473 research outputs found
A Kleene Theorem for Forest Languages
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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 BĂĽchi-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 BĂĽchi-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 BĂĽchi 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
A Survey on the Local Divisor Technique
Local divisors allow a powerful induction scheme on the size of a monoid. We
survey this technique by giving several examples of this proof method. These
applications include linear temporal logic, rational expressions with Kleene
stars restricted to prefix codes with bounded synchronization delay,
Church-Rosser congruential languages, and Simon's Factorization Forest Theorem.
We also introduce the notion of localizable language class as a new abstract
concept which unifies some of the proofs for the results above
A survey on the local divisor technique
© 2015 Elsevier B.V. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of a localizable language class as a new abstract concept which unifies some of the proofs for the results above
The submonoid and rational subset membership problems for graph groups
We show that the membership problem in a finitely generated submonoid of a
graph group (also called a right-angled Artin group or a free partially
commutative group) is decidable if and only if the independence graph
(commutation graph) is a transitive forest. As a consequence we obtain the
first example of a finitely presented group with a decidable generalized word
problem that does not have a decidable membership problem for finitely
generated submonoids. We also show that the rational subset membership problem
is decidable for a graph group if and only if the independence graph is a
transitive forest, answering a question of Kambites, Silva, and the second
author. Finally we prove that for certain amalgamated free products and
HNN-extensions the rational subset and submonoid membership problems are
recursively equivalent. In particular, this applies to finitely generated
groups with two or more ends that are either torsion-free or residually finite
Symbolic Algorithms for Language Equivalence and Kleene Algebra with Tests
We first propose algorithms for checking language equivalence of finite
automata over a large alphabet. We use symbolic automata, where the transition
function is compactly represented using a (multi-terminal) binary decision
diagrams (BDD). The key idea consists in computing a bisimulation by exploring
reachable pairs symbolically, so as to avoid redundancies. This idea can be
combined with already existing optimisations, and we show in particular a nice
integration with the disjoint sets forest data-structure from Hopcroft and
Karp's standard algorithm. Then we consider Kleene algebra with tests (KAT), an
algebraic theory that can be used for verification in various domains ranging
from compiler optimisation to network programming analysis. This theory is
decidable by reduction to language equivalence of automata on guarded strings,
a particular kind of automata that have exponentially large alphabets. We
propose several methods allowing to construct symbolic automata out of KAT
expressions, based either on Brzozowski's derivatives or standard automata
constructions. All in all, this results in efficient algorithms for deciding
equivalence of KAT expressions
On factorisation forests
The theorem of factorisation forests shows the existence of nested
factorisations -- a la Ramsey -- for finite words. This theorem has important
applications in semigroup theory, and beyond. The purpose of this paper is to
illustrate the importance of this approach in the context of automata over
infinite words and trees. We extend the theorem of factorisation forest in two
directions: we show that it is still valid for any word indexed by a linear
ordering; and we show that it admits a deterministic variant for words indexed
by well-orderings. A byproduct of this work is also an improvement on the known
bounds for the original result. We apply the first variant for giving a
simplified proof of the closure under complementation of rational sets of words
indexed by countable scattered linear orderings. We apply the second variant in
the analysis of monadic second-order logic over trees, yielding new results on
monadic interpretations over trees. Consequences of it are new caracterisations
of prefix-recognizable structures and of the Caucal hierarchy.Comment: 27 page
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Automated verification of refinement laws
Demonic refinement algebras are variants of Kleene algebras. Introduced by von Wright as a light-weight variant of the refinement calculus, their intended semantics are positively disjunctive predicate transformers, and their calculus is entirely within first-order equational logic. So, for the first time, off-the-shelf automated theorem proving (ATP) becomes available for refinement proofs. We used ATP to verify a toolkit of basic refinement laws. Based on this toolkit, we then verified two classical complex refinement laws for action systems by ATP: a data refinement law and Back's atomicity refinement law. We also present a refinement law for infinite loops that has been discovered through automated analysis. Our proof experiments not only demonstrate that refinement can effectively be automated, they also compare eleven different ATP systems and suggest that program verification with variants of Kleene algebras yields interesting theorem proving benchmarks. Finally, we apply hypothesis learning techniques that seem indispensable for automating more complex proofs
Exhaustible sets in higher-type computation
We say that a set is exhaustible if it admits algorithmic universal
quantification for continuous predicates in finite time, and searchable if
there is an algorithm that, given any continuous predicate, either selects an
element for which the predicate holds or else tells there is no example. The
Cantor space of infinite sequences of binary digits is known to be searchable.
Searchable sets are exhaustible, and we show that the converse also holds for
sets of hereditarily total elements in the hierarchy of continuous functionals;
moreover, a selection functional can be constructed uniformly from a
quantification functional. We prove that searchable sets are closed under
intersections with decidable sets, and under the formation of computable images
and of finite and countably infinite products. This is related to the fact,
established here, that exhaustible sets are topologically compact. We obtain a
complete description of exhaustible total sets by developing a computational
version of a topological Arzela--Ascoli type characterization of compact
subsets of function spaces. We also show that, in the non-empty case, they are
precisely the computable images of the Cantor space. The emphasis of this paper
is on the theory of exhaustible and searchable sets, but we also briefly sketch
applications
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