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We consider the images of the initial algebra semantics of weighted tree automata over strong bimonoids (hence also over semirings). These images are subsets of the carrier set of the underlying strong bimonoid. We consider locally finite, weakly locally finite, and bi-locally finite strong bimonoids. We show that there exists a strong bimonoid which is weakly locally finite and not locally finite. We also show that if the ranked alphabet contains a binary symbol, then for any finitely generated strong bimonoid, weighted tree automata can generate, via their initial algebra semantics, all elements of the strong bimonoid. As a consequence of these results, for weakly locally finite strong bimonoids which are not locally finite, weighted tree automata can generate infinite images provided that the input ranked alphabet contains at least one binary symbol. This is in sharp contrast to the setting of weighted string automata, where each such image is known to be finite. As a further consequence, for any finitely generated semiring, there exists a weighted tree automaton which generates, via its run semantics, all elements of the semiring., Comment: 20 pages, 2 figures. arXiv admin note: text overlap with arXiv:2212.05529

This is a book on weighted tree automata. We present the basic definitions and some of the important results in a coherent form with full proofs. The concept of weighted tree automata is part of Automata Theory and it touches the area of Universal Algebra. It originated from two sources: weighted string automata and finite-state tree automata.

We consider weighted tree automata over strong bimonoids (for short: wta). A wta $\mathcal{A}$ has the finite-image property if its recognized weighted tree language $[\![\mathcal{A}]\!]$ has finite image; moreover, $\mathcal{A}$ has the preimage property if the preimage under $[\![\mathcal{A}]\!]$ of each element of the underlying strong bimonoid is a recognizable tree language. For each wta $\mathcal{A}$ over a past-finite monotonic strong bimonoid we prove the following results. In terms of $\mathcal{A}$'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 $\mathcal{A}$ it is decidable whether $\mathcal{A}$ has the finite-image property. In particular, the finite-image property is decidable for wta over past-finite monotonic semirings. Moreover, we prove that $\mathcal{A}$ has the preimage property. All our results also hold for weighted string automata., Comment: 42 pages, 4 figures, 1 algorithm

We consider weighted tree automata (wta) over strong bimonoids and their initial algebra semantics and their run semantics. There are wta for which these semantics are different; however, for bottom-up deterministic wta and for wta over semirings, the difference vanishes. A wta is crisp-deterministic if it is bottom-up deterministic and each transition is weighted by one of the unit elements of the strong bimonoid. We prove that the class of weighted tree languages recognized by crisp-deterministic wta is the same as the class of recognizable step mappings. Moreover, we investigate the following two crisp-determinization problems: for a given wta ${\cal A}$, (a) does there exist a crisp-deterministic wta which computes the initial algebra semantics of ${\cal A}$ and (b) does there exist a crisp-deterministic wta which computes the run semantics of ${\cal A}$? We show that the finiteness of the Nerode algebra ${\cal N}({\cal A})$ of ${\cal A}$ implies a positive answer for (a), and that the finite order property of ${\cal A}$ implies a positive answer for (b). We show a sufficient condition which guarantees the finiteness of ${\cal N}({\cal A})$ and a sufficient condition which guarantees the finite order property of ${\cal A}$. Also, we provide an algorithm for the construction of the crisp-deterministic wta according to (a) if ${\cal N}({\cal A})$ is finite, and similarly for (b) if ${\cal A}$ has finite order property. We prove that it is undecidable whether an arbitrary wta ${\cal A}$ is crisp-determinizable. We also prove that both, the finiteness of ${\cal N}({\cal A})$ and the finite order property of ${\cal A}$ are undecidable.

We develop a general framework for weighted parsing which is built on top of grammar-based language models and employs multioperator monoids as weight algebras. It generalizes previous work in that area (semiring parsing, weighted deductive parsing) and also covers applications outside the classical scope of parsing, e.g., algebraic dynamic programming. We show an algorithm for weighted parsing and, for a large class of weighted grammar-based language models, we prove formally that it terminates and is correct., Comment: 59 pages (89 with appendix)

We introduce MSO graph storage types, and call a storage type MSO-expressible if it is isomorphic to some MSO graph storage type. An MSO graph storage type has MSO-definable sets of graphs as storage configurations and as storage transformations. We consider sequential automata with MSO graph storage and associate with each such automaton a string language (in the usual way) and a graph language; a graph is accepted by the automaton if it represents a correct sequence of storage configurations for a given input string. For each MSO graph storage type, we define an MSO logic which is a subset of the usual MSO logic on graphs. We prove a B\"uchi-Elgot-Trakhtenbrot theorem, both for the string case and the graph case. Moreover, we prove that (i) each MSO graph transduction can be used as storage transformation in an MSO graph storage type, (ii) every automatic storage type is MSO-expressible, and (iii) the pushdown operator on storage types preserves the property of MSO-expressibility. Thus, the iterated pushdown storage types are MSO-expressible.

We introduce weighted regular tree grammars with storage as combination of (a) regular tree grammars with storage and (b) weighted tree automata over multioperator monoids. Each weighted regular tree grammar with storage generates a weighted tree language, which is a mapping from the set of trees to the multioperator monoid. We prove that, for multioperator monoids canonically associated to particular strong bi-monoids, the support of the generated weighted tree languages can be generated by (unweighted) regular tree grammars with storage. We characterize the class of all generated weighted tree languages by the composition of three basic concepts. Moreover, we prove results on the elimination of chain rules and of finite storage types, and we characterize weighted regular tree grammars with storage by a new weighted MSO-logic., Comment: added errata

We consider symbolic tree automata (sta) and symbolic tree transducers (stt). We characterize s-recognizable tree languages (which are the tree languages recognizable by sta) in terms of (classical) recognizable tree languages and relabelings. We prove that sta and the recently introduced variable tree automata are incomparable with respect to their recognition power. We define symbolic regular tree grammars and characterize s-regular tree languages in terms of regular tree languages and relabelings. As a consequence, we obtain that s-recognizable tree languages are the same as s-regular tree languages. We show that the syntactic composition of two stt computes the composition of the tree transformations computed by each stt, provided that (1) the first one is deterministic or the second one is linear and (2) the first one is total or the second is nondeleting. We consider forward application and backward application of stt and prove that the backward application of an stt to any s-recognizable tree language yields an s-recognizable tree language. We give a linear stt of which the range is not an s-recognizable tree language. We show that the forward application of simple and linear stt preserves s-recognizability. As a corollary, we obtain that the type checking problem of simple and linear stt and the inverse type checking problem of arbitrary stt is decidable.

We introduce the branching transitive closure operator on weighted monadic second-order logic formulas where the branching corresponds in a natural way to the branching inherent in trees. For arbitrary commutative semirings, we prove that weighted monadic second order logics on trees is equivalent to the definability by formulas which start with one of the following operators: (i) a branching transitive closure or (ii) an existential second-order quantifier followed by one universal first-order quantifier; in both cases the operator is applied to step-formulas over (a) Boolean first-order logic enriched by modulo counting or (b) Boolean monadic-second order logic.

Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices, and more. In our main result, we derive the fundamental Chomsky-Sch\"utzenberger theorem for such quantitative context-free languages, showing that each arises as the image of a Dyck language and a regular language under a suitable morphism. Moreover, we show that quantitative context-free language are expressively equivalent to a model of weighted pushdown automata. This generalizes results previously known only for semirings. We also investigate when quantitative context-free languages assume only finitely many values., Comment: This new version combines a conference and a journal paper of the authors on the same topic, see references [15,16], and supplements them by a few additional examples and more detailed proofs. It also corrects a mistake in Theorem 7.7 of the first arxiv version (the property sequential was missing)