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John Bernier
(@john-bernier)
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Joined: 5 days ago
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Hello, I am a software engineer and computer scientist. My role in this broader community is to integrate topos theory and computer science. You can find me on github: https://github.com/jhuni


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Olivia Caramello
(@ocaramello)
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Joined: 5 years ago
Posts: 75
 

@john-bernier Hi John, welcome to the forum! Can you tell us which specific topics at the interface of topos theory and computer science you plan to address?  


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John Bernier
(@john-bernier)
New Member
Joined: 5 days ago
Posts: 3
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I think topos theory's basic correctness comes from the principle of locality. This means that reality is massively parallel and local, with multiple events happening all at once at the same time down to the quantum scale. The advancement of computer technology has forced us to face physical realities. Instead of making cores faster we can only make them in higher and higher quantities. The holy grail of the computing industry is now parallelism, which means making computers that better adapt to the principle of locality.

The problem is that all existing programming languages are based upon the idea of global time, not the pervasive reality of local time. This is unnatural. To partially solve this problem, computer scientists have developed the technology of automatic parallelism. This requires complex program analysis to determine how programs can be broken down into parts that run side by side. An example of this kind of automatic program analysis is Bernstein's conditions. This says that if we have two operations that write to separate memory areas and don't affect their respective inputs then we can perform automatic parallelization. The broader subject of dataflow analysis helps us to reason about computer programs in terms of their local effects.

It occurs to me that breaking up computer programs into parts needs algebraic semantics. The very word algebra means "the reunion of broken parts." A colleague recently informed me of the excellent text Algebraic Structure Theory of Sequential Machines (1966). This comes closer than any prior work I've seen to describing a local structure theory of machines, and it even approaches the formalism we need, but we can do better.

My recent work on studying information flows in the Sierpinski topos generalizes the previous model of Hartmanis and Stearns. The result is a dataflow model that is more powerful and elegant than any other I have seen. This convinces me that I am on the right track in using topos theory. Grothendieck studied the local properties of algebraic varieties using local ringed spaces. All I want to do is reason about the local properties of machines to achieve greater parallelism. That is the topic I hope to address.

This post was modified 4 days ago by John Bernier

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