Automata with Nested Pebbles Capture First-Order Logic with Transitive Closure

TL;DR

This paper characterizes the computational power of automata with nested pebbles in terms of first-order logic with transitive closure, establishing a precise correspondence based on the number of heads and the arity of the transitive closure.

Contribution

It extends the understanding of automata with nested pebbles by matching the number of heads to the arity of transitive closure in first-order logic, for various graph families and structures.

Findings

Automata with k heads and nested pebbles correspond to first-order logic with k-ary deterministic transitive closure.

Single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure.

The results apply to strings, trees, and other graph families like grids and mazes.

Abstract

String languages recognizable in (deterministic) log-space are characterized either by two-way (deterministic) multi-head automata, or following Immerman, by first-order logic with (deterministic) transitive closure. Here we elaborate this result, and match the number of heads to the arity of the transitive closure. More precisely, first-order logic with k-ary deterministic transitive closure has the same power as deterministic automata walking on their input with k heads, additionally using a finite set of nested pebbles. This result is valid for strings, ordered trees, and in general for families of graphs having a fixed automaton that can be used to traverse the nodes of each of the graphs in the family. Other examples of such families are grids, toruses, and rectangular mazes. For nondeterministic automata, the logic is restricted to positive occurrences of transitive closure. The special case of k=1 for trees, shows that single-head deterministic tree-walking automata with nested pebbles are characterized by first-order logic with unary deterministic transitive closure. This refines our earlier result that placed these automata between first-order and monadic second-order logic on trees.