Bisimulation Minimization of Tree Automata

Abstract

We extend an algorithm by Paige and Tarjan that solves the coarsest stable refinement problem to the domain of trees. The algorithm is used to minimize nondeterministic tree automata (NTA) with respect to bisimulation. We show that our algorithm has an overall complexity of [Formula: see text], where [Formula: see text] is the maximum rank of any symbol in the input alphabet, m is the total size of the transition table, and n is the number of states.

Bisimulation Minimisation for Weighted Tree Automata

“…Containing the restrictions of both, it is less efficient than backward bisimulation when applied to the minimisation of nondeterministic automata, but just as expensive to calculate, and unlike forward bisimulation it does not yield the standard algorithm when applied to deterministic automata. The pair of algorithms presented in this paper thus supersedes that of [16].…”

“…The bisimulation introduced in [16] can be seen as a combination of backward and forward bisimulation. Containing the restrictions of both, it is less efficient than backward bisimulation when applied to the minimisation of nondeterministic automata, but just as expensive to calculate, and unlike forward bisimulation it does not yield the standard algorithm when applied to deterministic automata.…”

Backward and Forward Bisimulation Minimisation of Tree Automata

“…The problem must thus be restricted to allow algorithms of practical value, and one possibility is to settle for a partial minimisation. This was done in [4] for non-deterministic tree automata (nta), which are a generalisation of nfa that recognise tree languages and are used in applications such as model checking [5] and natural language processing [6].…”

“…The minimisation algorithm in [4] was inspired by a partitioning algorithm due to Paige and Tarjan [7], and relies heavily on bisimulation; a concept introduced by R. Milner as a formal tool for investigating transition systems. Intuitively, two states are bisimilar if they can simulate each other, or equivalently, the observable behaviour of the two states must coincide.…”

“…Identifying bisimilar states allows us to reduce the size of the input automaton, but we are not guaranteed to obtain the smallest possible automaton. In this work we extend the approach of [4] in two ways: (i) we relax the constraints for state equivalence, and (ii) we introduce a new bisimulation relation that (with effect) can be applied to deterministic (bottom-up) tree automata (dta) [8]. Note that [4] is ineffective on dta.…”