The Diagonal Problem for Higher-Order Recursion Schemes is Decidable

Abstract

A non-deterministic recursion scheme recognizes a language of finite trees. This very expressive model can simulate, among others, higher-order pushdown automata with collapse. We show decidability of the diagonal problem for schemes. This result has several interesting consequences. In particular, it gives an algorithm that computes the downward closure of languages of words recognized by schemes. In turn, this has immediate application to separability problems and reachability analysis of concurrent systems.

Intersection Types for Unboundedness Problems

“…Deciding this property is known under the names simultaneous unboundedness problem (SUP) and diagonal problem (these are two different names for the same problem). SUP for recursion schemes was first solved in Clemente, Parys, Salvati, and Walukiewicz [8], in a different way. The advantage of solving SUP using the type system presented here is twofold.…”

“…Intersection type systems were intensively used in the context of recursion schemes, for several purposes like model-checking [18,21,5,29], pumping [19,2], transformations of HORSes [20,1,8], etc. Interestingly, constructions very similar to intersection types were used also on the side of collapsible pushdown systems; they were alternating stack automata [4], and types of stacks [23,16].…”

Intersection Types and Counting

“…Intersection type systems were intensively used in the context of HORSes, for several purposes like model-checking [13,16,5,21], pumping [14], transformations of HORSes [15,6], etc. Interestingly, constructions very similar to intersection types were used also on the side of collapsible pushdown systems; they were alternating stack automata [4], and types of stacks [19,11].…”

“…The language finiteness problem was first attacked in [1] (for safe HORSes only), but their algorithm turned out to be incorrect [2]. To our knowledge, the only known solution of this problem follows from a recent decidability result for the diagonal problem [9,6]. This problem asks, given a nondeterministic HORS and a set of letters Σ, whether for every n ∈ N the HORS generates a finite tree in which every letter from Σ appears at least n times.…”

“…Our type system is, to some extent, motivated by the algorithm of [6] solving the diagonal problem. This algorithm works by repeating two kinds of transformations of HORSes.…”