We use the theory of algebraic effects to give a complete equational axiomatization for dynamic threads. Our method is based on parameterized algebraic theories, which give a concrete syntax for strong monads on functor categories, and are a convenient framework for names and binding. Our programs are built from the key primitives ‘fork’ and ‘wait’. ‘Fork’ creates a child thread and passes its name (thread ID) to the parent thread. ‘Wait’ allows us to wait for given child threads to finish. We provide a parameterized algebraic theory built from fork and wait, together with basic atomic actions and laws such as associativity of ‘fork’. Our equational axiomatization is complete in two senses. First, for closed expressions, it completely captures equality of labelled posets (pomsets), an established model of concurrency: model complete. Second, any two open expressions are provably equal if they are equal under all closing substitutions: syntactically complete. The benefit of algebraic effects is that the semantic analysis can focus on the algebraic operations of fork and wait. We then extend the analysis to a simple concurrent programming language by giving operational and denotational semantics. The denotational semantics is built using the methods of parameterized algebraic theories and we show that it is sound, adequate, and fully abstract at first order for labelled-poset observations.

We adapt Fiore, Plotkin, and Turi’s treatment of abstract syntax with binding, substitution, and holes to account for languages with second-class sorts. These situations include programming calculi such as the Call-by-Value lambda-calculus (CBV) and Levy’s Call-by-Push-Value (CBPV). Prohibiting second-class sorts from appearing in variable contexts changes the characterisation of the abstract syntax from monoids in monoidal categories to actions in actegories. We reproduce much of the development through bicategorical arguments. We apply the resulting theory by proving substitution lemmata for varieties of CBV.

We present a new design for an algebraic simplification library structured around concepts from universal algebra: theories, models, homomorphisms, and universal properties of free algebras and free extensions of algebras. The library’s dependently typed interface guarantees that both built-in and user-defined simplification modules are terminating, sound, and complete with respect to a well-specified class of equations. We have implemented the design in the Idris 2 and Agda dependently typed programming languages and shown that it supports modular extension to new theories, proof extraction and certification, goal extraction via reflection, and interactive development.

We present a compositional denotational semantics for a functional language with first-class parallel composition and shared-memory operations whose operational semantics follows the Release/Acquire weak memory model (RA). The semantics is formulated in Moggi’s monadic approach and is based on Brookes-style traces. To do so we adapt Brookes’s traces to view-based machine for RA by Kang et al., and supplement Brookes’s mumble and stutter closure operations with additional operations, specific to RA. The latter provides a more nuanced understanding of traces that uncouples them from operational interrupted executions. We show that our denotational semantics is adequate and use it to validate various program transformations of interest. This is the first work to put weak memory models on the same footing as many other programming effects in Moggi’s standard monadic approach.

We define a two sorted equational theory of algebraic effects that models concurrent shared state with preemptive interleaving, recovering Brookes’s seminal 1996 trace-based model precisely. The decomposition allows us to analyse Brookes’s model algebraically in terms of separate but interacting components. The multiple sorts partition terms into layers. We use two sorts: a “hold” sort for layers that disallow interleaving of environment memory accesses, analogous to holding a global lock on the memory; and a “cede” sort for the opposite. The algebraic signature comprises of independent interlocking components: two new operators that switch between these sorts, delimiting the atomic layers, thought of as acquiring and releasing the global lock; non-deterministic choice; and state-accessing operators. The axioms similarly divide cleanly: the delimiters behave as a closure pair; all operators are strict, and distribute over non-empty non-deterministic choice; and non-deterministic global state obeys Plotkin and Power’s presentation of global state. Our representation theorem expresses the free algebras over a two-sorted family of variables as sets of traces with suitable closure conditions. When the held sort has no variables, we recover Brookes’s trace semantics. We define several other single-and two-sorted theories to elucidate the connection to Brookes’s model via translation embeddings and equivalences.

Dependent pattern matching is a key feature in dependently typed programming. However, there is a theory-practice disconnect: while many proof assistants implement pattern matching as primitive, theoretical presentations give semantics to pattern matching by elaborating to eliminators. Though theoretically convenient, eliminators can be awkward and verbose, particularly for complex combinations of patterns. This work aims to bridge the theory-practice gap by presenting a direct categorical semantics for pattern matching, which does not elaborate to eliminators. This is achieved using sheaf theory to describe when sets of arrows (terms) can be amalgamated into a single arrow. We present a language with top-level dependent pattern matching, without specifying which sets of patterns are considered covering for a match. Then, we give a sufficient criterion for which pattern-sets admit a sound model: patterns should be in the canonical coverage for the category of contexts. Finally, we use sheaf-theoretic saturation conditions to devise some allowable sets of patterns. We are able to express and exceed the status quo, giving semantics for datatype constructors, nested patterns, absurd patterns, propositional equality, and dot patterns.

Rewriting is a versatile and powerful technique used in many domains. Strategic rewriting allows programmers to control the application of rewrite rules by composing individual rewrite rules into complex rewrite strategies. These strategies are semantically complex, as they may be nondeterministic, they may raise errors that trigger backtracking, and they may not terminate. Given such semantic complexity, it is necessary to establish a formal understanding of rewrite strategies and to enable reasoning about them in order to answer questions like: How do we know that a rewrite strategy terminates? How do we know that a rewrite strategy does not fail because we compose two incompatible rewrites? How do we know that a desired property holds after applying a rewrite strategy? In this paper, we introduce Shoggoth: a formal foundation for understanding, analysing and reasoning about strategic rewriting that is capable of answering these questions. We provide a denotational semantics of System S, a core language for strategic rewriting, and prove its equivalence to our big-step operational semantics, which extends existing work by explicitly accounting for divergence. We further define a location-based weakest precondition calculus to enable formal reasoning about rewriting strategies, and we prove this calculus sound with respect to the denotational semantics. We show how this calculus can be used in practice to reason about properties of rewriting strategies, including termination, that they are well-composed, and that desired postconditions hold. The semantics and calculus are formalised in Isabelle/HOL and all proofs are mechanised.

We present a compositional denotational semantics for a functional language with first-class parallel composition and shared-memory operations whose operational semantics follows the Release/Acquire weak memory model (RA). The semantics is formulated in Moggi’s monadic approach, and is based on Brookes-style traces. To do so we adapt Brookes’s traces to Kang et al.’s view-based machine for RA, and supplement Brookes’s mumble and stutter closure operations with additional operations, specific to RA. The latter provides a more nuanced understanding of traces that uncouples them from operational interrupted executions. We show that our denotational semantics is adequate and use it to validate various program transformations of interest. This is the first work to put weak memory models on the same footing as many other programming effects in Moggi’s standard monadic approach.

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming.

We give an adequate denotational semantics for languages with recursive higher-order types, continuous probability distributions, and soft constraints. These are expressive languages for building Bayesian models of the kinds used in computational statistics and machine learning. Among them are untyped languages, similar to Church and WebPPL, because our semantics allows recursive mixed-variance datatypes. Our semantics justifies important program equivalences including commutativity. Our new semantic model is based on ‘quasi-Borel predomains’. These are a mixture of chain-complete partial orders (cpos) and quasi-Borel spaces. Quasi-Borel spaces are a recent model of probability theory that focuses on sets of admissible random elements. Probability is traditionally treated in cpo models using probabilistic powerdomains, but these are not known to be commutative on any class of cpos with higher order functions. By contrast, quasi-Borel predomains do support both a commutative probabilistic powerdomain and higher-order functions. As we show, quasi-Borel predomains form both a model of Fiore’s axiomatic domain theory and a model of Kock’s synthetic measure theory.

We compare the expressive power of three programming abstractions for user-defined computational effects: Plotkin and Pretnar’s effect handlers, Filinski’s monadic reflection, and delimited control. This comparison allows a precise discussion about the relative expressiveness of each programming abstraction. It also demonstrates the sensitivity of the relative expressiveness of user-defined effects to seemingly orthogonal language features. We present three calculi, one per abstraction, extending Levy’s call-by-push-value. For each calculus, we present syntax, operational semantics, a natural type-and-effect system, and, for effect handlers and monadic reflection, a set-theoretic denotational semantics. We establish their basic metatheoretic properties: safety, termination, and, where applicable, soundness and adequacy. Using Felleisen’s notion of a macro translation, we show that these abstractions can macro express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro expressed while preserving typeability either by monadic reflection or by delimited control. Our argument fails with simple changes to the type system such as polymorphism and inductive types. We supplement our development with a mechanised Abella formalisation.

Partially-static data structures are a well-known technique for improving binding times. However, they are often defined in an ad-hoc manner, without a unifying framework to ensure full use of the equations associated with each operation. We present a foundational view of partially-static data structures as free extensions of algebras for suitable equational theories, i.e. the coproduct of an algebra and a free algebra in the category of algebras and their homomorphisms. By precalculating these free extensions, we construct a high-level library of partially-static data representations for common algebraic structures. We demonstrate our library with common use-cases from the literature: string and list manipulation, linear algebra, and numerical simplification.

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock’s synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.