown, and stack automata

The complexity of predicates on several classes of formal languages is studied. For finite automata, pushdown automata, and several classes of stack automata, every nontrivial predicate on the languages accepted by the 1-way devices requires as much time and space as the recognition problem for any language accepted by the corresponding 2-way devices. Moreover there are nontrivial predicates on the languages accepted by the 1-way devices such that is the accepted language of some corresponding one or two head 2-way device. Thus our lower bounds are fairly tight.