Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem

The paper uses the formalism of indexed categories to recover the proof of a standard final coalgebra theorem, thus showing existence of final coalgebras for a special class of functors on categories with finite limits and colimits. As an instance of this result, we build the final coalgebra for the powerclass functor, in the context of a Heyting pretopos with a class of small maps. This is then proved to provide a model for various non-well-founded set theories, depending on the chosen axiomatisation for the class of small maps

W-types in Homotopy Type Theory

We will give a detailed account of why the simplicial sets model of the univalence axiom due to Voevodsky also models W-types. In addition, we will discuss W-types in categories of simplicial presheaves and an application to models of set theory.Comment: We have corrected the statement of Theorem 3.4. We thank Christian Sattler for alerting us to the error in the original versio

Structural operational semantics for stochastic and weighted transition systems

We introduce weighted GSOS, a general syntactic framework to specify well-behaved transition systems where transitions are equipped with weights coming from a commutative monoid. We prove that weighted bisimilarity is a congruence on systems defined by weighted GSOS specifications. We illustrate the flexibility of the framework by instantiating it to handle some special cases, most notably that of stochastic transition systems. Through examples we provide weighted-GSOS definitions for common stochastic operators in the literature

Modeling Fresh Names in the π-calculus Using Abstractions

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First steps in synthetic guarded domain theory: step-indexing in the topos of trees

We present the topos S of trees as a model of guarded recursion. We study the internal dependently-typed higher-order logic of S and show that S models two modal operators, on predicates and types, which serve as guards in recursive definitions of terms, predicates, and types. In particular, we show how to solve recursive type equations involving dependent types. We propose that the internal logic of S provides the right setting for the synthetic construction of abstract versions of step-indexed models of programming languages and program logics. As an example, we show how to construct a model of a programming language with higher-order store and recursive types entirely inside the internal logic of S. Moreover, we give an axiomatic categorical treatment of models of synthetic guarded domain theory and prove that, for any complete Heyting algebra A with a well-founded basis, the topos of sheaves over A forms a model of synthetic guarded domain theory, generalizing the results for S

The Epstein-Glaser approach to pQFT: graphs and Hopf algebras

The paper aims at investigating perturbative quantum field theory (pQFT) in the approach of Epstein and Glaser (EG) and, in particular, its formulation in the language of graphs and Hopf algebras (HAs). Various HAs are encountered, each one associated with a special combination of physical concepts such as normalization, localization, pseudo-unitarity, causality and an associated regularization, and renormalization. The algebraic structures, representing the perturbative expansion of the S-matrix, are imposed on the operator-valued distributions which are equipped with appropriate graph indices. Translation invariance ensures the algebras to be analytically well-defined and graded total symmetry allows to formulate bialgebras. The algebraic results are given embedded in the physical framework, which covers the two recent EG versions by Fredenhagen and Scharf that differ with respect to the concrete recursive implementation of causality. Besides, the ultraviolet divergences occuring in Feynman's representation are mathematically reasoned. As a final result, the change of the renormalization scheme in the EG framework is modeled via a HA which can be seen as the EG-analog of Kreimer's HA.Comment: 52 pages, 5 figure

Guarded Kleene algebra with tests: verification of uninterpreted programs in nearly linear time

Guarded Kleene Algebra with Tests (GKAT) is a variation on Kleene Algebra with Tests (KAT) that arises by restricting the union (+) and iteration (*) operations from KAT to predicate-guarded versions. We develop the (co)algebraic theory of GKAT and show how it can be efficiently used to reason about imperative programs. In contrast to KAT, whose equational theory is PSPACE-complete, we show that the equational theory of GKAT is (almost) linear time. We also provide a full Kleene theorem and prove completeness for an analogue of Salomaa’s axiomatization of Kleene Algebra

The Post‐Modern Transcendental of Language in Science and Philosophy

In this chapter I discuss the deep mutations occurring today in our society and in our culture, the natural and mathematical sciences included, from the standpoint of the “transcendental of language”, and of the primacy of language over knowledge. That is, from the standpoint of the “completion of the linguistic turn” in the foundations of logic and mathematics using Peirce’s algebra of relations. This evolved during the last century till the development of the Category Theory as universal language for mathematics, in many senses wider than set theory. Therefore, starting from the fundamental M. Stone’s representation theorem for Boolean algebras, computer scientists developed a coalgebraic first-order semantics defined on Stone’s spaces, for Boolean algebras, till arriving to the definition of a non-Turing paradigm of coalgebraic universality in computation. Independently, theoretical physicists developed a coalgebraic modelling of dissipative quantum systems in quantum field theory, interpreted as a thermo-field dynamics. The deep connection between these two coalgebraic constructions is the fact that the topologies of Stone spaces in computer science are the same of the C*-algebras of quantum physics. This allows the development of a new class of quantum computers based on coalgebras. This suggests also an intriguing explanation of why one of the most successful experimental applications of this coalgebraic modelling of dissipative quantum systems is just in cognitive neuroscience