On Matching Concurrent Traces

Concurrent traces are sequences of computational steps where independent steps can be permuted and executed in any order. We study the problem of matching on concurrent traces. We give a sound and complete algorithm for matching traces with one variable standing for an unknown subtrace. 1 State-Transition Concurrency Computation in a concurrent system results from the interactions of computing units such as threads or agents. One popular approach to modeling concurrent computation views each such agent as being able to perform local transformations on a global state, possibly in parallel. This is the approach embraced by Petri nets [5]. This is also the approach underlying propositional multiset rewriting, which we will now describe in further detail. A multiset rewriting system is determined by a set of rules of the form ã→ b̃, where ã and b̃ are multisets over some support set S. We write “·” for the empty multiset and “ã, b̃” for the union of multisets ã and b̃. Multiset union is associative and commutative, and has the empty multiset as its unit. Therefore, the set of multisets over S is a commutative monoid with respect to “,” and “·”. We give each rule ã→ b̃ with a unique name, r, writing the association as r : ã→ b̃. We write R for a set of labeled rules. A state s is a set of pairs x : a where a ∈ S is an element of the support set and x is a unique name. Abusing notation, we occasionally write such a state as x̃ : ã, where ã is the multiset of the support set element occurrences in s and x̃ the corresponding names. We also use “,” for set union in the context of states. A rule r : ã → b̃ in a multiset rewriting system R is enabled in a state s if s = s′, (x̃ : ã), i.e., if s contains its antecedent. In this circumstance, the application of r to s results in the state s′′ = s′, (ỹ : b̃) where ỹ are fresh names: the portion of s matching ã has been replaced with new state elements for b̃. The triple r(x̃; ỹ), generically denoted t, is a rule instance, or transition. We call x̃ : ã the pre-condition of t and denote it as •t and ỹ : b̃ its post-condition, denoted t•. We formalize the state transformation embodied by the application of a rule 2 J.L. Sacchini, I. Cervesato, F. Pfenning, C. Schürmann, R.J. Simmons by means of the multiset rewriting judgment R ` s t −→ s′′. Then, application is defined simply as R, (r : ã→ b̃) ` s′, x̃ : ã } {{ } s r(x̃;ỹ) −→ s′, ỹ : b̃ } {{ } s′′ or more succinctly as R ` s′, •t t −→ s′, t• for t an instance of a rule r ∈ R. Multistep computation is obtained by iterating application. We write R ` s t =⇒ s′ for the reflexive and transitive closure of our base judgment, where t records the rules that have been applied to go from s to s′. It is defined by the following grammar: t ::= · | r(x̃; ỹ) | t1; t2 We call t a trace. Here, “·” represents the empty trace and “t1; t2” is the concatenation of t1 and t2. A concurrent trace t can be interpreted as a bipartite directed acyclic graph (BDAG) G = (N1, N2, E) where N1 is the set of transitions t in t and N2 is the set of state elements x : a mentioned in t. There is an edge from x : a to t = r(x̃; ỹ) if and only if x occurs in x̃, and there is an edge from t to y : b iff y occurs in ỹ. These BDAGs are exactly what we get by graphically unfolding the computation of a place/transition Petri net.