Finite-image property of weighted tree automata over past-finite monotonic strong bimonoids

We consider weighted tree automata over strong bimonoids (for short: wta). A wta A has the finite-image property if its recognized weighted tree language [[A]] has finite image; moreover, A has the preimage property if the preimage under [[A]] of each element of the underlying strong bimonoid is a recognizable tree language. For each wta A over a past-finite monotonic strong bimonoid we prove the following results. In terms of A’s structural properties, we characterize whether it has the finite-image property. We characterize those past-finite monotonic strong bimonoids such that for each wta A it is decidable whether A has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that A has the preimage property. All our results also hold for weighted string automata. Research of this author was supported by grant TUDFO/47138-1/2019-ITM of the Ministry for Innovation and Technology, Hungary. Supported by the ÚNKP-20-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund.

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