Approximation metrics for discrete and continuous systems

A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category

REVIEWS-Sketches of an elephant: A topos theory compendium

Functional Programming Languages and Computer Architecture

We prove that the modal mu-calculus model-checking problem for (ranked and ordered) node-labelled trees that are generated by order- recursion schemes (whether safe or not, and whether homogeneously typed or not) is - EXPTIME complete, for every n \geqslant 0. It follows that the monadic second-order theories of these trees are decidable. There are three major ingredients. The first is a certain transference principle from the tree generated by the scheme - the value tree - to an auxiliary computation tree, which is itself a tree generated by a related order-0 recursion scheme (equivalently, a regular tree). Using innocent game semantics in the sense of Hyland and Ong, we establish a strong correspondence between paths in the value tree and traversals in the computation tree. This allows us to prove that a given alternating parity tree automaton (APT) has an (accepting) run-tree over the value tree iff it has an (accepting) traversal-tree over the computation tree. The second ingredient is the simulation of an (accepting) traversal-tree by a certain set of annotated paths over the computation tree; we introduce traversal-simulating APT as a recognising device for the latter. Finally, for the complexity result, we prove that traversal-simulating APT enjoy a succinctness property: for deciding acceptance, it is enough to consider run-trees that have a reduced branching factor. The desired bound is then obtained by analysing the complexity of solving an associated (finite) acceptance parity game.

/pdf/on-model-checking-trees-generated-by-higher-order-recursion-2yxdzkagib.pdf

On Model-Checking Trees Generated by Higher-Order Recursion Schemes

Separation logic is an extension of Hoare's logic which supports a local way of reasoning about programs that mutate memory. We present a study of the semantic structures lying behind the logic. The core idea is of a local action, a state transformer that mutates the state in a local way. We formulate local actions for a class of models called separation algebras, abstracting from the RAM and other specific concrete models used in work on separation logic. Local actions provide a semantics for a generalized form of (sequential) separation logic. We also show that our conditions on local actions allow a general soundness proof for a separation logic for concurrency, interpreted over arbitrary separation algebras.

/pdf/local-action-and-abstract-separation-logic-4o6sm9x09l.pdf

Local Action and Abstract Separation Logic

Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science

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.

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.

We present guarded dependent type theory, \(\mathsf {gDTT}\), an extensional dependent type theory with a ‘later’ modality and clock quantifiers for programming and proving with guarded recursive and coinductive types. The later modality is used to ensure the productivity of recursive definitions in a modular, type based, way. Clock quantifiers are used for controlled elimination of the later modality and for encoding coinductive types using guarded recursive types. Key to the development of \(\mathsf {gDTT}\) are novel type and term formers involving what we call ‘delayed substitutions’. These generalise the applicative functor rules for the later modality considered in earlier work, and are crucial for programming and proving with dependent types. We show soundness of the type theory with respect to a denotational model.

http://cs.ioc.ee/types15/slides/bizjak-grathwohl-clouston-mogelberg-birkedal-slides.pdf

Guarded Dependent Type Theory with Coinductive Types

Guarded recursive functions and types are useful for giving semantics to advanced programming languages and for higher-order programming with infinite data types, such as streams, e.g., for modeling reactive systems. We propose an extension of intensional type theory with rules for forming fixed points of guarded recursive functions. Guarded recursive types can be formed simply by taking fixed points of guarded recursive functions on the universe of types. Moreover, we present a general model construction for constructing models of the intensional type theory with guarded recursive functions and types. When applied to the groupoid model of intensional type theory with the universe of small discrete groupoids, the construction gives a model of guarded recursion for which there is a one-to-one correspondence between fixed points of functions on the universe of types and fixed points of (suitable) operators on types. In particular, we find that the functor category Grpdωop from the preordered set of natural numbers to the category of groupoids is a model of intensional type theory with guarded recursive types.

http://www.cs.au.dk/~birke/papers/sgdtuniverse-conf.pdf

Rasmus Ejlers Møgelberg

Papers

Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science

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

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

Guarded Dependent Type Theory with Coinductive Types

Intensional Type Theory with Guarded Recursive Types qua Fixed Points on Universes