Detlef Wotschke

Amounts of nondeterminism in finite automata

SummaryThe amount of nondeterminism in a nondeterministic finite automaton (NFA) is measured by counting the minimal number of “guessing points” a string w has to pass through on its way to an accepting state. NFAs with more nondeterminism can achieve greater savings in the number of states over their deterministic counterparts than NFAs with less nondeterminism. On the other hand, for some nontrivial infinite regular languages a deterministic finite automaton (DFA) can already be quite succinct in the sense that NFAs need as many states (and even context-free grammars need as many nonterminals) as the minimal DFA has states.

Concise description of finite languages

Abstract This paper is a contribution to the theory of grammatical complexity, in particular to the following basic question: consider some language L and grammars of some type X; what is the smallest number of productions of a type X grammar required to generate L? This complexity measure, the so-called X complexity of L, has been investigated before. We study the more basic case when the languages L considered are finite (a case which has been neglected, so far). We obtain a number of results and insights suggesting that such study is of importance. In addition to a number of ‘expected’ (but not necessarily easy to prove) results that with type X grammars more productions are necessary than with some other type Y grammars (even if types X and Y define the same family of languages) we show that if the limit of certain sequences of finite languages is of type X1 then the type X complexity of each of the finite languages involved must be low.

Nondeterminism and boolean operations in pda's

Abstract There are nondeterministic context-free languages that cannot be expressed as a Boolean formula over deterministic context-free languages. The closure of the context-free languages under intersection does not yield closure under complementation.

On strict interpretations of grammar forms

Strict interpretations of grammar forms are studied with respect to parsing, ambiguity, and decidability for intersection and containment. In particular, a parsing procedure for an arbitrary strict interpretation grammar is given which is based on a given parsing method for the master grammar. Time and space bounds on the new procedure are then obtained. Each ambiguous interpretation grammar of an unambiguous grammar form can be converted to an equivalent unambiguous interpretation of the same grammar form. Unambiguity is always decidable for strict interpretation grammars of unambiguous grammar forms. Also, for languages obtained from “compatible” strict interpretations of an unambiguous grammar form, the following problems are solvable: empty intersection, finite intersection, containment, and equality.

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.

Degree-languages: A new concept of acceptance

The concept of acceptance degree is introduced. A string is accepted by a recognition device if and only if its acceptance degree is not less than a preselected rational number. A degree-language is the set of all those strings. The class of degree-languages over push-down automata (linear bounded automata) contains the intersection-closure and the complements of the context-free (context-sensitive) languages and it is contained in the Boolean closure of the context-free (context-sensitive) languages. Consequently degree-languages over pushdown automata can be recognized as fast as context-free languages. The class of context-sensitive languages is closed under complementation if and only if every degree-language over a linear bounded automaton is either context-sensitive or is the complement of a context-sensitive language.

Degree-languages, polynomial time recognition, and the LBA problem

The so-called Chomsky hierarchy [5], consisting of regular, context-free, context-sensitive, and recursively enumerable languages, does not account for many “real world” classes of languages, e.g., programming languages and natural languages [4]. This is one of the reasons why many attempts have been made to “refine” the original Chomsky classification. The main goal has been to describe languages which, for instance, are not context-free but are still context-sensitive, without using the powerful and complex concept of context-sensitive grammars.

States Can Sometimes Do More Than Stack Symbols in PDA's

In pushdown automata, states can sometimes do more than stack symbols. More precisely, reducing the state set by a factor of k may require an increase in the stack alphabet by a factor of k2. These results are based on the observation that the triple construction for converting a pushdown automaton into a context-free grammar is optimal.

The influence of productions on derivations and parsing (Extended Abstract)

The concept of grammar forms [4,5] provides evidence that there seems to be no way to base the definitions of many grammar types used in parsing and compiling solely on the concept of productions.n Strict interpretations, as introduced in [3,5], of unambiguous or LR(k) grammar forms generate unambiguous or LR(k) languages, respectively. This is not true in the LL(k) case.n It is decidable whether a strict interpretation of an unambiguous grammar form is unambiguous. For any two compatible strict interpretations G<subscrpt>1</subscrpt> and G<subscrpt>2</subscrpt> of an unambiguous grammar form it is decidable whether L(G<subscrpt>1</subscrpt>)@@@@L(G<subscrpt>2</subscrpt>), L(G<subscrpt>1</subscrpt>)@@@@L(G<subscrpt>2</subscrpt>)&equil;&thgr;, finite, or infinite.n For every grammar form F<subscrpt>1</subscrpt> there exists a grammar form F<subscrpt>2</subscrpt> such that the grammatical family of F<subscrpt>1</subscrpt> under unrestricted interpretations is equal to the grammatical family of F<subscrpt>2</subscrpt> under strict interpretations.

A note on classes of complements and the LBA problem

SummaryThe complements of an AFL ℒ form an AFL if and only if ℒ is closed under “length-preserving” universal quantification. The complements of the context-sensitive languages form a principal AFL with a hardest set L1. The context-sensitive languages are closed under complementation if and only if L1 is context-sensitive.