20 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
String Diagram Rewriting Modulo Commutative (Co)Monoid Structure
String diagrams constitute an intuitive and expressive graphical syntax that has found application in a very diverse range of fields including concurrency theory, quantum computing, control theory, machine learning, linguistics, and digital circuits. Rewriting theory for string diagrams relies on a combinatorial interpretation as double-pushout rewriting of certain hypergraphs. As previously studied, there is a “tension” in this interpretation: in order to make it sound and complete, we either need to add structure on string diagrams (in particular, Frobenius algebra structure) or pose restrictions on double-pushout rewriting (resulting in “convex” rewriting). From the string diagram viewpoint, imposing a full Frobenius structure may not always be natural or desirable in applications, which motivates our study of a weaker requirement: commutative monoid structure. In this work we characterise string diagram rewriting modulo commutative monoid equations, via a sound and complete interpretation in a suitable notion of double-pushout rewriting of hypergraphs
Open System Categorical Quantum Semantics in Natural Language Processing
Originally inspired by categorical quantum mechanics (Abramsky and Coecke, LiCS'04), the categorical compositional distributional model of natural language meaning of Coecke, Sadrzadeh and Clark provides a conceptually motivated procedure to compute the meaning of a sentence, given its grammatical structure within a Lambek pregroup and a vectorial representation of the meaning of its parts. The predictions of this first model have outperformed that of other models in mainstream empirical language processing tasks on large scale data. Moreover, just like CQM allows for varying the model in which we interpret quantum axioms, one can also vary the model in which we interpret word meaning. In this paper we show that further developments in categorical quantum mechanics are relevant to natural language processing too. Firstly, Selinger's CPM-construction allows for explicitly taking into account lexical ambiguity and distinguishing between the two inherently different notions of homonymy and polysemy. In terms of the model in which we interpret word meaning, this means a passage from the vector space model to density matrices. Despite this change of model, standard empirical methods for comparing meanings can be easily adopted, which we demonstrate by a small-scale experiment on real-world data. This experiment moreover provides preliminary evidence of the validity of our proposed new model for word meaning. Secondly, commutative classical structures as well as their non-commutative counterparts that arise in the image of the CPM-construction allow for encoding relative pronouns, verbs and adjectives, and finally, iteration of the CPM-construction, something that has no counterpart in the quantum realm, enables one to accommodate both entailment and ambiguity
Bialgebraic foundations for the operational semantics of 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) 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 semantics as a distributive law for that monad. As a proof of concept, we provide bialgebraic semantics for a versatile string diagrammatic language which has been used to model both signal flow graphs (control theory) and Petri nets (concurrency theory)
Diagrammatic Algebra: from Linear to Concurrent Systems
We introduce the resource calculus, a string diagrammatic language for concurrent systems. Significantly, it
uses the same syntax and operational semantics as the signal flow calculus — an algebraic formalism for signal
flow graphs, which is a combinatorial model of computation of interest in control theory. Indeed, our approach
stems from the simple but fruitful observation that, by replacing real numbers (modelling signals) with natural
numbers (modelling resources) in the operational semantics, concurrent behaviour patterns emerge.
The resource calculus is canonical: we equip it and its stateful extension with equational theories that
characterise the underlying space of definable behaviours—a convex algebraic universe of additive relations—
via isomorphisms of categories. Finally, we demonstrate that our calculus is sufficiently expressive to capture
behaviour definable by classical Petri net
Graphical affine algebra
Graphical linear algebra is a diagrammatic language
allowing to reason compositionally about different types of linear
computing devices. In this paper, we extend this formalism with
a connector for affine behaviour. The extension, which we call
graphical affine algebra, is simple but remarkably powerful: it
can model systems with richer patterns of behaviour such as
mutual exclusion—with modules over the natural numbers as
semantic domain—or non-passive electrical components—when
considering modules over a certain field. Our main technical
contribution is a complete axiomatisation for graphical affine
algebra over these two interpretations. We also show, as case
studies, how graphical affine algebra captures electrical circuits
and the calculus of stateless connectors—a coordination language
for distributed system
Contextual Equivalence for Signal Flow Graphs
We extend the signal flow calculus—a compositional account of the classical signal flow graph model of computation—to encompass affine behaviour, and furnish it with a novel operational semantics. The increased expressive power allows us to define a canonical notion of contextual equivalence, which we show to coincide with denotational equality. Finally, we characterise the realisable fragment of the calculus: those terms that express the computations of (affine) signal flow graphs
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 for 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, à la Hoare and à la Milner
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 for 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, Ă la Hoare and Ă la Milner
Contextual Equivalence for Signal Flow Graphs
We extend the signal flow calculus—a compositional account of the classical signal flow graph model of computation—to encompass affine behaviour, and furnish it with a novel operational semantics. The increased expressive power allows us to define a canonical notion of contextual equivalence, which we show to coincide with denotational equality. Finally, we characterise the realisable fragment of the calculus: those terms that express the computations of (affine) signal flow graphs