Jonathan Goldstine

On measuring nondeterminism in regular languages

Abstract It is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language. This paper studies situations intermediate between forbidding nondeterminism and allowing it. The amount of nondeterminism used by a finite automaton is quantified, so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied. These intermediate situations are shown always to lie between two extremes: (1) there are no savings as the amount of nondeterminism increases incrementally, so that savings occur only when the amount of nondeterminism becomes unlimited; (2) each increment of nondeterminism results in additional savings, the number s of states decreasing approximately as s 1 i , until exponential savings have been achieved after about i = log s log log s increments.

On the relation between ambiguity and nondeterminism in finite automata

Abstract Nondeterminism in a finite automaton is measured dynamically by counting the number of guesses that the automaton has to make in order to recognize an input string. When the amount of nondeterminism is small (bounded) or large (linear in the input length), nothing can be concluded about the amount of ambiguity in the automaton. But when the amount of nondeterminism is intermediate between these extremes, the degree of ambiguity must be infinite.

A pushdown automation or a context-free grammar—which is more economical?

Abstract For every pair of positive integers n and p, there is a language accepted by a real-time deterministic pushdown automaton with n states and p stack symbols and size O(np), for which every context-free grammar needs at least n2p+1 nonterminals if n>1 (or p non-terminals if n = 1). It follows that there are context-free languages which can be recognized by pushdown automata of size O(np), but which cannot be generated by context-free grammars of size smaller than O(n2p); and that the standard construction for converting a pushdown automaton to a context-free grammar is optimal in the sense that it infinitely often produces grammars with the fewest number of nonterminals possible.

A simplified proof of parikh's theorem☆

Abstract A strengthened form of the pumping lemma for context-free languages is used to give a simple proof of Parikhs Theorem.

FORMAL LANGUAGES AND THEIR RELATION TO AUTOMATA: WHAT HOPCROFT & ULLMAN DIDN'T TELL US

Publisher Summary This chapter discusses the relationship between formal languages and automata. The relationship is a weak one and proceeds in only one direction. Automata are used as acceptors to define languages; therefore, the languages can be considered the external behavior of their acceptors and that end the relationship. There is no application of results from language theory to the study of automata. The chapter explains that there are deficiencies in the theory of automata and the deficiencies are reparable. The automata may be more intuitive than that of grammars; however, automata are so much more complicated that they would be used in formal proofs only with great awkwardness and when a proof by means of grammars is for some reason not feasible. A finite-state automaton is not always the best way to describe a regular set. There is no single method of representing regular sets that is always the most convenient to use.

On reducing the number of stack symbols in a PDA

Two transformations are presented which, for any pushdown automaton (PDA)M withn states andp stack symbols, reduce the number of stack symbols to any desired numberp′ greater than one. The first transformation preserves deterministic behavior and produces an equivalent PDA witho(np/p′) states. The second construction, using a technique which introduces nondeterminism, produces an equivalent PDA withO(n√p/p′) states. Both transformations are essentially optimal, the former among determinism-preserving transformations, the latter among all transformations.

On reducing the number of states in a PDA

A transformation is presented which converts any pushdown automaton (PDA)M0 withn0 states andp0 stack symbols into an equivalent PDAM withn states and ⌈n0/n⌉2p0 stack symbols into an equivalent ofn, 1⩽n<n0. This transformation preserves realtime behavior but not derterminism. The transformation is proved to be the best possible one in the following sense: for each choice of the parametersn0 + 1 stack symbols for any desired value realtime PDAM0 such that any equivalent PDAM (whether realtime or not) havingn states must have at least ⌈(n0/n)2 p0⌉ stack symbols. Furthermore, the loss of deterministic behavior cannot be avoided, since for each choice ofn0 andp0, there is a deterministic PDAM0 such that no equivalent PDAM with fewer states can be deterministic.

Measuring Nondeterminism in Pushdown Automata

The amount of nondeterminism that a pushdown automaton requires to recognize an input string can be measured by the minimum number of guesses that it must make to accept the string, where guesses are measured in bits of information. When this quantity is unbounded, the rate at which it grows as the length of the string increases serves as a measure of the pushdown automatons “rate of consumption” of non-determinism. We show that this measure is similar to other complexity measures in that it gives rise to an infinite hierarchy of complexity classes of context-free languages differing in the amount of this resource (nondeterminism) that they require. In addition, we show that there are contextfree languages that can only be recognized by a pushdown automaton whose nondeterminism grows linearly, resolving an open problem in the literature. In particular, {ww R : we {a,b}}*} is such a language.

Intersection-closed full AFL and the recursively enumerable languages

A study is made of conditions on a language L which ensure that the smallest intersection-closed full AFL containing L (written @@@@ @@@@(L)) does or does not contain all recursively enumerable languages. For example, it is shown that if L &euil; {a<supscrpt>n</supscrpt><subscrpt>i</subscrpt>/i @@@@ 0}and [equation]inf <supscrpt>n</supscrpt>i+1/n<subscrpt>i</subscrpt>>1, then [equation](L) contains all recursively enumerable languages. On the other hand, it is shown that if L @@@@ a<supscrpt>*</supscrpt> and the ratio of the number of words in L of length less than n to n goes to 1 as [equation], then [equation] does not contain all recursively enumerable languages.

On the equality of grammatical families

Abstract A decision procedure involving the testing for containment is presented for determining whether two grammatical families (families generated by context-free grammar forms) are equal. It is also shown that every nontrivial grammatical family which is a proper subset of the family of context-free languages can be constructed in a unique canonical way from the family of regular sets by certain operations. Thus, two such grammatical families are equal iff their canonical representations are identical.