E. H. Spanier

Finite-Turn Pushdown Automata

Abstract : A finite-turn pda is a pda in which the length of the pushdown tape alternatively increases and decreases at most a fixed bounded number of times during any sweep of the automation. This paper is a study of these finite-turn pda and the context free languages they recognize. These context free languages are characterized both in terms of grammars (two ways) and in terms of generation from finite sets by three operations. A decision procedure is given for determining if an arbitrary pda is a finite-turn pda. There is no decision procedure for determining if an arbitrary context free language is accepted by some finite-turn pda. (Author)

Derivation-bounded languages

A derivation in a phrase-structure grammar is said to be k-bounded if each word in the derivation contains at most k occurrences of nonterminals. A set L is said to be derivation bounded if there exists a phrase-structure grammar G and a positive integer k such that L is the set of words in the language generated by G which have some k-bounded derivation. The main result is that every derivation-bounded set is a contextfree language. Various characterizations of the derivation-bounded languages are then given. For example, the derivation-bounded languages coincide with the standard matching-choice sets discussed by Yntema. They also coincide with the smallest family of sets containing the linear context-free languages and closed under arbitrary substitution.

THE THEORY OF CARRIERS AND S-THEORY

Publisher Summary This chapter discusses the general theory of carriers, which provides a method of treating simultaneously the various relativizations that occur in homotopy theory. Moreover, the chapter discusses the suspension category, referred as S theory. The suspension category is obtained by passing to the direct limit with respect to suspension. It has the property that suspension is always an isomorphism that makes it simpler and easier to handle than the category of homotopy classes. Moreover, the set of mappings from one object in the suspension category to another is always an Abelian group. The chapter also presents the way to obtain an exact couple from an ordered sequence of carriers and to use this to obtain obstructions for extension and compression problems in the sense of S-theory. There are two obstruction theories, one leading to obstructions involving cohomology of the space being mapped, with coefficients in S-homotopy groups of the image space, and the other involving homology of the image space with coefficients in S-cohomotopy groups of the antecedent space.

DUALITY IN HOMOTOPY THEORY

This chapter discusses the principle of duality in the S -theory of finite polyhedra. It is analogous to the Alexander duality as it is primarily defined for subsets of polyhedral spheres. An n -dual of a subpolyhedron X ⊂ S n is a subpolyhedron D n X ⊂ S n –X which is an S -deformation retract of S–X . An S -map α:X→Y , where Y is also a subpolyhedron of S n , has a dual D n α:D n Y→D n X , and the map α→D n α is an isomorphism { X, Y } ≈{ D n Y, D n X }. When D n X is n -dual to X , then X is n -dual to D n X, an isomorphism of { D n Y, D n X } onto { X, Y } is present. The duality is expressed by the statement that these two isomorphisms are inverse to each other. Among other things, the chapter presents that how the construction of a finite CW -complex by the successive attaching of cells can be dualized.