IA Scholar Query: A Formalisation of the Myhill-Nerode Theorem Based on Regular Expressions.
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Internet Archive Scholar query results feedeninfo@archive.orgMon, 22 Aug 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A Complexity Approach to Tree Algebras: the Polynomial Case
https://scholar.archive.org/work/o24s3rmzyrh4xg3xb7tqqu7vky
In this paper, we consider infinitely sorted tree algebras recognising regular language of finite trees. We pursue their analysis under the angle of their asymptotic complexity, i.e. the asymptotic size of the sorts as a function of the number of variables involved. Our main result establishes an equivalence between the languages recognised by algebras of polynomial complexity and the languages that can be described by nominal word automata that parse linearisation of the trees. On the way, we show that for such algebras, having polynomial complexity corresponds to having uniformly boundedly many orbits under permutation of the variables, or having a notion of bounded support (in a sense similar to the one in nominal sets). We also show that being recognisable by an algebra of polynomial complexity is a decidable property for a regular language of trees.Thomas Colcombet, Arthur Jaquard, Stefan Szeider, Robert Ganian, Alexandra Silvawork_o24s3rmzyrh4xg3xb7tqqu7vkyMon, 22 Aug 2022 00:00:00 GMTFast Computations on Ordered Nominal Sets
https://scholar.archive.org/work/eurvmukoyrgc7nas4vcswc5bgi
We show how to compute efficiently with nominal sets over the total order symmetry, by developing a direct representation of such nominal sets and basic constructions thereon. In contrast to previous approaches, we work directly at the level of orbits, which allows for an accurate complexity analysis. The approach is implemented as the library ONS (Ordered Nominal Sets). Our main motivation is nominal automata, which are models for recognising languages over infinite alphabets. We evaluate ONS in two applications: minimisation of automata and active automata learning. In both cases, ONS is competitive compared to existing implementations and outperforms them for certain classes of inputs.David Venhoek, Joshua Moerman, Jurriaan Rotwork_eurvmukoyrgc7nas4vcswc5bgiTue, 16 Aug 2022 00:00:00 GMTGuarded Kleene Algebra with Tests: Automata Learning
https://scholar.archive.org/work/xyjchpguj5agpp6dzpixycsn5q
Guarded Kleene Algebra with Tests (GKAT) is the fragment of Kleene Algebra with Tests (KAT) that arises by replacing the union and iteration operations of KAT with predicate-guarded variants. GKAT is more efficiently decidable than KAT and expressive enough to model simple imperative programs, making it attractive for applications to e.g. network verification. In this paper, we further explore GKAT's automata theory, and present GL*, an algorithm for learning the GKAT automaton representation of a black-box, by observing its behaviour. A complexity analysis shows that it is more efficient to learn a representation of a GKAT program with GL* than with Angluin's existing L* algorithm. We implement GL* and L* in OCaml and compare their performances on example programs.Stefan Zetzsche, Alexandra Silva, Matteo Sammartinowork_xyjchpguj5agpp6dzpixycsn5qThu, 23 Jun 2022 00:00:00 GMTA Categorical Framework for Learning Generalised Tree Automata
https://scholar.archive.org/work/vk4lu5j57fbglokig7zfqvtjwy
Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.Gerco van Heerdt, Tobias Kappé, Jurriaan Rot, Matteo Sammartino, Alexandra Silvawork_vk4lu5j57fbglokig7zfqvtjwyMon, 02 May 2022 00:00:00 GMTComputability in constructive type theory
https://scholar.archive.org/work/6nindq52jjdxfe3jnudoeds4yi
We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Rice's theorem, the Myhill isomorphism theorem, and the existence of Post's simple and hypersimple predicates relying on no other axioms such as Markov's principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type → is L-computable. This thesis is a product of more than seven years of research. Since then, my journey through academia has brought me to many great places and introduced me to many great people. I feel that the final product of the journey -this thesis -is a result of a big collaboration of all the people who journeyed with me. Ein riesiger Dank geht an meine Familie. Papa, ich hab die letzten Jahre häufig gemerkt wie viel ich von dir gelernt habe, und wie viel ich noch lernen kann. Die Leichtigkeit, die du zeigst, während du jeden Morgen so früh aufstehst, und die Energie, die du trotzdem für dein Umfeld hast, sind mir ein Vorbild. Mama, deine Liebe, dein Interesse, deine Unterstützung, dein Verständnis und dein Vertrauen waren unabdingbar für mich. Joshua, danke dass du mich in FIFA bisweilen hast gewinnen lassen. Opa, es macht mich immer noch traurig, dass du meinen Werdegang nicht begleiten konntest. Danke für das Fördern meiner Neugier und dein Vertrauen in meine Fähigkeiten ab dem ersten Tag. Ich wäre ohne euch alle nie an den Punkt gekommen, überhaupt diese Dissertation zu schreiben. Mi familia peruana, gracias por todo.Yannick Forster, Universität Des Saarlandeswork_6nindq52jjdxfe3jnudoeds4yiTue, 05 Apr 2022 00:00:00 GMTMembership and separation problems inside two-variable first order logic
https://scholar.archive.org/work/cv3jfs6mdncsrloghnfugdzsbm
This thesis studies the expressive power of restricted fragments of first order logic on words with the order predicate. In particular, we consider particular instances of the following two questions.Given a regular language L, and a fragment of first order logic, can L be described by formulae in this fragment?Given two regular languages L1 and L2, and a fragment of first order logic, can we find a language L which can be described by formulae in this fragment such that L1 is a subset of L and the intersection of L2 and L is empty.The former is known as the membership problem, and the latter the separation problem. These problems are well studied in the theory of regular languages, and their solution has often relied on tools from algebra, such as finite monoids and pointlikes.We consider formulae using only two variables, a fragment known as FO2[For languages of finite words, membership for these fragments has already been solved. We generalise these results in multiple directions.First, we solve the membership problem for the aforementioned fragments for inf-regular languages, i.e. languages consisting of both infinitely and finitely long words. This in particular solves the membership problem for omega-regular languages, i.e. those containing only infinitely long words. Our main tool is to impose algebraic conditions on the syntactic monoid: a monoid which can be computed from any major presentation of regular languages. For the lower levels, we also need to consider topological properties of the languages.The above shows that membership for inf- and omega-regular languages is decidable. However, given a language presented by an automata, calculating the syntactic monoid is generally not efficient. Our second main contribution is therefore an efficient way to decide membership for languages presented by deterministic finite automata (for regular languages) and Carton--Michel automata (for omega-regular languages). We give forbidden patterns for these fragments; that is, we specify [...]Viktor Henrikssonwork_cv3jfs6mdncsrloghnfugdzsbmThu, 03 Feb 2022 00:00:00 GMTCone Types, Automata, and Regular Partitions in Coxeter Groups
https://scholar.archive.org/work/xb46p66cwngxtlfho5orzzalm4
In this article we introduce the notion of a regular partition of a Coxeter group. We develop the theory of these partitions, and show that the class of regular partitions is essentially equivalent to the class of automata (not necessarily finite state) recognising the language of reduced words in the Coxeter group. As an application of this theory we prove that each cone type in a Coxeter group has a unique minimal length representative. This result can be seen as an analogue of Shi's classical result that each component of the Shi arrangement of an affine Coxeter group has a unique minimal length element. We further develop the theory of cone types in Coxeter groups by identifying the minimal set of roots required to express a cone type as an intersection of half-spaces. This set of boundary roots is closely related to the elementary inversion sets of Brink and Howlett, and also to the notion of the base of an inversion set introduced by Dyer.James Parkinson, Yeeka Yauwork_xb46p66cwngxtlfho5orzzalm4Mon, 13 Dec 2021 00:00:00 GMTCanonical automata via distributive law homomorphisms
https://scholar.archive.org/work/rxy4fmoacbgsznt62ipef7vbcm
The classical powerset construction is a standard method converting a nondeterministic automaton into a deterministic one recognising the same language. Recently, the powerset construction has been lifted to a more general framework that converts an automaton with side-effects, given by a monad, into a deterministic automaton accepting the same language. The resulting automaton has additional algebraic properties, both in the state space and transition structure, inherited from the monad. In this paper, we study the reverse construction and present a framework in which a deterministic automaton with additional algebraic structure over a given monad can be converted into an equivalent succinct automaton with side-effects. Apart from recovering examples from the literature, such as the canonical residual finite-state automaton and the \'atomaton, we discover a new canonical automaton for a regular language by relating the free vector space monad over the two element field to the neighbourhood monad. Finally, we show that every regular language satisfying a suitable property parametric in two monads admits a size-minimal succinct acceptor.Stefan Zetzsche, Gerco van Heerdt, Alexandra Silva, Matteo Sammartinowork_rxy4fmoacbgsznt62ipef7vbcmTue, 16 Nov 2021 00:00:00 GMT