Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

Abstract

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved eciently for inputs from C. The prime example is Courcelle’s Theorem, which states that monadic second-order (mso) definable problems are linear-time solvable on graphs of bounded tree width. We contribute new algorithmic meta theorems, which state that mso-definable problems are (a) solvable by uniform constant-depth circuit families (AC 0 for decision problems and TC 0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC 1 for decision problems and #NC 1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC 0 -completeness proof for the unary version of integer linear programming with a fixed number of equations and extensions of a recent result that counting the number of accepting paths of a visible pushdown automaton lies in #NC 1 . Our main technical contributions are a new tree automata model for unordered, unranked, labeled trees; a method for representing the tree automata’s computations algebraically using convolution circuits; and a lemma on computing balanced width-3 tree decompositions of trees in TC 0 , which encapsulates most of the technical diculties surrounding earlier results connecting tree automata and NC 1 . 1998 ACM Subject Classification F.1.3 Complexity Measures and Classes