Deque languages, automata and planar graphs

TL;DR

This paper explores the theoretical properties of deque automata and languages, introducing a characteristic language, CDL, and representing computations using planar deque graphs, linking automata theory with graph theory.

Contribution

It defines the characteristic deque language CDL, characterizes deque automata using multiple methods, and introduces deque graphs as a novel representation of deque computations.

Findings

Deque languages form an AFL but not a full AFL.

The characteristic deque language CDL combines Dyck and AntiDyck languages.

Deque computations can be represented by planar Hamiltonian graphs on a cylinder.

Abstract

The memory of a deque (double ended queue) automaton is more general than a queue or two stacks; to avoid overgeneralization, we consider quasi-real-time operation. Normal forms of such automata are given. Deque languages form an AFL but not a full one. We define the characteristic deque language, CDL, which combines Dyck and AntiDyck (or FIFO) languages, and homomorphically characterizes the deque languages. The notion of deque graph, from graph theory, well represents deque computation by means of a planar hamiltonian graph on a cylinder, with edges visualizing producer-consumer relations for deque symbols. We give equivalent definitions of CDL by labelled deque graphs, by cancellation rules, and by means of shuffle and intersection of simpler languages. The labeled deque graph of a sentence generalizes traditional syntax trees. The layout of deque computations on a cylinder is remindful of 3D models used in theoretical (bio)chemistry.