31 research outputs found
Automata Minimization: a Functorial Approach
In this paper we regard languages and their acceptors - such as deterministic
or weighted automata, transducers, or monoids - as functors from input
categories that specify the type of the languages and of the machines to
categories that specify the type of outputs. Our results are as follows:
A) We provide sufficient conditions on the output category so that
minimization of the corresponding automata is guaranteed.
B) We show how to lift adjunctions between the categories for output values
to adjunctions between categories of automata.
C) We show how this framework can be instantiated to unify several phenomena
in automata theory, starting with determinization, minimization and syntactic
algebras. We provide explanations of Choffrut's minimization algorithm for
subsequential transducers and of Brzozowski's minimization algorithm in this
setting.Comment: journal version of the CALCO 2017 paper arXiv:1711.0306
Forward and Backward Steps in a Fibration
Distributive laws of various kinds occur widely in the theory of coalgebra, for instance to model automata constructions and trace semantics, and to interpret coalgebraic modal logic. We study steps, which are a general type of distributive law, that allow one to map coalgebras along an adjunction. In this paper, we address the question of what such mappings do to well known notions of equivalence, e.g., bisimilarity, behavioural equivalence, and logical equivalence.
We do this using the characterisation of such notions of equivalence as (co)inductive predicates in a fibration. Our main contribution is the identification of conditions on the interaction between the steps and liftings, which guarantees preservation of fixed points by the mapping of coalgebras along the adjunction. We apply these conditions in the context of lax liftings proposed by Bonchi, Silva, Sokolova (2021), and generalise their result on preservation of bisimilarity in the construction of a belief state transformer. Further, we relate our results to properties of coalgebraic modal logics including expressivity and completeness
Automata Minimization: a Functorial Approach
In this paper we regard languages and their acceptors - such as deterministic or weighted automata, transducers, or monoids - as
functors from input categories that specify the type of the
languages and of the machines to categories that specify the type of
outputs.
Our results are as follows: a) We provide sufficient conditions on the output category so that minimization of the corresponding automata is guaranteed. b) We show how to lift adjunctions between the categories for output values to adjunctions between categories of automata. c) We show how this framework can be applied to several phenomena in automata theory, starting with determinization and
minimization (previously studied from a coalgebraic and duality theoretic perspective). We apply in particular these techniques to
Choffrut\u27s minimization algorithm for subsequential transducers and revisit Brzozowski\u27s minimization algorithm
Minimisation in Logical Form
Stone-type dualities provide a powerful mathematical framework for studying
properties of logical systems. They have recently been fruitfully explored in
understanding minimisation of various types of automata. In Bezhanishvili et
al. (2012), a dual equivalence between a category of coalgebras and a category
of algebras was used to explain minimisation. The algebraic semantics is dual
to a coalgebraic semantics in which logical equivalence coincides with trace
equivalence. It follows that maximal quotients of coalgebras correspond to
minimal subobjects of algebras. Examples include partially observable
deterministic finite automata, linear weighted automata viewed as coalgebras
over finite-dimensional vector spaces, and belief automata, which are
coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowski's
double-reversal minimisation algorithm for deterministic finite automata was
described categorically and its correctness explained via the duality between
reachability and observability. This work includes generalisations of
Brzozowski's algorithm to Moore and weighted automata over commutative
semirings.
In this paper we propose a general categorical framework within which such
minimisation algorithms can be understood. The goal is to provide a unifying
perspective based on duality. Our framework consists of a stack of three
interconnected adjunctions: a base dual adjunction that can be lifted to a dual
adjunction between coalgebras and algebras and also to a dual adjunction
between automata. The approach provides an abstract understanding of
reachability and observability. We illustrate the general framework on range of
concrete examples, including deterministic Kripke frames, weighted automata,
topological automata (belief automata), and alternating automata
Separation and Renaming in Nominal Sets
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions
Coinduction in Flow: The Later Modality in Fibrations
This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search.
Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results