Quantization via Linear homotopy types

In the foundational logical framework of homotopy-type theory we discuss a natural formalization of secondary integral transforms in stable geometric homotopy theory. We observe that this yields a process of non-perturbative cohomological quantization of local pre-quantum field theory; and show that quantum anomaly cancellation amounts to realizing this as the boundary of a field theory that is given by genuine (primary) integral transforms, hence by linear polynomial functors. Recalling that traditional linear logic has semantics in symmetric monoidal categories and serves to formalize quantum mechanics, what we consider is its refinement to linear homotopy-type theory with semantics in stable infinity-categories of bundles of stable homotopy types (generalized cohomology theories) formalizing Lagrangian quantum field theory, following Nuiten and closely related to recent work by Haugseng and Hopkins-Lurie. For the reader interested in technical problems of quantization we provide non-perturbative quantization of Poisson manifolds and of the superstring; and find insight into quantum anomaly cancellation, the holographic principle and motivic structures in quantization. For the reader inclined to the interpretation of quantum mechanics we exhibit quantum superposition and interference as existential quantification in linear homotopy-type theory. For the reader inclined to foundations we provide a refinement of the proposal by Lawvere for a formal foundation of physics, lifted from classical continuum mechanics to local Lagrangian quantum gauge field theory.Comment: 89 pages; these are expanded talk notes for three talks that I gave in February 2014 at Paris Diderot and at ESI in Vienna, one together with Joost Nuite

Formalizing the Metatheory of Logical Calculi and Automatic Provers in Isabelle/HOL (Invited Talk)

International audienceIsaFoL (Isabelle Formalization of Logic) is an undertaking that aims at developing formal theories about logics, proof systems, and automatic provers, using Isabelle/HOL. At the heart of the project is the conviction that proof assistants have become mature enough to actually help researchers in automated reasoning when they develop new calculi and tools. In this paper, I describe and reflect on three verification subprojects to which I contributed: a first-order resolution prover, an imperative SAT solver, and generalized term orders for λ-free higher-order logic

Higher-order architectural connectors

We develop a notion of higher-order connector towards supporting the systematic construction of architectural connectors for software design. A higher-order connector takes connectors as parameters and allows for services such as security protocols and fault-tolerance mechanisms to be superposed over the interactions that are handled by the connectors passed as actual arguments. The notion is first illustrated over CommUnity, a parallel program design language that we have been using for formalizing aspects of architectural design. A formal, algebraic semantics is then presented which is independent of any Architectural Description Language. Finally, we discuss how our results can impact software design methods and tools