Aleš Bizjak

Step-Indexed Logical Relations for Probability

It is well-known that constructing models of higher-order probabilistic programming languages is challenging. We show how to construct step-indexed logical relations for a probabilistic extension of a higher-order programming language with impredicative polymorphism and recursive types. We show that the resulting logical relation is sound and complete with respect to the contextual preorder and, moreover, that it is convenient for reasoning about concrete program equivalences. Finally, we extend the language with dynamically allocated first-order references and show how to extend the logical relation to this language. We show that the resulting relation remains useful for reasoning about examples involving both state and probabilistic choice.

Guarded Cubical Type Theory: Path Equality for Guarded Recursion

This paper improves the treatment of equality in guarded dependent type theory (GDTT), by combining it with cubical type theory (CTT). GDTT is an extensional type theory with guarded recursive types, which are useful for building models of program logics, and for programming and reasoning with coinductive types. We wish to implement GDTT with decidable type-checking, while still supporting non-trivial equality proofs that reason about the extensions of guarded recursive constructions. CTT is a variation of Martin-Lof type theory in which the identity type is replaced by abstract paths between terms. CTT provides a computational interpretation of functional extensionality, is conjectured to have decidable type checking, and has an implemented type-checker. Our new type theory, called guarded cubical type theory, provides a computational interpretation of extensionality for guarded recursive types. This further expands the foundations of CTT as a basis for formalisation in mathematics and computer science. We present examples to demonstrate the expressivity of our type theory, all of which have been checked using a prototype type-checker implementation, and present semantics in a presheaf category.

A Model of Guarded Recursion With Clock Synchronisation

Guarded recursion is an approach to solving recursive type equations where the type variable appears guarded by a modality to be thought of as a delay for one time step. Atkey and McBride proposed a calculus in which guarded recursion can be used when programming with coinductive data, allowing productivity to be captured in types. The calculus uses clocks representing time streams and clock quantifiers which allow limited and controlled elimination of modalities. The calculus has since been extended to dependent types by Mogelberg. Both works give denotational semantics but no rewrite semantics.In previous versions of this calculus, different clocks represented separate time streams and clock synchronisation was prohibited. In this paper we show that allowing clock synchronisation is safe by constructing a new model of guarded recursion and clocks. This result will greatly simplify the type theory by removing freshness restrictions from typing rules, and is a necessary step towards defining rewrite semantics, and ultimately implementing the calculus.

A Model of Countable Nondeterminism in Guarded Type Theory

We show how to construct a logical relation for countable nondeterminism in a guarded type theory, corresponding to the internal logic of the topos Sh ω 1 of sheaves over ω 1. In contrast to earlier work on abstract step-indexed models, we not only construct the logical relations in the guarded type theory, but also give an internal proof of the adequacy of the model with respect to standard contextual equivalence. To state and prove adequacy of the logical relation, we introduce a new propositional modality. In connection with this modality we show why it is necessary to work in the logic of bf Shω 1.

The Guarded Lambda-Calculus: Programming and Reasoning with Guarded Recursion for Coinductive Types

We present the guarded lambda-calculus, an extension of the simply typed lambda-calculus with guarded recursive and coinductive types. The use of guarded recursive types ensures the productivity of well-typed programs. Guarded recursive types may be transformed into coinductive types by a type-former inspired by modal logic and Atkey-McBride clock quantification, allowing the typing of acausal functions. We give a call-by-name operational semantics for the calculus, and define adequate denotational semantics in the topos of trees. The adequacy proof entails that the evaluation of a program always terminates. We introduce a program logic with Lob induction for reasoning about the contextual equivalence of programs. We demonstrate the expressiveness of the calculus by showing the definability of solutions to Ruttens behavioural differential equations.

ModuRes: A Coq Library for Modular Reasoning About Concurrent Higher-Order Imperative Programming Languages

It is well-known that it is challenging to build semantic models of type systems or logics for reasoning about concurrent higher-order imperative programming languages. One of the key challenges is that such semantic models often involve constructing solutions to certain kinds of recursive domain equations, which in practice has been a barrier to formalization efforts. Here we present the ModuRes Coq library, which provides an easy way to solve such equations. We show how the library can be used to construct models of type systems and logics for reasoning about concurrent higher-order imperative programming languages.

Step-Indexed Relational Reasoning for Countable Nondeterminism

Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {\omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {\omega}.

Guarded Cubical Type Theory

This paper improves the treatment of equality in guarded dependent type theory (

Relational Reasoning for Markov Chains in a Probabilistic Guarded Lambda Calculus

\mathsf {GDTT}