Abstract
In this paper we present a basic framework for weighted Petri net transducers (PNTs) for the translation of partial languages (consisting of partial words) as a natural generalisation of finite state transducers (FSTs).
Concerning weights, we use the algebraic structure of continuous concurrent semirings which is based on bisemirings and induces a natural order on its elements. Using the operations of this algebra, it is possible to define the weight of sequential parallel partial words in a standard way. We define the weight of a general partial word as the supremum of the weights of all of its sequential parallel extensions. As a fundamental result we show that concurrent semirings are the least restrictive idempotent bisemiring structure such that partial words with fewer dependencies have bigger weights. Moreover, the weight definition turns out to be compositional, i.e. the weight of (sequential or parallel) composed partial words equals the corresponding bisemiring composition of the weights of its components.
To be able to create complex PNTs through composition of simple PNTs, we introduce clean PNTs and the composition operations union, product, closure, parallel product and language composition on clean PNTs, lifting standard composition operations on FSTs. Composed PNTs yield a compositional computation of weights, where in the case of language composition such a compositional computation is possible only in restricted cases. Moreover, we give definitions for equivalent PNTs and show that all composition operations preserve equivalence. We also show that under certain conditions concerning the algebraic weight structure an FST can be represented by an equivalent PNT.
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Lorenz, R., Huber, M., Wirsching, G. (2014). On Weighted Petri Net Transducers. In: Ciardo, G., Kindler, E. (eds) Application and Theory of Petri Nets and Concurrency. PETRI NETS 2014. Lecture Notes in Computer Science, vol 8489. Springer, Cham. https://doi.org/10.1007/978-3-319-07734-5_13
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DOI: https://doi.org/10.1007/978-3-319-07734-5_13
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