1,107 research outputs found
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
Formal logic: Classical problems and proofs
Not focusing on the history of classical logic, this book provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational
Diagrammatic Polyhedral Algebra
We extend the theory of Interacting Hopf algebras with an order primitive, and give a sound and complete axiomatisation of the prop of polyhedral cones. Next, we axiomatise an affine extension and prove soundness and completeness for the prop of polyhedra
A General Framework for Sound and Complete Floyd-Hoare Logics
This paper presents an abstraction of Hoare logic to traced symmetric
monoidal categories, a very general framework for the theory of systems. Our
abstraction is based on a traced monoidal functor from an arbitrary traced
monoidal category into the category of pre-orders and monotone relations. We
give several examples of how our theory generalises usual Hoare logics (partial
correctness of while programs, partial correctness of pointer programs), and
provide some case studies on how it can be used to develop new Hoare logics
(run-time analysis of while programs and stream circuits).Comment: 27 page
An Introduction to String Diagrams for Computer Scientists
This document is an elementary introduction to string diagrams. It takes a
computer science perspective: rather than using category theory as a starting
point, we build on intuitions from formal language theory, treating string
diagrams as a syntax with its semantics. After the basic theory, pointers are
provided to contemporary applications of string diagrams in various fields of
science
A compositional theory of digital circuits
A theory is compositional if complex components can be constructed out of
simpler ones on the basis of their interfaces, without inspecting their
internals. Digital circuits, despite being studied for nearly a century and
used at scale for about half that time, have until recently evaded a fully
compositional theoretical understanding. The sticking point has been the need
to avoid feedback loops that bypass memory elements, the so called
'combinational feedback' problem. This requires examining the internal
structure of a circuit, defeating compositionality. Recent work remedied this
theoretical shortcoming by showing how digital circuits can be presented
compositionally as morphisms in a freely generated Cartesian traced (or
dataflow) category. The focus was to support a better syntactical understanding
of digital circuits, culminating in the formulation of novel operational
semantics for digital circuits. In this paper we shift the focus onto the
denotational theory of such circuits, interpreting them as functions on streams
with to certain properties. These ensure that the model is fully abstract, i.e.
the equational theory and the semantic model are in perfect agreement. To
support this result we introduce two key equations: the first can reduce
circuits with combinational feedback to circuits without combinational feedback
via finite unfoldings of the loop, and the second can translate between open
circuits with the same behaviour syntactically by reducing the problem to
checking a finite number of closed circuits. The most important consequence of
this new semantics is that we can now give a recipe that ensures a circuit
always produces observable output, thus using the denotational model to inform
and improve the operational semantics.Comment: Restructured and refined presentation, 21 page
Constructor Theory as Process Theory
Constructor theory is a meta-theoretic approach that seeks to characterise
concrete theories of physics in terms of the (im)possibility to implement
certain abstract "tasks" by means of physical processes. Process theory, on the
other hand, pursues analogous characterisation goals in terms of the
compositional structure of said processes, concretely presented through the
lens of (symmetric monoidal) category theory. In this work, we show how to
formulate fundamental notions of constructor theory within the canvas of
process theory. Specifically, we exploit the functorial interplay between the
symmetric monoidal structure of the category of sets and relations, where the
abstract tasks live, and that of symmetric monoidal categories from physics,
where concrete processes can be found to implement said tasks. Through this, we
answer the question of how constructor theory relates to the broader body of
process-theoretic literature, and provide the impetus for future collaborative
work between the fields.Comment: In Proceedings ACT 2023, arXiv:2312.0813
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