Reachability in Unions of Commutative Rewriting Systems Is Decidable

We consider commutative string rewriting systems (Vector Addition Systems, Petri nets), i.e., string rewriting systems in which all pairs of letters commute. We are interested in reachability: given a rewriting system R and words v and w, can v be rewritten to w by applying rules from R? A famous result states that reachability is decidable for commutative string rewriting systems. We show that reachability is decidable for a union of two such systems as well. We obtain, as a special case, that if h : U → S and g : U → T are homomorphisms of commutative monoids, then their pushout has a decidable word problem. Finally, we show that, given commutative monoids U , S and T satisfying S ∩ T = U , it is decidable whether there exists a monoid M such that S ∪ T ⊆ M ; we also show that the problem remains decidable if we require M to be commutative, too. Topic classification: Logic in computer science – rewriting 1 Summary of results A string rewriting system R over a finite alphabet Σ is simply a finite set of rules of the form v 7→ w, where v and w are words over Σ (string rewriting systems are also called semi-Thue systems). Such a system defines a one-step rewriting relation →R and a multistep rewriting relation → ∗ R on words over Σ: a word v rewrites in one step to a word w if there exist words t, v0, u, w0 ∈ Σ ∗ such that v = tv0u, w = tw0u and v0 7→ w0 is a rule of R; the multistep rewriting relation is the reflexive-transitive closure of the one-step relation. In the sequel, the statement “v rewrites to w in R” shall mean that v →∗R w. The (uniform) reachability problem is defined as follows: Given a string rewriting system R and words v and w in the alphabet of that system, answer whether v rewrites to w in R? This problem is one of the most basic undecidable problems. However, for appropriate restrictions on the form of the rewriting system R, the problem may become decidable. A string rewriting system is said to be commutative if for any two letters a and b of the alphabet it contains the rule ab 7→ ba. Commutative string rewriting 1 First author supported by the EC Research Training Network Games, second author by EC project Sensoria (No. 016004). systems are also called Vector Addition Systems or Multiset Rewriting Systems, since they treat words as multisets of letters — or elements of N , where Σ is the alphabet. These systems are equivalent to Petri nets. The following is a famous result [1, 2]: Theorem 1. Reachability in commutative string rewriting systems is decidable. If RΣ and RΓ are rewriting systems over alphabets Σ and Γ , which may be distinct and may have a non-empty intersection, then one may consider the union system RΣ ∪RΓ over the alphabet Σ ∪ Γ . This system is constructed by simply taking both the rules from RΣ , as well as the rules from RΓ . Note that the union of string rewriting systems may be much more complex than its parts. This is shown by the following example. A string rewriting system is said to be symmetric if for any rule v 7→ w in the system, the system also contains the rule w 7→ v. Such systems are called Thue systems. Sapir [3] (for an outline of the proof see [4]) constructed two symmetric string rewriting systems R1 and R2 such that the sets {v#w : v rewrites to w in R1} {v#w : v rewrites to w in R2} are both regular languages, but reachability in the (symmetric) union R1 ∪ R2 is undecidable! In this paper, we consider unions of commutative string rewriting systems. We do not make any assumptions on how the alphabets Σ and Γ of these systems relate to each other, whether they are disjoint or not, etc. Notice that the union RΣ ∪ RΓ of commutative systems RΣ and RΓ will not be commutative itself: if a is a letter from Σ \ Γ and b a letter from Γ \ Σ, then ab 7→ ba will not be in the union system; moreover, ab will in general not rewrite to ba in the union system. The main contribution of the paper is the following theorem: Theorem 2. Let RΣ and RΓ be commutative string rewriting systems. Then the reachability problem in the union RΣ ∪RΓ is decidable. This theorem properly extends Th. 1. Its proof is quite complex, and in the next two sections we only outline it. The full proof will be found in the full version of this paper. The same applies to other omitted proofs. In Sec. 4, we present results related to amalgamations [5] of commutative monoids. Some of these results are straightforward consequences of Th. 2, while others do not depend on it. The following is obtained easily from Th. 2: Corollary 1. Let h : U → S and g : U → T be homomorphisms of commutative monoids, and let h : S → P and g : T → P form a pushout of h and g (i.e. h ◦ h = g ◦ g and any homomorphisms h : S → P ′ and g : T → P ′ with h ◦ h = g ◦ g can be factored as u ◦ h = h and u ◦ g = g for a unique homomorphism u : P → P ). Then P has a decidable word problem. If the homomorphisms h and g above are injective, then without loss of generality one may assume that S∩T = U and that h and g are inclusions. In this case the