Second-Order Finite Automata

Abstract

Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinite sets of sets of strings. Our main result states that second-order finite automata can be canonized with respect to the second-order language they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable.

We provide two algorithmic applications for second-order automata. First, we show that they can be used to show that several width and size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width and size minimization problems for ordered binary decision diagrams with a fixed variable ordering. The previous best algorithms for these problems were exponential on the size of the input ODD even for ODDs of constant width. Second, we show that for each $k$ and $w$ one can count the number of distinct functions accepted by ODDs of width $w$ and length $k$ in time $h_{\varSigma }(w)\cdot k^{O(1)}$. This improves exponentially on the time necessary to explicitly enumerate all distinct functions, which take time exponential in both the width parameter $w$ and in the length $k$ of the ODDs.

Alexsander Andrade de Melo acknowledges support from the Brazilian agencies CNPq/GD 140399/2017-8 and CAPES/PDSE 88881.187636/2018-01. Mateus de Oliveira Oliveira acknowledges support from the Bergen Research Foundation and from the Research Council of Norway (Grant Nr. 288761).