ABSTRACT. In type theory, an oracle may be specified abstractly by a predicate whose domain is the type of queries asked of the oracle, and whose proofs are the oracle answers. Such a specification induces an oracle modality that captures a computational intuition about oracles: at each step of reasoning we either know the result, or we ask the oracle a query and proceed upon receiving an answer. We characterize an oracle modality as the least one forcing the given predicate. We establish an adjoint retraction between modalities and propositional containers, from which it follows that every modality is an oracle modality. The left adjoint maps sums to suprema, which makes suprema of modalities easy to compute when they are given in terms of oracle modalities. We also study sheaves for oracle modalities. We describe sheafification in terms of a quotient-inductive type of computation trees, and describe sheaves as algebras for the corresponding monad. We also introduce equifoliate trees, an intensional notion of oracle computation given by a (non-propositional) container. Equifoliate trees descend to sheaves, and lift from sheaves in case the container is projective. As an application, we give a concrete description of all Lawvere-Tierney topologies in a realizability topos, closely related to a game-theoretic characterization by Takayuki Kihara.

ABSTRACT. To ensure decidability and consistency of its type theory, a proof assistant should only accept terminating recursive functions and productive corecursive functions. Most proof assistants enforce this through syntactic conditions, which can be restrictive and non-modular. Sized types are a type-based alternative where (co)inductive types are annotated with additional size information. Well-founded induction on sizes can then be used to prove termination and productivity. An implementation of sized types exists in Agda, but it is currently inconsistent due to the addition of a largest size. We investigate an alternative approach, where intensional type theory is extended with a large type of sizes and parametric quantifiers over sizes. We show that inductive and coinductive types can be constructed in this theory, which improves on earlier work where this was only possible for the finitely-branching inductive types. The consistency of the theory is justified by an impredicative realisability model, which interprets the type of sizes as an uncountable ordinal.

ABSTRACT. The axiom of choice (AC) states that every total relation contains a function. It enjoys a pivotal role in both classical and constructive dialects of mathematics. In the former, it is seen as a useful closure property invoked especially in set-theoretic contexts, in the latter it is seen either as a tautology, following from a constructive reading of totality proofs, or as a taboo, as by an extensional reading of totality proofs it enforces full classical logic. It has therefore been debated how much of AC should be accepted in constructive foundations and authors like Richman argued for ``Constructive mathematics without choice'' where even countable choice, not immediately jeopardising constructive reasoning, is avoided. With this paper, we propose a continuation of Richman's programme of more radical extent and systematically study constructive foundations absent of countable, unique, or quantifier-free choice principles as well as the spurious fragments of (the actual) AC in form of extensionality principles:``Constructive mathematics without \textit{any} choice'' We argue that such a minimalistic setting is advantageous, for instance for studies in constructive reverse mathematics and synthetic computability theory. Apart from these programmatic considerations and a careful encyclopedia of choice principles, we revisit and refine several results from the literature: We show that already the partition principle (a consequence of AC of unknown strength) implies the excluded middle, that already decidable equality of propositions implies proof irrelevance, and that function inversion principles such as the Cantor-Bernstein theorem not only rely on the excluded middle but also on unique choice. To the best of our knowledge, the latter is the first reverse mathematics result regarding the full axiom of unique choice, enabled by our minimal setting. Implementing such a minimalistic foundation, the proofs of all our results have been mechanised with the Rocq prover.

ABSTRACT. The correctness of type-checking implementations for proof assistants based on dependent type theory relies on metatheoretical properties that ensure the decidability of typing, some of which require substantial logical strength. Recent mechanizations of such algorithms have highlighted the importance of separating the algorithmic components of the proof---often intricate but requiring relatively low logical strength---from the logical components, which depend on stronger metatheoretical properties, such as normalization or the injectivity of type constructors. In this work, we revisit the logical relations technique and show how it can be used to derive these metatheoretical properties in a direct and uniform way for a core dependent type theory featuring Pi-types, nat, empty and a universe. Our presentation yields a compact and conceptually simplified argument that isolates the logically strong reasoning from the algorithmic core. We argue that this approach scales smoothly to richer type theories, and demonstrate this by extending our construction to Exceptional Type Theory (ExcTT), obtaining the first mechanized canonicity proof for this theory.

ABSTRACT. The stabiliser fragment of quantum theory is a foundational building block for quantum error correction, and hence for the fault-tolerant compilation of quantum programs. In this article, we develop a sound, universal, and complete denotational semantics for stabiliser operations, including measurement, classically controlled Pauli operators, and affine classical computation; supporting an explicit treatment of quantum error-correcting codes. We interpret stabiliser operations as \emph{affine relations} over finite fields, yielding a semantics that reflects the algebraic structure underlying stabiliser quantum error correction. Because stabiliser quantum mechanics has a well-behaved algebraic structure, our relational semantics is conceptually transparent and computationally tractable when compared to standard denotational models for general quantum programs. We demonstrate the resulting semantics by describing a small, low-level assembly language for stabiliser programs with fully-abstract denotational semantics.

ABSTRACT. Equational reasoning is central to quantum circuit optimisation and verification: one replaces subcircuits by provably equivalent ones using a fixed set of rewrite rules viewed as equations. We study such reasoning through finite equational theories, presenting restricted quantum gate fragments as symmetric monoidal categories (PROPs), where wire permutations are treated as structural and separated cleanly from fragment-specific gate axioms. For six widely used near-Clifford fragments: qubit Clifford, real Clifford, Clifford+T (up to two qubits), Clifford+CS (up to three qubits), CNOT-dihedral, we transfer the completeness results of prior work into our PROP framework. Beyond completeness, we address minimality (axiom independence). Using uniform separating interpretations into simple semantic targets, we prove minimality for several fragments (including all arities for qubit Clifford, real Clifford, and CNOT-dihedral), and bounded minimality for the remaining cases. Overall, our presentations significantly reduce rule counts compared to prior work and provide a reusable categorical framework for constructing complete and often minimal rewrite systems for quantum circuit fragments.

ABSTRACT. We introduce a family of diagrammatic equational theories unifying two research programs: categorical quantum mechanics and graphical linear algebra. We prove their completeness with respect to denotational semantics described in terms of relations between vector spaces equipped with symplectic structure. This provides versatile graphical languages encompassing both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser and Gaussian quantum circuits. Terms are described by labelled graphs with input and output interfaces, and the languages are equipped with equational theories amenable to standard graph rewriting techniques. In order to reason about large composite systems, we introduce a compact scalable notation where the vertices are themselves labelled by graphs. This notation allows us to state new and powerful rewrite rules which operate on diagrams at a large scale. We also show how this notation neatly captures some important constructions, such as graph states of quantum computing and the impedance and admittance matrices of electrical networks

ABSTRACT. BV-categories are a recent development that aims to give categorical semantics to proofs in the logic BV. However, due to the absence of a coherence theorem on one side and a well-defined notion of proof identity for BV on the other side, the precise relation between BV-categories and the logic BV is still not clear. To improve on this situation, we define in this paper a notion of proof identity for BV, based on the notion of atomic flows, which can be seen as a special form of string diagrams. Based on this notion of proof identity, we then strengthen the existing notion of BV-category and prove that it is sound with respect to the logic.

ABSTRACT. The Kripke semantics of various logics arise via categorical dualities between a category of relational frames and their maps, and a category of algebras and logical homomorphisms. When the relational frames are considered as computational systems (e.g. the states of a machine), the corresponding algebra is one of logical predicates on these systems (e.g. predicates on these states, i.e. program logics). Our aim is to extend this phenomenon to relations, putting well-behaved relations between systems (e.g. bisimulations) in correspondence with relations between predicates. This is achieved by constructing particular relational extensions of Tarski duality (for infinitary classical propositional logic) and Thomason duality (for infinitary classical modal logic). We sketch how these dualities give rise to a proof system that relates formulae between different systems.

ABSTRACT. It is an established idea in concurrency theory that every Petri net admits an unfolding semantics. This is a denotational object that represents its domain of possible executions. Unfoldings play an important role in formal reasoning and verification. This paper is concerned with the following well-known problem: while the unfolding resembles a universal construction in the category of Petri nets, it fails in general to satisfy the expected universal property because the construction overlooks the net's internal symmetries. There are two solutions: make these symmetries explicit to obtain a weak universal property (``up to symmetry''); or break the symmetries by assigning individual identities to components of the net, to restore a strict universal property. We review these two solutions in light of recent developments, and show that a universal unfolding to event structures--the canonical domain for Petri net semantics--can be established in each case. This paper additionally demonstrates a 2-categorical approach to Petri net unfoldings. We show that each unfolding semantics determines a 2-categorical relative adjunction involving Petri nets and event structures. Viewed in this way, the above two solutions can be related formally via an appropriate morphism of adjunctions. Finally we exhibit a 2-density property of event structures which implies that unfolding functors are essentially unique.