28 research outputs found
String Diagrammatic Trace Theory
We extend the theory of formal languages in monoidal categories to the
multi-sorted, symmetric case, and show how this theory permits a graphical
treatment of topics in concurrency. In particular, we show that Mazurkiewicz
trace languages are precisely symmetric monoidal languages over monoidal
distributed alphabets. We introduce symmetric monoidal automata, which define
the class of regular symmetric monoidal languages. Furthermore, we prove that
Zielonka's asynchronous automata coincide with symmetric monoidal automata over
monoidal distributed alphabets. Finally, we apply the string diagrams for
symmetric premonoidal categories to derive serializations of traces.Comment: Paper accepted for MFCS 202
Commutativity
We describe a general framework for notions of commutativity based on
enriched category theory. We extend Eilenberg and Kelly's tensor product for
categories enriched over a symmetric monoidal base to a tensor product for
categories enriched over a normal duoidal category; using this, we re-find
notions such as the commutativity of a finitary algebraic theory or a strong
monad, the commuting tensor product of two theories, and the Boardman-Vogt
tensor product of symmetric operads.Comment: 48 pages; final journal versio
Resource modalities in game semantics
The description of resources in game semantics has never achieved the
simplicity and precision of linear logic, because of a misleading conception:
the belief that linear logic is more primitive than game semantics. We advocate
instead the contrary: that game semantics is conceptually more primitive than
linear logic. Starting from this revised point of view, we design a categorical
model of resources in game semantics, and construct an arena game model where
the usual notion of bracketing is extended to multi- bracketing in order to
capture various resource policies: linear, affine and exponential
Promonads and String Diagrams for Effectful Categories
Premonoidal and Freyd categories are both generalized by non-cartesian Freyd
categories: effectful categories. We construct string diagrams for effectful
categories in terms of the string diagrams for a monoidal category with a
freely added object. We show that effectful categories are pseudomonoids in a
monoidal bicategory of promonads with a suitable tensor product.Comment: In Proceedings ACT 2022, arXiv:2307.1551
Categorical structure of continuation passing style
Laboratory for Foundations of Computer ScienceThis thesis attempts to make precise the structure inherent in Continuation Passing Style (CPS).
We emphasize that CPS translates lambda-calculus into a very basic calculus that does not have functions as primitive.
We give an abstract categorical presentation of continuation semantics by taking the continuation type constructor (cont in Standard ML of New Jersey) as primitive. This constructor on types extends to a contravariant functor on terms which is adjoint to itself on the left; restricted to the subcategory of those programs that do not manipulate the current continuation, it is adjoint to itself on the right.
The motivating example of such a category is built from (equivalence classes of typing judgements for) continuation passing style (CPS) terms. The categorical approach suggests a notion of effect-free term as well as some operators for manipulating continuations. We use these for writing programs that illustrate our categorical approach and refute some conjectures about control effects.
A call-by-value lambda-calculus with the control operator callcc can be interpreted. Arrow types are broken down into continuation types for argument/result-continuations pairs, reflecting the fact that CPS compiles functions into a special case of continuations. Variant translations are possible, among them lazy call-by-name, which can be derived by way of argument thunking, and a genuinely call-by-name transform. Specialising the semantics to the CPS term model allows a rational reconstruction of various CPS transforms
Rewriting for Monoidal Closed Categories
This paper develops a formal string diagram language for monoidal closed categories. Previous work has shown that string diagrams for freely generated symmetric monoidal categories can be viewed as hypergraphs with interfaces, and the axioms of these categories can be realized by rewriting systems. This work proposes hierarchical hypergraphs as a suitable formalization of string diagrams for monoidal closed categories. We then show double pushout rewriting captures the axioms of these closed categories
Substructural Simple Type Theories for Separation and In-place Update
This thesis studies two substructural simple type theories, extending
the "separation" and "number-of-uses" readings of the basic
substructural simply typed lambda-calculus with exchange.
The first calculus, lambda_sep, extends the alpha lambda-calculus of
O'Hearn and Pym by directly considering the representation of separation
in a type system. We define type contexts with separation relations and
introduce new type constructors of separated products and separated
functions. We describe the basic metatheory of the calculus, including a
sound and complete type-checking algorithm. We then give new categorical
structure for interpreting the type judgements, and prove that it
coherently, soundly and completely interprets the type theory. To show
how the structure models separation we extend Day's construction of
closed symmetric monoidal structure on functor categories to our
categorical structure, and describe two instances dealing with the
global and local separation.
The second system, lambda_inplc, is a re-presentation of substructural
calculus for in-place update with linear and non-linear values, based on
Wadler's Linear typed system with non-linear types and Hofmann's LFPL.
We identify some problems with the metatheory of the calculus, in
particular the failure of the substitution rule to hold due to the
call-by-value interpretation inherent in the type rules. To resolve this
issue, we turn to categorical models of call-by-value computation,
namely Moggi's Computational Monads and Power and Robinson's
Freyd-Categories. We extend both of these to include additional
information about the current state of the computation, defining
Parameterised Freyd-categories and Parameterised Strong Monads. These
definitions are equivalent in the closed case. We prove that by adding a
commutativity condition they are a sound class of models for
lambda_inplc. To obtain a complete class of models for lambda_inplc we
refine the structure to better match the syntax. We also give a direct
syntactic presentation of Parameterised Freyd-categories and prove that
it is soundly and completely modelled by the syntax. We give a concrete
model based on Day's construction, demonstrating how the categorical
structure can be used to model call-by-value computation with in-place
update and bounded heaps