Degree of Sequentiality of Weighted Automata

Weighted automata (WA) are an important formalism to describe quantitative properties. Obtaining equivalent deterministic machines is a longstanding research problem. In this paper we consider WA with a set semantics, meaning that the semantics is given by the set of weights of accepting runs. We focus on multi-sequential WA that are defined as finite unions of sequential WA. The problem we address is to minimize the size of this union. We call this minimum the degree of sequentiality of (the relation realized by) the WA. For a given positive integer k, we provide multiple characterizations of relations realized by a union of k sequential WA over an infinitary finitely generated group: a Lipschitz-like machine independent property, a pattern on the automaton (a new twinning property) and a subclass of cost register automata. When possible, we effectively translate a WA into an equivalent union of k sequential WA. We also provide a decision procedure for our twinning property for commutative computable groups thus allowing to compute the degree of sequentiality. Last, we show that these results also hold for word transducers and that the associated decision problem is PSPACE -complete

Containment and equivalence of weighted automata: Probabilistic and max-plus cases

This paper surveys some results regarding decision problems for probabilistic and max-plus automata, such as containment and equivalence. Probabilistic and max-plus automata are part of the general family of weighted automata, whose semantics are maps from words to real values. Given two weighted automata, the equivalence problem asks whether their semantics are the same, and the containment problem whether one is point-wise smaller than the other one. These problems have been studied intensively and this paper will review some techniques used to show (un)decidability and state a list of open questions that still remain

Finite Sequentiality of Finitely Ambiguous Max-Plus Tree Automata

We show that the finite sequentiality problem is decidable for finitely ambiguous max-plus tree automata. A max-plus tree automaton is a weighted tree automaton over the max-plus semiring. A max-plus tree automaton is called finitely ambiguous if the number of accepting runs on every tree is bounded by a global constant. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton

Finite Sequentiality of Unambiguous Max-Plus Tree Automata

We show the decidability of the finite sequentiality problem for unambiguous max-plus tree automata. A max-plus tree automaton is called unambiguous if there is at most one accepting run on every tree. The finite sequentiality problem asks whether for a given max-plus tree automaton, there exist finitely many deterministic max-plus tree automata whose pointwise maximum is equivalent to the given automaton

A Robust Class of Linear Recurrence Sequences

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers

Register complexity and determinisation of max-plus automata

We survey some results about the sequentiality problem for max-plus automata and its generalisation, the register complexity problem for cost register automata. We compare classes of functions computed by maxplus automata and by cost register automata with respect to the notion of ambiguity. The two models are introduced gently, so the novice reader is welcome

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