Asynchronous Multi-Tape Automata Intersection: Undecidability and Approximation

When their reading heads are allowed to move completely asynchronously, finite-state automata with multiple tapes achieve a significant expressive power, but also lose useful closure properties—closure under intersection, in particular. This paper investigates to what extent it is still feasible to use multi-tape automata as recognizers of polyadic predicates on words. On the negative side, determining whether the intersection of asynchronous multi-tape automata is expressible is not even semidecidable. On the positive side, we present an algorithm that computes under-approximations of the intersection; and discuss simple conditions under which it can construct complete intersections. A prototype implementation and a few non-trivial examples demonstrate the algorithm in practice. 1 Automata As Decision Procedures Standard finite-state automata are simple computing devices widely used in computer science. They define a robust class of language acceptors, as each automaton instance A identifies a set L(A) of words that it accepts as input. The connection between finitestate automata and predicate logic has been well-known since the work of Büchi [4, 5] and others [33, 10], and is widely used in applications such as model-checking: each automaton AP can be seen as implementing a monadic (that is, unary) predicate P (x), in the sense that the set L(AP ) of words accepted by the automaton corresponds to the set {x | x |= P (x)} of models of the predicate. Logic connectives (negation ¬, conjunction ∧, etc.) translate into composition operations on automata (complement, intersection ∩, etc.), so that finite-state automata can capture the semantics of arbitrary first-order monadic formulas whose interpreted atomic predicates are implementable. This gives a very efficient way to decide the satisfiability of monadic logic formulas representable by finite-state automata: unsatisfiability of a formula corresponds to emptiness of its automaton, which is testable efficiently in time linear in the automaton size. It is natural to extend this framework [1, 32] to represent n-ary predicates, for n > 1, by means of multi-tape finite-state automata. An n-tape automaton AR is a device that accepts n-tuples of words, corresponding to the set of models of a predicate R(x1, . . . , xn) over n variables. Section 2 defines multi-tape automata and summarizes some of their fundamental properties. It turns out that the class of multi-tape automata (in their most expressive asynchronous variant) is not as robust as one-tape automata.