On the Upward/Downward Closures of Petri Nets

Abstract

We study the size and the complexity of computing finite state automata (FSA) representing and approximating the downward and the upward closure of Petri net languages with coverability as the acceptance condition. We show how to construct an FSA recognizing the upward closure of a Petri net language in doubly-exponential time, and therefore the size is at most doubly exponential. For downward closures, we prove that the size of the minimal automata can be non-primitive recursive. In the case of BPP nets, a well-known subclass of Petri nets, we show that an FSA accepting the downward/upward closure can be constructed in exponential time. Furthermore, we consider the problem of checking whether a simple regular language is included in the downward/upward closure of a Petri net/BPP net language. We show that this problem is EXPSPACE-complete (resp. NP-complete) in the case of Petri nets (resp. BPP nets). Finally, we show that it is decidable whether a Petri net language is upward/downward closed. To this end, we prove that one can decide whether a given regular language is a subset of a Petri net coverability language. ACM Subject ClassificationF.1.1 Models of Computation Keywords and phrases Petri nets, BPP nets, downward closure, upward closure 1 * The conference version of this paper has been published in the proceedings of the 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017 [8]. † A part of this work was carried out when the author was at Aalto University. 4 On the Upward/Downward Closures of Petri NetsLet Γ be a subset of Σ. Given a word u ∈ Σ * , we denote by π Γ (u) the projection of u over Γ, i.e. the word obtained from u by erasing all the letters that are not in Γ.The Parikh image of a word [39] counts the number of occurrences of all letters while forgetting about their positioning. Formally, the function Ψ : Σ * → N Σ takes a word w ∈ Σ * and gives the function Ψ(w) : Σ → N defined by (Ψ(w))(a) = π {a} (w) for all a ∈ Σ.The subword relation ⊆ Σ * × Σ * [25] between words is defined as follows: A word u = a 1 . . . a n is a subword of v, denoted u v, if u can be obtained by deleting letters from v or, equivalentlyLet L be a language over Σ. The upward closure of L consists of all words that have a subword in the language, L ↑= {v ∈ Σ * | ∃u ∈ L : u v}. The downward closure of L contains all words that are dominated by a word in the language, L ↓= {u ∈ Σ * | ∃v ∈ L : u v}. Higman showed that the subword relation is a well-quasi ordering [25], which means that every set of words L ⊆ Σ * has a finite basis, a finite set of minimal elements v ∈ L such that ∄u ∈ L : u = v, u v. With finite bases, L ↑ and L ↓ are guaranteed to be regular for every language L ⊆ Σ * [24]. Indeed, they can be expressed using the subclass of simple regular languages defined by so-called simple regular expressions [1].These SREs are choices among products p that interleave single letters a or (a + ε) with iterations over letters from subsets Γ ⊆ Σ of the alphabet: sre ::= p sre + sre p ::= a ...