### Reachability in Unions of Commutative Rewriting Systems Is Decidable [chapter]

Mikołaj Bojańczyk, Piotr Hoffman
STACS 2007
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
more » ... 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 → 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, v 0 , u, w 0 ∈ Σ * such that v = tv 0 u, w = tw 0 u and v 0 → w 0 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 → ba. Commutative string rewriting systems are also called Vector Addition Systems or Multiset Rewriting Systems, since they treat words as multisets of letters -or elements of N Σ , where Σ is the alphabet. These systems are equivalent to Petri nets. The following is a famous result [1, 2]: