Automatic constraints solved efficiently with new algebraic methods
Automatic constraints with few subpowers and graphoid recognition
Logic in Computer ScienceFormal Languages and Automata Theory
Summary
Some complex problems involve finding solutions that fit many conditions described by machines called finite automata. The authors show new ways to solve these problems quickly when the conditions have special algebraic properties, like certain fixed operations on a limited set of values. Their approach also helps identify when these algebraic structures exist, even for complicated cases. This work answers longstanding questions about how efficiently these problems can be solved and provides methods to represent solutions in compact forms.
What this means in practice
- •For software engineers: Optimize algorithms that solve constraint problems represented by automata using new polynomial-time methods.
- •For database administrators: Use polynomial-time recognition of graph-based relations for efficient query optimization and data integrity checking.
A theory result. No direct application yet.
Authors
Antonios Kalampakas
Abstract
Finite automata can describe relations of unbounded arity that are exponentially larger than their descriptions. We prove that constraint satisfaction for such relations is solvable in polynomial time whenever their length slices are preserved by a common fixed edge operation on a finite domain. The algorithm computes compact representations of the complete solution relation and its projections. Its main ingredient is a polynomial-time compilation of nondeterministic finite automata into the fork witnesses and small projections required by the few-subpowers algorithm. In the Mal'tsev case, a direct proof is polynomial also when the domain and operation table are supplied as input, answering the Mal'tsev tractability question for automatic constraint satisfaction. We also characterize all invariant relations of a family of 3-edge algebras with neither Mal'tsev nor near-unanimity terms. Their normal forms combine Boolean activity constraints with affine value spaces and yield canonical quadratic-bit representations constructible from NFAs or arbitrary generators. For graphoid automata, these results give polynomial-time recognition without a graph-width restriction, effective boundary composition, and comparison of finite graph relations. The quadratic boundary bounds are optimal in the worst case. A fixed three-state example separates polynomial-time recognition from hard exact counting.