Minimizing the number of transitions with respect to observation equivalence

Abstract

Labeled transition systems (Its) provide an operational semantics for many specification languages. In order to abstract unrelevant details of Its's, many behavioural equivalences have been defined; here observation equivalence is considered. We are interested in the following problem: Given afinite lts, which is the minimal observation equivalent Its corresponding to it?It is well known that the number of states of an Its can be minimized by applying a relational coarsest partition algorithm, However, the obtained lts is not unique (up to the renaming of the states): for an Its there may exist several observation equivalent lts's which have the minimal number of states but varying number of transitions. In this paper we show how the number of transitions can be minimized, obtaining a unique lts.

An algorithm for compositional nonblocking verification using special events

“…Firstly, observation equivalence is used to remove redundant transitions [25]. A transition x σ → y is redundant by observation equivalence if the automaton contains another path connecting x and y by event σ, for example x…”

“…Then G ≈ w H. A maximal set of observation equivalent redundant transitions can be removed by a cubic-complexity algorithm [25]. This algorithm can be modified to take selfloop-only events into account.…”

“…Alternatively, the algorithm can be modified to assume the presence of selfloops as it executes. Neither of these changes affects the worst-case time complexity of the algorithm, which is dominated by transitive closure computations and runs in O(|Q| 3 |A|) time [25]. The most important use of weak observation equivalence is to reduce the state space of an automaton by computing a weak observation equivalence relation and replacing the automaton by the associated automaton quotient.…”

Compositional Nonblocking Verification with Always Enabled Events and Selfloop-Only Events

“…First, τ-loops (definition 8) and observation equivalent redundant transitions [9] are removed from the automaton. This is followed by the Only Silent Incoming Rule (proposition 6), the Only Silent Outgoing Rule [10], the Limited Certain Conflicts Rule (proposition 7), Observation Equivalence (proposition 1), the Non-α Determinisation Rule [16], the Active Events Rule [10], and the Silent Continuation Rule (proposition 5).…”

Modular Nonblocking Verification Using Conflict Equivalence

“…Thus, it is possible to use existing algorithms [10]- [12] for observation equivalence straight away for minimisation with respect to conflict equivalence. This alone gives a significant reduction of the state space.…”