We prove normalization for MTT, a general multimodal dependent type theory capable of expressing modal type theories for guarded recursion, internalized parametricity, and various other prototypical modal situations. We prove that deciding type checking and conversion in MTT can be reduced to deciding the equality of modalities in the underlying modal situation, immediately yielding a type checking algorithm for all instantiations of MTT in the literature. This proof uses a generalization of synthetic Tait computability – an abstract approach to gluing proofs – to account for modalities. This extension is based on MTT itself, so that this proof also constitutes a significant case study of MTT.

Simplicial type theory (STT) was introduced by Riehl and Shulman to leverage homotopy type theory to prove results about (∞,1)-categories. Initial work on simplicial type theory focused on “formal” arguments in higher category theory and, in particular, no non-trivial examples of ∞-category theory were constructible within STT. More recent work has changed this state of affairs by applying techniques developed initially for cubical type theory to construct the ∞-category of spaces. We complete this process by constructing the ∞-category of ∞-categories, recovering one of the main foundational results of ∞-category theory (straightening-unstraightening) purely type-theoretically. We also show how this construction enables new examples of the directed version of the structure identity principle: the structure homomorphism principle.

Categorical gluing is a powerful technique for proving meta-theorems of type theories such as canonicity and normalization. Synthetic Tait Computability (STC) provides an abstract treatment of the complex gluing models by internalizing the gluing category into a modal dependent type theory with a phase distinction. This work presents a mechanization of STC in the Istari proof assistant. Istari is a Martin-Löf-style extensional type theory with equality reflection, which avoids much of the explicit transport reasoning typically found in intensional proof assistants. This work develops a reusable library for synthetic phase distinction, including modalities, extension types, and strict glue types, and applies it to two case studies: (1) a canonicity model for dependent type theory with dependent products and booleans with large elimination, and (2) a Kripke canonicity model for the cost-aware logical framework. Our results demonstrate that the core STC constructions can be formalized essentially verbatim in Istari, preserving the elegance of the on-paper arguments while ensuring machine-checked correctness.

We present a novel analysis of the fundamental Löb induction principle from guarded recursion. Taking advantage of recent work in modal type theory and univalent foundations, we derive Löb induction from a simpler and more conceptual set of primitives. We then capitalize on these insights to present Gatsby, the first guarded type theory capturing the rich modal structure of the topos of trees alongside Löb induction without immediately precluding canonicity or normalization. We show that Gatsby can recover many prior approaches to guarded recursion and use its additional power to improve on prior examples. We crucially rely on homotopical insights and Gatsby constitutes a new application of univalent foundations to the theory of programming languages.

We study a cost-aware programming language for higher-order recursion dubbed ${\text{𝐏𝐂𝐅}}_{\text{𝖼𝗈𝗌𝗍}}$ in the setting of synthetic domain theory (SDT). Our main contribution relates the denotational cost semantics of ${\text{𝐏𝐂𝐅}}_{\text{𝖼𝗈𝗌𝗍}}$ to its computational cost semantics, a new kind of dynamic semantics for program execution that serves as a mathematically natural alternative to operational semantics in SDT. In particular we prove an internal, cost-sensitive version of Plotkin’s computational adequacy theorem, giving a precise correspondence between the denotational and computational semantics for complete programs at base type. The constructions and proofs of this paper take place in the internal dependent type theory of an SDT topos extended by a phase distinction in the sense of Sterling and Harper. By controlling the interpretation of cost structure via the phase distinction in the denotational semantics, we show that ${\text{𝐏𝐂𝐅}}_{\text{𝖼𝗈𝗌𝗍}}$ programs also evince a noninterference property of cost and behavior. We verify the axioms of the type theory by means of a model construction based on relative sheaf models of SDT.

Simplicial type theory extends homotopy type theory with a directed path type which internalizes the notion of a homomorphism within a type. This concept has significant applications both within mathematics – where it allows for synthetic (higher) category theory – and programming languages – where it leads to a directed version of the structure identity principle. In this work, we construct the first types in simplicial type theory with non-trivial homomorphisms. We extend simplicial type theory with modalities and new reasoning principles to obtain triangulated type theory in order to construct the universe of discrete types $\mathrm{𝒮︀}$. We prove that homomorphisms in this type correspond to ordinary functions of types i.e., that $\mathrm{𝒮︀}$ is directed univalent. The construction of $\mathrm{𝒮︀}$ is foundational for both of the aforementioned applications of simplicial type theory. We are able to define several crucial examples of categories and to recover important results from category theory. Using $\mathrm{𝒮︀}$, we are also able to define various types whose usage is guaranteed to be functorial. These provide the first complete examples of the proposed directed structure identity principle.

We present decalf , a d irected, e ffectful c ost- a ware l ogical f ramework for studying quantitative aspects of functional programs with effects. Like calf , the language is based on a formal phase distinction between the extension and the intension of a program, its pure behavior as distinct from its cost measured by an effectful step-counting primitive. The type theory ensures that the behavior is unaffected by the cost accounting. Unlike calf , the present language takes account of effects , such as probabilistic choice and mutable state. This extension requires a reformulation of calf ’s approach to cost accounting: rather than rely on a “separable” notion of cost, here a cost bound is simply another program . To make this formal, we equip every type with an intrinsic preorder, relaxing the precise cost accounting intrinsic to a program to a looser but nevertheless informative estimate. For example, the cost bound of a probabilistic program is itself a probabilistic program that specifies the distribution of costs. This approach serves as a streamlined alternative to the standard method of isolating a cost recurrence and readily extends to higher-order, effectful programs. The development proceeds by first introducing the decalf type system, which is based on an intrinsic ordering among terms that restricts in the extensional phase to extensional equality, but in the intensional phase reflects an approximation of the cost of a program of interest. This formulation is then applied to a number of illustrative examples, including pure and effectful sorting algorithms, simple probabilistic programs, and higher-order functions. Finally, we justify decalf via a model in the topos of augmented simplicial sets.

We develop a denotational semantics for general reference types in an impredicative version of guarded homotopy type theory, an adaptation of synthetic guarded domain theory to Voevodsky’s univalent foundations. We observe for the first time the profound impact of univalence on the denotational semantics of mutable state. Univalence automatically ensures that all computations are invariant under symmetries of the heap - a bountiful source of program equivalences. In particular, even the most simplistic univalent model enjoys many new equations that do not hold when the same constructions are carried out in the universes of traditional set-level (extensional) type theory.

We present calf, a cost-aware logical framework for studying quantitative aspects of functional programs. Taking inspiration from recent work that reconstructs traditional aspects of programming languages in terms of a modal account of phase distinctions, we argue that the cost structure of programs motivates a phase distinction between intension and extension. Armed with this technology, we contribute a synthetic account of cost structure as a computational effect in which cost-aware programs enjoy an internal noninterference property: input/output behavior cannot depend on cost. As a full-spectrum dependent type theory, calf presents a unified language for programming and specification of both cost and behavior that can be integrated smoothly with existing mathematical libraries available in type theoretic proof assistants. We evaluate calf as a general framework for cost analysis by implementing two fundamental techniques for algorithm analysis: the method of recurrence relations and physicist’s method for amortized analysis. We deploy these techniques on a variety of case studies: we prove a tight, closed bound for Euclid’s algorithm, verify the amortized complexity of batched queues, and derive tight, closed bounds for the sequential and parallel complexity of merge sort, all fully mechanized in the Agda proof assistant. Lastly we substantiate the soundness of quantitative reasoning in calf by means of a model construction.

Guarded type theory extends type theory with a handful of modalities and constants to encode productive recursion. While these theories have seen widespread use, the metatheory of guarded type theories, particularly guarded dependent type theories remains underdeveloped. We show that integrating Löb induction is the key obstruction to unifying guarded recursion and dependence in a well-behaved type theory and prove a no-go theorem sharply bounding such type theories. Based on these results, we introduce GuTT: a stratified guarded type theory. GuTT is properly two type theories, sGuTT and dGuTT. The former contains only propositional rules governing Löb induction but enjoys decidable type-checking while the latter extends the former with definitional equalities. Accordingly, dGuTT does not have decidable type-checking. We prove, however, a novel guarded canonicity theorem for dGuTT, showing that programs in dGuTT can be run. These two type theories work in concert, with users writing programs in sGuTT and running them in dGuTT.

Hofmann and Streicher famously showed how to lift Grothendieck universes into presheaf topoi, and Streicher has extended their result to the case of sheaf topoi by sheafification. In parallel, van den Berg and Moerdijk have shown in the context of algebraic set theory that similar constructions continue to apply even in weaker metatheories. Unfortunately, sheafification seems not to preserve an important realignment property enjoyed by the presheaf universes that plays a critical role in models of univalent type theory as well as synthetic Tait computability, a recent technique to establish syntactic properties of type theories and programming languages. In the context of multiple universes, the realignment property also implies a coherent choice of codes for connectives at each universe level, thereby interpreting the cumulativity laws present in popular formulations of Martin-Löf type theory. We observe that a slight adjustment to an argument of Shulman constructs a cumulative universe hierarchy satisfying the realignment property at every level in any Grothendieck topos. Hence one has direct-style interpretations of Martin-Löf type theory with cumulative universes into all Grothendieck topoi. A further implication is to extend the reach of recent synthetic methods in the semantics of cubical type theory and the syntactic metatheory of type theory and programming languages to all Grothendieck topoi.

The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction, computational effects, and type abstraction. We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis.

Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure. Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof-relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.”

Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types, by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.

The implementation and semantics of dependent type theories can be studied in a syntax-independent way: the objective metatheory of dependent type theories exploits the universal properties of their syntactic categories to endow them with computational content, mathematical meaning, and practical implementation (normalization, type checking, elaboration). The semantic methods of the objective metatheory inform the design and implementation of correct-by-construction elaboration algorithms, promising a principled interface between real proof assistants and ideal mathematics. In this dissertation, I add synthetic Tait computability to the arsenal of the objective metatheorist. Synthetic Tait computability is a mathematical machine to reduce difficult problems of type theory and programming languages to trivial theorems of topos theory. First employed by Sterling and Harper to reconstruct the theory of program modules and their phase separated parametricity, synthetic Tait computability is deployed here to resolve the last major open question in the syntactic metatheory of cubical type theory: normalization of open terms.

We prove normalization for (univalent, Cartesian) cubical type theory, closing the last major open problem in the syntactic metatheory of cubical type theory. Our normalization result is reduction-free, in the sense of yielding a bijection between equivalence classes of terms in context and a tractable language of β/η-normal forms. As corollaries we obtain both decidability of judgmental equality and the injectivity of type constructors.

Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the “synthetic” development of homotopy theory within type theory, as well as in formalising ordinary set-level mathematics in type theory. In this paper, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters : a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has weakly stable typal initial algebras for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specialises to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed (∞, 1)-category is presented by some model category to which our results apply.

We prove the conjecture that any Grothendieck $(\infty ,1)$-topos can be presented by a Quillen model category that interprets homotopy type theory with strict univalent universes. Thus, homotopy type theory can be used as a formal language for reasoning internally to $(\infty ,1)$-toposes, just as higher-order logic is used for 1-toposes. As part of the proof, we give a new, more explicit, characterization of the fibrations in injective model structures on presheaf categories. In particular, we show that they generalize the coflexible algebras of 2-monad theory.

We combine homotopy type theory with axiomatic cohesion, expressing the latter internally with a version of ‘adjoint logic’ in which the discretization and codiscretization modalities are characterized using a judgemental formalism of ‘crisp variables.’ This yields type theories that we call ‘spatial’ and ‘cohesive,’ in which the types can be viewed as having independent topological and homotopical structure. These type theories can then be used to study formally the process by which topology gives rise to homotopy theory (the ‘fundamental ∞-groupoid’ or ‘shape’), disentangling the ‘identifications’ of homotopy type theory from the ‘continuous paths’ of topology. In a further refinement called ‘real-cohesion,’ the shape is determined by continuous maps from the real numbers, as in classical algebraic topology. This enables us to reproduce formally some of the classical applications of homotopy theory to topology. As an example, we prove Brouwer’s fixed-point theorem.