Groupoid Semantics for Thermal Computing

A groupoid semantics is presented for systems with both logical and thermal degrees of freedom. We apply this to a syntactic model for encryption, and obtain an algebraic characterization of the heat produced by the encryption function, as predicted by Landauer's principle. Our model has a linear representation theory that reveals an underlying quantum semantics, giving for the first time a functorial classical model for quantum teleportation and other quantum phenomena.Comment: We describe a groupoid model for thermodynamic computation, and a quantization procedure that turns encrypted communication into quantum teleportation. Everything is done using higher category theor

Venus Homotopically

The identity concept developed in the Homotopy Type theory (HoTT) supports an analysis of Frege's famous Venus example, which explains how empirical evidences justify judgements about identities. In the context of this analysis we consider the traditional distinction between the extension and the intension of concepts as it appears in HoTT, discuss an ontological significance of this distinction and, finally, provide a homotopical reconstruction of a basic kinematic scheme, which is used in the Classical Mechanics, and discuss its relevance in the Quantum Mechanics

Computational Paths -- A Weak Groupoid

We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type. We also define a term rewriting system that is used to make computations between these computational paths, establishing equalities between equalities. We use a labelled deduction system based on the concept of computational paths (sequences of rewrites) as our tool, to perform in algebraic topology an approach of computational paths. This makes it possible to build the fundamental groupoid of a type $X$ connected by paths. Then, we will establish the morphism between these groupoid structures, getting the concept of isomorphisms between types and to constitute the category of computational paths, which will be called $\mathcal{C}_{paths}$. Finally, we will conclude that the weak category $\mathcal{C}_{paths}$ determines a weak groupid.Comment: 37 pages. arXiv admin note: substantial text overlap with arXiv:1906.0910

On generalized algebraic theories and categories with families

We give a syntax independent formulation of finitely presented generalized algebraic theories as initial objects in categories of categories with families (cwfs) with extra structure. To this end, we simultaneously define the notion of a presentation Σ of a generalized algebraic theory and the associated category CwFΣ of small cwfs with a Σ-structure and cwf-morphisms that preserve Σ-structure on the nose. Our definition refers to the purely semantic notion of uniform family of contexts, types, and terms in CwFΣ. Furthermore, we show how to syntactically construct an initial cwf with a Σ-structure. This result can be viewed as a generalization of Birkhoff’s completeness theorem for equational logic. It is obtained by extending Castellan, Clairambault, and Dybjer’s construction of an initial cwf. We provide examples of generalized algebraic theories for monoids, categories, categories with families, and categories with families with extra structure for some type formers of Martin-Löf type theory. The models of these are internal monoids, internal categories, and internal categories with families (with extra structure) in a small category with families. Finally, we show how to extend our definition to some generalized algebraic theories that are not finitely presented, such as the theory of contextual cwfs.publishedVersio

On the strength of proof-irrelevant type theories

We present a type theory with some proof-irrelevance built into the conversion rule. We argue that this feature is useful when type theory is used as the logical formalism underlying a theorem prover. We also show a close relation with the subset types of the theory of PVS. We show that in these theories, because of the additional extentionality, the axiom of choice implies the decidability of equality, that is, almost classical logic. Finally we describe a simple set-theoretic semantics.Comment: 20 pages, Logical Methods in Computer Science, Long version of IJCAR 2006 pape

Finitary Higher Inductive Types in the Groupoid Model

A higher inductive type of level 1 (a 1-hit) has constructors for points and paths only, whereas a higher inductive type of level 2 (a 2-hit) has constructors for surfaces too. We restrict attention to finitary higher inductive types and present general schemata for the types of their point, path, and surface constructors. We also derive the elimination and equality rules from the types of constructors for 1-hits and 2-hits. Moreover, we construct a groupoid model for dependent type theory with 2-hits and point out that we obtain a setoid model for dependent type theory with 1-hits by truncating the groupoid model