Geometry of Interaction is based on the idea that the ultimate explanation of logical rules is through the cut-elimination procedure. This is achieved by means of a pure geometric interpretation of normalization: 


proofs are operators on the Hilbert space describing I/O dependencies


cut-elimination is the solution of an I/O equation 



(the cut σ expressing a feedback of some output of the proof U to some input of U)


termination is nilpotency of the operator σU


execution is expressed by

We propose a new technique for hardware synthesis from higher-order functional languages with imperative features based on Reynolds's Syntactic Control of Interference. The restriction on contraction in the type system is useful for managing the thorny issue of sharing of physical circuits. We use a semantic model inspired by game semantics and the geometry of interaction, and express it directly as a certain class of digital circuits that form a cartesian, monoidal-closed category. A soundness result is given, which is also a correctness result for the compilation technique.

http://www.cs.bham.ac.uk/~drg/papers/popl07x.pdf

Geometry of Interaction (Abstract)

Geometry of synthesis: a structured approach to VLSI design

We propose a new technique for hardware synthesis from higher- order functional languages with imperative features based on Reynolds's Syntactic Control of Interference. The restriction on contraction in the type system is useful for managing the thorny issue of sharing of physical circuits. We use a semantic model in- spired by game semantics and the geometry of interaction, and express it directly as a certain class of digital circuits that form a cartesian, monoidal-closed category. A soundness result is given, which is also a correctness result for the compilation technique. We present a class of digital circuits which we call handshake circuits, and which are shown to form a closed monoidal category, and therefore provide the appropriate structure for interpreting the affine features of the language. We then show how a further refined class of circuits which we call simple-handshake circuits forms a Cartesian sub-category of the previous, and therefore has the right structure for the interpretation of products. This feature is needed for modelling sharing of resources in the language. We show that this model of bSCI is sound relative to an operational definition of the language. This is also a proof of correctness for the synthesis of digital circuits from bSCI programs. The most striking feature of this synthesis technique is the simplic- ity of the resulting circuitry. Abstraction and application are syn- thesised simply as wiring of the circuit representing the body of the function to that of the argument. Some of the other operations, such as sequential composition, assignment and dereferencing of variables reduce also to wiring. The rest of the constructs of the language require only a handful of basic circuits. An important issue in hardware synthesis is that of concurrency. Because hardware is naturally concurrent, the implementation of concurrent programs is no more expensive than that of sequen- tial programs. Concurrency in the framework of bSCI does not al- low shared-variable inter-process communication, and we examine several pragmatic options for overcoming this limitation. Some of these techniques are potentially unsound. As a proof of concept, we have implemented a prototype compiler from bSCI to Verilog specifications of VLSI circuits, based on a naive realisation of handshake circuits.

A structured approach to VLSI design

Starting from works aimed at extending the Curry-Howard correspondence beyond the functional world, in particular to process calculi, thorugh linear logic, we give another Curry-Howard counterpart for Milner's Calculus of Communicating Systems (CCS), by taking Girard’s ludics as the target system.

https://tel.archives-ouvertes.fr/tel-02613539/document

Process Algebras inside Ludics : an interpretation of the Calculus of Communicating Systems

We outline the logic of current approaches to the so-called “frame problem” (that is, the problem of predicting change in the physical world by using logical inference), and we show that these approaches are not completely extensional since none of them is closed under uniform substitution. The underlying difficulty is something known, in the philosophical community, as Goodman’s “new riddle of induction” or the “Grue paradox”. Although it seems, from the philosophical discussion, that this paradox cannot be solved in purely a priori terms and that a solution will require some form of real-world data, it nevertheless remains obscure both what the logical form of this real-world data might be, and also how such data actually interacts with logical deduction. We show, using work of McCain and Turner, that this data can be captured using the semantics of classical proofs developed by Bellin, Hyland and Robinson, and, consequently, that the appropriate arena for solutions of the frame problem lies in proof theory. We also give a very explicit model for the categorical semantics of classical proof theory using techniques derived from work on the frame problem.

/pdf/the-frame-problem-and-the-semantics-of-classical-proofs-2fpy5zef2z.pdf

Geometry of Interaction (Abstract)

Citations

Geometry of synthesis: a structured approach to VLSI design

A structured approach to VLSI design

Process Algebras inside Ludics : an interpretation of the Calculus of Communicating Systems

The Frame Problem and the Semantics of Classical Proofs

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