Structural Complexity of Multi-Valued Partial Functions Computed by Nondeterministic Pushdown Automata

TL;DR

This paper explores the structural properties and hierarchies of multi-valued partial functions computed by nondeterministic pushdown automata, revealing differences from context-free languages and analyzing complexity aspects.

Contribution

It provides a systematic study of CFL functions, including their containments, separations, and hierarchies via relativization and Boolean operations, and examines their computational complexity.

Findings

Turing relativization constructs a hierarchy of CFL functions.

Distinct behaviors of CFL functions compared to context-free languages.

Analysis of optimization functions' complexity and their language relationships.

Abstract

This paper continues a systematic and comprehensive study on the structural properties of CFL functions, which are in general multi-valued partial functions computed by one-way one-head nondeterministic pushdown automata equipped with write-only output tapes (or pushdown transducers), where CFL refers to a relevance to context-free languages. The CFL functions tend to behave quite differently from their corresponding context-free languages. We extensively discuss containments, separations, and refinements among various classes of functions obtained from the CFL functions by applying Boolean operations, functional composition, many-one relativization, and Turing relativization. In particular, Turing relativization helps construct a hierarchy over the class of CFL functions. We also analyze the computational complexity of optimization functions, which are to find optimal values of CFL functions, and discuss their relationships to the associated languages.