Backward and Forward Bisimulation Minimisation of Tree Automata

Abstract

Abstract. We improve an existing bisimulation minimisation algorithm for tree automata by introducing backward and forward bisimulations and developing minimisation algorithms for them. Minimisation via forward bisimulation is also effective for deterministic automata and faster than the previous algorithm. Minimisation via backward bisimulation generalises the previous algorithm and is thus more effective but just as fast. We demonstrate implementations of these algorithms on a typical task in natural language processing.

Bisimulation Minimisation for Weighted Tree Automata

“…One type of bisimulation, called forward bisimulation in [9,10], restricts bisimilar states to have identical futures. The future of a state q is the tree series of contexts that is recognised by the wta if the computation starts with the state q and weight 1 at the unique position of the special symbol 2 in the context.…”

“…In contrast to the Myhill-Nerode congruence, a forward bisimulation requires a local condition on the tree representation. The condition is strong enough to enforce equivalent futures, but not too strong which is shown by the fact that, on a deterministic all-accepting [13] wta M over a field [14] or a wta M over the Boolean semiring [10], minimisation via forward bisimulation yields the unique (up to isomorphism) minimal deterministic wta that recognises the same tree series as M .…”

“…The other type of bisimulation we will consider is called backward bisimulation in [9,10]. Backward bisimulation also uses a local condition on the tree representation that enforces that the past of any two bisimilar states is equal.…”

“…The O n 2 log n minimisation algorithm for nfa by Paige & Tarjan [15] could be called a forward bisimulation minimisation. Bisimulation minimisation of tree automata is discussed in [10]. The paper [10] presents two minimisation algorithms that are based on forward and backward bisimulation and run in time O(rn r+1 log n) and O(r 2 n r+1 log n), respectively, where r is the maximal rank of the input symbols and n is the number of states.…”

“…Bisimulation minimisation of tree automata is discussed in [10]. The paper [10] presents two minimisation algorithms that are based on forward and backward bisimulation and run in time O(rn r+1 log n) and O(r 2 n r+1 log n), respectively, where r is the maximal rank of the input symbols and n is the number of states. In this paper, we generalise these results to weighted tree automata and obtain minimisation algorithms that work for arbitrary semirings in O(rn r+2 ) and O r 2 n r+2 for the forward and backward approach, respectively.…”