17 research outputs found
A Finite Axiomatisation of Finite-State Automata Using String Diagrams
We develop a fully diagrammatic approach to finite-state automata, based on
reinterpreting their usual state-transition graphical representation as a
two-dimensional syntax of string diagrams. In this setting, we are able to
provide a complete equational theory for language equivalence, with two notable
features. First, the proposed axiomatisation is finite. Second, the Kleene star
is a derived concept, as it can be decomposed into more primitive algebraic
blocks.Comment: arXiv admin note: text overlap with arXiv:2009.1457
A FINITE AXIOMATISATION OF FINITE-STATE AUTOMATA USING STRING DIAGRAMS
We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks
Categorical Modelling of Logic Programming: Coalgebra, Functorial Semantics, String Diagrams
Logic programming (LP) is driven by the idea that logic subsumes computation. Over the
past 50 years, along with the emergence of numerous logic systems, LP has also grown into a
large family, the members of which are designed to deal with various computation scenarios.
Among them, we focus on two of the most influential quantitative variants are probabilistic
logic programming (PLP) and weighted logic programming (WLP).
In this thesis, we investigate a uniform understanding of logic programming and its quan-
titative variants from the perspective of category theory. In particular, we explore both a
coalgebraic and an algebraic understanding of LP, PLP and WLP.
On the coalgebraic side, we propose a goal-directed strategy for calculating the probabilities
and weights of atoms in PLP and WLP programs, respectively. We then develop a coalgebraic
semantics for PLP and WLP, built on existing coalgebraic semantics for LP. By choosing
the appropriate functors representing probabilistic and weighted computation, such coalgeraic
semantics characterise exactly the goal-directed behaviour of PLP and WLP programs.
On the algebraic side, we define a functorial semantics of LP, PLP, and WLP, such that they
three share the same syntactic categories of string diagrams, and differ regarding to the semantic
categories according to their data/computation type. This allows for a uniform diagrammatic
expression for certain semantic constructs. Moreover, based on similar approaches to Bayesian
networks, this provides a framework to formalise the connection between PLP and Bayesian
networks. Furthermore, we prove a sound and complete aximatization of the semantic category
for LP, in terms of string diagrams. Together with the diagrammatic presentation of the
fixed point semantics, one obtain a decidable calculus for proving the equivalence between
propositional definite logic programs
Vector Semantics
This open access book introduces Vector semantics, which links the formal theory of word vectors to the cognitive theory of linguistics. The computational linguists and deep learning researchers who developed word vectors have relied primarily on the ever-increasing availability of large corpora and of computers with highly parallel GPU and TPU compute engines, and their focus is with endowing computers with natural language capabilities for practical applications such as machine translation or question answering. Cognitive linguists investigate natural language from the perspective of human cognition, the relation between language and thought, and questions about conceptual universals, relying primarily on in-depth investigation of language in use. In spite of the fact that these two schools both have ‘linguistics’ in their name, so far there has been very limited communication between them, as their historical origins, data collection methods, and conceptual apparatuses are quite different. Vector semantics bridges the gap by presenting a formal theory, cast in terms of linear polytopes, that generalizes both word vectors and conceptual structures, by treating each dictionary definition as an equation, and the entire lexicon as a set of equations mutually constraining all meanings
String diagram rewrite theory II: Rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
Foundations of Software Science and Computation Structures
This open access book constitutes the proceedings of the 22nd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2019, which took place in Prague, Czech Republic, in April 2019, held as part of the European Joint Conference on Theory and Practice of Software, ETAPS 2019. The 29 papers presented in this volume were carefully reviewed and selected from 85 submissions. They deal with foundational research with a clear significance for software science
A String Diagrammatic Axiomatisation of Finite-State Automata
We develop a fully diagrammatic approach to the theory of finite-state
automata, based on reinterpreting their usual state-transition graphical
representation as a two-dimensional syntax of string diagrams. Moreover, we
provide an equational theory that completely axiomatises language equivalence
in this new setting. This theory has two notable features. First, the Kleene
star is a derived concept, as it can be decomposed into more primitive
algebraic blocks. Second, the proposed axiomatisation is finitary -- a result
which is provably impossible to obtain for the one-dimensional syntax of
regular expressions.Comment: Minor corrections, in particular in the proof of completeness
(including the ordering of the steps of Brzozowski's algorithm
Bialgebraic Semantics for String Diagrams
Turi and Plotkin's bialgebraic semantics is an abstract approach to
specifying the operational semantics of a system, by means of a distributive
law between its syntax (encoded as a monad) and its dynamics (an endofunctor).
This setup is instrumental in showing that a semantic specification (a
coalgebra) satisfies desirable properties: in particular, that it is
compositional.
In this work, we use the bialgebraic approach to derive well-behaved
structural operational semantics of string diagrams, a graphical syntax that is
increasingly used in the study of interacting systems across different
disciplines. Our analysis relies on representing the two-dimensional operations
underlying string diagrams in various categories as a monad, and their
bialgebraic semantics in terms of a distributive law over that monad.
As a proof of concept, we provide bialgebraic compositional semantics for a
versatile string diagrammatic language which has been used to model both signal
flow graphs (control theory) and Petri nets (concurrency theory). Moreover, our
approach reveals a correspondence between two different interpretations of the
Frobenius equations on string diagrams and two synchronisation mechanisms for
processes, \`a la Hoare and \`a la Milner.Comment: Accepted for publications in the proceedings of the 30th
International Conference on Concurrency Theory (CONCUR 2019