We introduce a family of (k, h)-interpretations for 2 ≤ k ≤ ∞ and 1 ≤ h ≤ ∞ of constructive set theory into type theory, in which sets and formulas are interpreted as types of homotopy level k and h, respectively. Depending on the values of the parameters k and h, we are able to interpret different theories, like Aczel’s CZF and Myhill’s CST. We also define a proposition-as-hproposition interpretation in the context of logic-enriched type theories. The rest of the paper is devoted to characterising and analysing the interpretations considered. The formulas valid in the prop-as-hprop interpretation are characterised in terms of the axiom of unique choice. We also analyse the interpretations of CST into homotopy type theory, providing a comparative analysis with Aczel’s interpretation. This is done by formulating in a logic-enriched type theory the key principles used in the proofs of the two interpretations. Finally, we characterise a class of sentences valid in the (k, ∞)-interpretations in terms of the ΠΣ axiom of choice.

We prove a strictification theorem for cartesian closed bicategories. First, we adapt Power’s proof of coherence for bicategories with finite bilimits to show that every bicategory with bicategorical cartesian closed structure is biequivalent to a 2-category with 2-categorical cartesian closed structure. Then we show how to extend this result to a Mac Lane-style “all pasting diagrams commute” coherence theorem: precisely, we show that in the free cartesian closed bicategory on a graph, there is at most one 2-cell between any parallel pair of 1-cells. The argument we employ is reminiscent of that used by Čubrić, Dybjer, and Scott to show normalisation for the simply-typed lambda calculus (Čubrić et al., 1998). The main results first appeared in a conference paper (Fiore and Saville, 2020) but for reasons of space many details are omitted there; here we provide the full development.

We formalize in Constructive Type Theory the Lambda Calculus in its classical first-order syntax, employing only one sort of names for both bound and free variables, and with α-conversion based upon name swapping. As a fundamental part of the formalization, we introduce principles of induction and recursion on terms which provide a framework for reproducing the use of the Barendregt Variable Convention as in pen-and-paper proofs within the rigorous formal setting of a proof assistant. The principles in question are all formally derivable from the simple principle of structural induction/recursion on concrete terms. We work out applications to some fundamental meta-theoretical results, such as the Church–Rosser Theorem and Weak Normalization for the Simply Typed Lambda Calculus. The whole development has been machine checked using the system Agda.

I recall how Martin Hofmann and I found the groupoid model of type theory in the early 1990s.

We develop the Scott model of the programming language PCF in univalent type theory. Moreover, we work constructively and predicatively. To account for the non-termination in PCF, we use the lifting monad (also known as the partial map classifier monad) from topos theory, which has been extended to univalent type theory by Escardó and Knapp. Our results show that lifting is a viable approach to partiality in univalent type theory. Moreover, we show that the Scott model can be constructed in a predicative and constructive setting. Other approaches to partiality either require some form of choice or quotient inductive-inductive types. We show that one can do without these extensions.

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.

In Hyland et al. (1980), Hyland, Johnstone and Pitts introduced the notion of tripos for the purpose of organizing the construction of realizability toposes in a way that generalizes the construction of localic toposes from complete Heyting algebras. In Pitts (2002), one finds a generalization of this notion eliminating an unnecessary assumption of Hyland et al. (1980). The aim of this paper is to characterize triposes over a base topos ${\cal S}$ in terms of so-called constant objects functors from ${\cal S}$ to some elementary topos. Our characterization is slightly different from the one in Pitts’s PhD Thesis (Pitts, 1981) and motivated by the fibered view of geometric morphisms as described in Streicher (2020). In particular, we discuss the question whether triposes over Set giving rise to equivalent toposes are already equivalent as triposes.

The ALEA Coq library formalizes measure theory based on a variant of the Giry monad on the category of sets. This enables the interpretation of a probabilistic programming language with primitives for sampling from discrete distributions. However, continuous distributions have to be discretized because the corresponding measures cannot be defined on all subsets of their carriers. This paper proposes the use of synthetic topology to model continuous distributions for probabilistic computations in type theory. We study the initial σ-frame and the corresponding induced topology on arbitrary sets. Based on these intrinsic topologies, we define valuations and lower integrals on sets and prove versions of the Riesz and Fubini theorems. We then show how the Lebesgue valuation, and hence continuous distributions, can be constructed.

We condense the theory of UTxO blockchains down to a simple and compact set of four type equations (Idealised EUTxO), and to an algebraic characterisation (abstract chunk systems), and exhibit an adjoint pair of functors between them. This gives a novel account of the essential mathematical structures underlying blockchain technology, such as Bitcoin.

The local solver TD is a generic fixpoint engine which explores a given system of equations on demand. It has been successfully applied to the interprocedural analysis of procedural languages. The solver TD gains efficiency by detecting dependencies between unknowns on the fly. This algorithm has been recently extended to deal with widening and narrowing as well. In particular, it has been equipped with an automatic detection of widening and narrowing points. That version, however, is only guaranteed to terminate under two conditions: only finitely many unknowns are encountered, and all right-hand sides are monotonic. While the first condition is unavoidable, the second limits the applicability of the solver. Another limitation is that the solver maintains the current abstract values of all encountered unknowns instead of a minimal set sufficient for performing the iteration. By consuming unnecessarily much space, interprocedural analyses may not succeed on seemingly small programs. In the present paper, we therefore extend the top-down solver TD in three ways. First, we indicate how the restriction to monotonic right-hand sides can be lifted without compromising termination. We then show how the solver can be tuned to store abstract values only when their preservation is inevitable. Finally, we also show how the solver can be extended to side-effecting equation systems. Right-hand sides of these may not only provide values for the corresponding left-hand side unknowns but at the same time produce contributions to other unknowns. This practical extension has successfully been used for a seamless combination of context-sensitive analyses (e.g., of local states) with flow-insensitive analyses (e.g., of globals).

Emerson and Halpern (1986, Journal of the Association for Computing Machinery33, 151–178) prove that the Computation Tree Logic (CTL) cannot express the existence of a path on which a proposition holds infinitely often (fairness for short).

The scope is widened from CTL to a general branching-time logic. A path quantifier is followed by a language with temporal descriptions. In this extended setting, the said inexpressiveness is strengthened in two aspects. First, universal path quantifiers are unrestricted. In this way, they are relieved of any temporal quantifiers such as of those in $\mathtt{AU}$ and $\mathtt{AR}$ from CTL. Second, existential path quantifiers are allowed with any countable language. Instances are the temporal quantifiers in $\mathtt{EU}$ and $\mathtt{ER}$ from CTL. By contrast, the fairness statement is an existential path quantifier with an uncountable language. Both aspects indicate that this inexpressiveness is optimal with respect to the polarity of path quantifiers and to the cardinality of their languages.