Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth

Michael Elberfeld , Andreas Jakoby , Till Tantau

An algorithmic meta theorem for a logic and a class C of structures states that all problems expressible in this logic can be solved efficiently 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 (AC0 for decision problems and TC0 for counting problems) when restricted to input structures of bounded tree depth and (b) solvable by uniform logarithmic-depth circuit families (NC1 for decision problems and #NC1 for counting problems) when a tree decomposition of bounded width in term representation is part of the input. Applications of our theorems include a TC0-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 #NC1. 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 TC0, which encapsulates most of the technical difficulties surrounding earlier results connecting tree automata and NC1.

Mots clés: algorithmic meta theorem; monadic second-order logic; circuit complexity; tree width; tree depth

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Michael Elberfeld, Andreas Jakoby, Till Tantau. Algorithmic Meta Theorems for Circuit Classes of Constant and Logarithmic Depth. STACS'12 (29th Symposium on Theoretical Aspects of Computer Science), Feb 2012, Paris, France. pp.66-77. ⟨hal-00678176⟩