20 research outputs found
Kaemika app, Integrating protocols and chemical simulation
Kaemika is an app available on the four major app stores. It provides
deterministic and stochastic simulation, supporting natural chemical notation
enhanced with recursive and conditional generation of chemical reaction
networks. It has a liquid-handling protocol sublanguage compiled to a virtual
digital microfluidic device. Chemical and microfluidic simulations can be
interleaved for full experimental-cycle modeling. A novel and unambiguous
representation of directed multigraphs is used to lay out chemical reaction
networks in graphical form
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
Alpha Equivalence Equalities
Programming languages and logics, which are pervasive in Computer Science, have syntax which involves variable binding constructors. As such, reasoning about such languages in general, and formal reasoning in particular (such as within a theorem prover), requires frameworks within which the syntax may be properly represented. One key requirement is a correct representation of α-equivalence. The current literature provides a number of different definitions of the notion of α-equivalence. The formal definitions may be nameless as in the approach of de Bruijn, or have explicit names, as in the approaches that use either a renaming/substitution axiom, or instead use a notion of variable swapping. The first contribution of this paper is to draw together five definitions of α-equivalence relations and to prove formally and in detail, but using mathematics, that the relations are all equal. There are two key reasons for doing this: Firstly, the literature has many examples of proofs of results involving α-equivalence which contain technical errors. Such examples concern both the application of α-equivalence, and the meta-theory of α-equivalence itself. Secondly, the literature does not currently contain detailed presentations of such results. The point of giving the detail is partly to avoid falling into common error-traps, but mainly to provide clear mathematical machinery that will be useful to those working in the area. This includes systems of inductive rules and proofs by induction, and clear accounts of the key lemmas that support the main proofs. The second contribution is to provide two definitions of α-equivalence relations over (program) contexts, namely expressions with a single meta-variable (or “hole”). One of the definitions is already in the literature, and the other is new. We prove some basic properties of α-equivalence on contexts, and show that the two definitions give rise to the same relation
Basic Category Theory for Models of Syntax
These notes form the basis of four lectures given at the Summer School on Generic Programming, Oxford, UK, which took place during August 2002. The aims of the notes are to provide an introduction to very elementary category theory, and to show how such category theory can be used to provide both abstract and concrete mathematical models of syntax. Much of the material is now standard, but some of the ideas which are used in the modeling of syntax involving variable binding are quite new. It is assumed that readers..
A Hybrid Encoding of Howe's Method for Establishing Congruence of Bisimilarity
We give a short description of Hybrid, a new tool for interactive theorem proving. It provides a form of Higher Order Abstract Syntax (HOAS) combined consistently with induction and coinduction. We present a case study illustrating the use of Hybrid for reasoning about the lazy lambda-calculus. In particular, we prove that the standard notion of simulation is a precongruence. Although such a proof is not new, the development is non-trivial, and we attempt to illustrate the advantages of using Hybrid, as well as some issues which are being addressed as further work