Characterisation Theorems for Weighted Tree Automaton Models

In this thesis, we investigate different theoretical questions concerning weighted automata models over tree-like input structures. First, we study exact and approximated determinisation and then, we turn to Kleene-like and Büchi-like characterisations. We consider multiple weighted automata models, including weighted tree automata over semirings (Chapters 3 and 4), weighted forest automata over M-monoids (Chapter 5), and rational weighted tree languages with storage (Chapter 6). For an explanation as to why the last class can be considered as a weighted automaton model, we refer to page 188 of the thesis. We will now summarise the main contributions of the thesis. In Chapter 3, we focus on the determinisation of weighted tree automata and present our determinisation framework, called M-sequentialisation, which can model different notions of determinisation from the existing literature. Then, we provide a positive M-sequentialisation result for the case of additively idempotent semirings or finitely M-ambiguous weighted tree automata. Another important contribution of Chapter 3 is Theorem 77, where we provide a blueprint theorem that can be used to find determini- sation results for more classes of semirings and weighted tree automata easily. In fact, instead of repeating an entire determinisation construction, Theorem 77 allows us to prove a determinisation result by finding certain finite equivalence relations. This is a very potent tool for future research in the area of determinisation. In Chapter 4, we move from exact determinisation towards approximate determini- sation. We lift the formalisms and the main results from one approach from the literature from the word case to the tree case. This successfully results in an approximated determinisation construction for weighted tree automata over the tropical semiring. We provide a formal mathematical description of the approximated determinisation construction, rather than an algorithmic description as found in the related approach from the literature. In Chapter 5, we turn away from determinisation and instead consider Kleene-like and Büchi-like characterisations of weighted recognisability. We introduce weighted forest automata over M-monoids, which are a generalisation of weighted tree automata over M-monoids and weighted forest automata over semirings. Then, we prove that our recognisable weighted forest languages can be decomposed into a finite product of recognisable weighted tree languages. We also prove that the initial algebra semantic and the run semantic for weighted forest automata are equivalent under certain conditions. Lastly, we define rational forest expressions and forest M-expressions and and prove that the classes of languages generated by these formalisms coincide with recognisable weighted forest languages under certain conditions. In Chapter 6, we consider rational weighted tree languages with storage, where the storage is introduced by composing rational weighted tree languages without storage with a storage map. It has been proven in the literature that rational weighted tree languages with storage are closed under the rational operations. In Chapter 6, we provide alternative proofs of these closure properties. In fact, we prove that our way of introducing storage to rational weighted tree languages preserves the closure properties from rational weighted tree languages without storage.:1 Introduction 2 Preliminaries 2.1 Languages 2.2 WeightedLanguages 2.3 Weighted Tree Automata 3 A Unifying Framework for the Determinisation of Weighted Tree Automata 3.1 Introduction 3.2 Preliminaries 3.3 Factorisation in Monoids 3.3.1 Ordering Multisets over Monoids 3.3.2 Cayley Graph and Cayley Distance 3.3.3 Divisors and Rests 3.3.4 Factorisation Properties 3.4 Weighted Tree Automata over M_fin(M) and the Twinning Property 3.4.1 Weighted Tree Automata over M_fin(M) 3.4.2 The Twinning Property 3.5 Sequentialisation of Weighted Tree Automata over M_fin(M) 3.5.1 The Sequentialisation Construction 3.5.2 The Finitely R-Ambiguous Case 3.6 Relating WTA over M_fin(M) and WTA over S 3.7 M-Sequentialisation of Weighted Tree Automata 3.7.1 Accumulation of D_B 3.7.2 M-Sequentialisation Results 3.8 Comparison of our Results to the Literature 3.8.1 Determinisation of Unweighted Tree Automata 3.8.2 The Free Monoid Case 3.8.3 The Group Case 3.8.4 The Extremal Case 3.9 Conclusion 4 Approximated Determinisation of Weighted Tree Automata 125 4.1 Introduction 4.2 Preliminaries 4.3 Approximated Determinisation 4.3.1 The Approximated Determinisation Construction 4.3.2 Correctness of the Construction 4.4 The Approximated Twinning Property 4.4.1 Implications for Approximated Determinisability 4.4.2 Decidability of the Twinning Property 4.5 Conclusion 5 Kleene and Büchi Theorems for Weighted Forest Languages over M-Monoids 5.1 Introduction 5.2 Preliminaries 5.3 WeightedForestAutomata 5.3.1 Forests 5.3.2 WeightedForestAutomata 5.3.3 Rectangularity 5.3.4 I-recognisable is R-recognisable 5.4 Kleene’s Theorem 5.4.1 Kleene’s Theorem for Trees 5.4.2 Kleene’s Theorem for Forests 5.4.3 An Inductive Approach 5.5 Büchi’s Theorem 5.5.1 Büchi’s Theorem for Trees 5.5.2 Büchi’s Theorem for Forests 5.6 Conclusion 6 Rational Weighted Tree Languages with Storage 6.1 Introduction 6.2 Preliminaries 6.3 Rational Weighted Tree Languages with Storage 6.4 The Kleene-Goldstine Theorem 6.5 Closure of Rat(S¢,Σ,S) under Rational Operations 6.5.1 Top-Concatenation, Scalar Multiplication, and Sum 6.5.2 α-Concatenation 6.5.3 α-Kleene Star 6.6 Conclusion 7 Outlook Reference

Recognizable tree series with discounting

We consider weighted tree automata with discounting over commutative semirings. For their behaviors we establish a Kleene theorem and an MSO-logic characterization. We introduce also weighted Muller tree automata with discounting over the max-plus and the min-plus semirings, and we show their expressive equivalence with two fragments of weighted MSO-sentences

Weighted Automata and Expressions over Pre-Rational Monoids

The Kleene theorem establishes a fundamental link between automata and expressions over the free monoid. Numerous generalisations of this result exist in the literature; on one hand, lifting this result to a weighted setting has been widely studied. On the other hand, beyond the free monoid, different monoids can be considered: for instance, two-way automata, and even tree-walking automata, can be described by expressions using the free inverse monoid. In the present work, we aim at combining both research directions and consider weighted extensions of automata and expressions over a class of monoids that we call pre-rational, generalising both the free inverse monoid and graded monoids. The presence of idempotent elements in these pre-rational monoids leads in the weighted setting to consider infinite sums. To handle such sums, we will have to restrict ourselves to rationally additive semirings. Our main result is thus a generalisation of the Kleene theorem for pre-rational monoids and rationally additive semirings. As a corollary, we obtain a class of expressions equivalent to weighted two-way automata, as well as one for tree-walking automata

Automaták, fák és logika = Automata, trees and logic

Elemi idejű exponenciális algoritmus adtunk meg reguláris szavak ekvivalenciájának eldönthetőségére. Általánosítottuk Kleene tételét végtelen szavakat is felismerő súlyozott automatákra. Kifejlesztettünk egy algebrai módszert, amellyel a CTL logika számos szegmense estén eldönthető, hogy egy reguláris fanyelv definiálható-e a szegmensben. Vizsgáltuk a faautomaták algebrai tulajdonságait, megadtuk a felismerhetőség egy algebrai jellemzését. Definiáltunk a multi-leszálló fatranszformátort és megmutattuk, hogy ekvivalens a determinisztikus reguláris szűkítésű felszálló fatranszformátorral. Meghatároztuk a lineáris multi-leszálló osztály számítási erejét. Megmutattuk, hogy az alakmegőrző leszálló fatranszformátorok ekvivalensek az átcímkézőkkel és bebizonyítottuk, hogy az alakmegőrző tulajdonság eldönthető. Megadtuk a kavics makró fatranszformációk egy felbontását és megmutattuk, hogy a különböző cirkularitási tulajdonságok eldönthetők. Ugyancsak megadtuk a felbontást erős kavics kezelés estén is. Általánosítottuk J. Engelfriet hiararchia tételét súlyozott fatranszformátorokra. Súlyozott faautomatákra definiáltuk a termátíró szemantikát és megmutattuk, hogy ekvivalens az algebari szenmatikával. Algoritmust adtunk annak eldöntésére, hogy egy polinomiálisan súlyozott faautomata véges költségű-e. Vizsgáltuk a súlyozott faautomata különböző változatait: fuzzy faautomata, multioperátor monoid feletti faautomata, Ez utóbbi esetre általánosítottuk a Kleene tételt. | We gave an elementary algorithm for deciding the equivalence of regular words. We generalized Kleene's theorem to weighted automata processing infinite words. We developed an algebraic method that, for several segments of the CTL logic, can be applied to decide if a regular tree language can be defined in that segment. We examined algebraic properties of tree automata, and gave an algebraic characterization of recognizability. We defined multi bottom-up tree transducers and showed that they are equivalent to top-down tree transducers with regular look-ahead. We determined the computation power of the linear subclass. We showed that shape preserving bottom-up tree transducers are equivalent to relabelings. We proved that the shape preserving property is decidable. We gave a decomposition for pebble macro tree transducers and showed that certain circularity properties are decidable. We also gave a decomposition for the strong pebble handling. We have generalized the hierarchy theorem of J. Engelfriet to weighted tree transducers. We defined the term rewrite semantics of weighted tree transducers and showed that it is equivalent to the algebraic semantics. We gave a decision algorithm for the finite cost property of a polynomially weighted tree automata. We defined different versions of weighted tree automata: fuzzy tree automata, weighted tree automata over a multioperator monoid. For the latter we generalized Kleene's theorem

Sound and complete axiomatizations of coalgebraic language equivalence

Coalgebras provide a uniform framework to study dynamical systems, including several types of automata. In this paper, we make use of the coalgebraic view on systems to investigate, in a uniform way, under which conditions calculi that are sound and complete with respect to behavioral equivalence can be extended to a coarser coalgebraic language equivalence, which arises from a generalised powerset construction that determinises coalgebras. We show that soundness and completeness are established by proving that expressions modulo axioms of a calculus form the rational fixpoint of the given type functor. Our main result is that the rational fixpoint of the functor $FT$, where $T$ is a monad describing the branching of the systems (e.g. non-determinism, weights, probability etc.), has as a quotient the rational fixpoint of the "determinised" type functor $\bar F$, a lifting of $F$ to the category of $T$-algebras. We apply our framework to the concrete example of weighted automata, for which we present a new sound and complete calculus for weighted language equivalence. As a special case, we obtain non-deterministic automata, where we recover Rabinovich's sound and complete calculus for language equivalence.Comment: Corrected version of published journal articl

*-Continuous Kleene ω\omega-Algebras for Energy Problems

Energy problems are important in the formal analysis of embedded or autonomous systems. Using recent results on star-continuous Kleene omega-algebras, we show here that energy problems can be solved by algebraic manipulations on the transition matrix of energy automata. To this end, we prove general results about certain classes of finitely additive functions on complete lattices which should be of a more general interest.Comment: In Proceedings FICS 2015, arXiv:1509.0282

Automata and rational expressions

This text is an extended version of the chapter 'Automata and rational expressions' in the AutoMathA Handbook that will appear soon, published by the European Science Foundation and edited by JeanEricPin

Root-Weighted Tree Automata and their Applications to Tree Kernels

In this paper, we define a new kind of weighted tree automata where the weights are only supported by final states. We show that these automata are sequentializable and we study their closures under classical regular and algebraic operations. We then use these automata to compute the subtree kernel of two finite tree languages in an efficient way. Finally, we present some perspectives involving the root-weighted tree automata

Parametrized Stochastic Grammars for RNA Secondary Structure Prediction

We propose a two-level stochastic context-free grammar (SCFG) architecture for parametrized stochastic modeling of a family of RNA sequences, including their secondary structure. A stochastic model of this type can be used for maximum a posteriori estimation of the secondary structure of any new sequence in the family. The proposed SCFG architecture models RNA subsequences comprising paired bases as stochastically weighted Dyck-language words, i.e., as weighted balanced-parenthesis expressions. The length of each run of unpaired bases, forming a loop or a bulge, is taken to have a phase-type distribution: that of the hitting time in a finite-state Markov chain. Without loss of generality, each such Markov chain can be taken to have a bounded complexity. The scheme yields an overall family SCFG with a manageable number of parameters.Comment: 5 pages, submitted to the 2007 Information Theory and Applications Workshop (ITA 2007