arXiv subjects

Dan R. Ghica

Publications and source records attributed to Dan R. Ghica.

At least 19 recordsLinked to original sources

arXiv2026

A Complete Theory of Sequential Digital Circuits: Denotational, Operational and Algebraic Semantics

Digital circuits, despite having been studied for nearly a century and used at scale for about half that time, have until recently evaded a fully compositional theoretical in which arbitrary circuits may be freely composed together without consulting their internals. Recent work remedied this theoretical shortcoming by showing how digital circuits can be presented compositionally as morphisms in a freely generated symmetric traced category. However, this was done informally; in this paper we refine and expand the previous work in several ways, culminating in the presentation of three sound and complete semantics for digital circuits: denotational, operational and algebraic. For the denotational semantics, we establish a correspondence between stream functions with certain properties and circuits constructed syntactically. For the operational semantics, we present the reductions required to model how a circuit processes a value, including the addition of a new reduction for eliminating non-delay-guarded feedback; this leads to an adequate notion of observational equivalence for digital circuits. Finally, we define a new family of equations for translating circuits into bisimilar circuits of a 'normal form', leading to a complete algebraic semantics for sequential circuits.

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arXiv2026

Categorical E-Graphs for Lambda Calculi

Equality saturation, a technique for program optimisation and reasoning, has gained attention due to the resurgence of equality graphs (e-graphs). E-graphs represent equivalence classes of terms under rewrite rules, enabling simultaneous rewriting across a family of terms. However, they struggle in domains like $λ$-calculus that involve variable binding, due to a lack of native support for bindings. Building on recent work interpreting e-graphs categorically as morphisms in semilattice-enriched symmetric monoidal categories, we extend this framework to closed symmetric monoidal categories to handle bindings. We provide a concrete combinatorial representation using hierarchical hypergraphs and introduce a corresponding double-pushout (DPO) rewriting mechanism. Finally, we establish the equivalence of term rewriting and DPO rewriting, with the key property that the combinatorial representation absorbs the equations of the symmetric monoidal category.

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arXiv2026

Rewriting Modulo Traced Comonoid Structure

In this paper we adapt previous work on rewriting string diagrams using hypergraphs to the case where the underlying category has a traced comonoid structure, in which wires can be forked and the outputs of a morphism can be connected to its input. Such a structure is particularly interesting because any traced Cartesian (dataflow) category has an underlying traced comonoid structure. We show that certain subclasses of hypergraphs are fully complete for traced comonoid categories: that is to say, every term in such a category has a unique corresponding hypergraph up to isomorphism, and from every hypergraph with the desired properties, a unique term in the category can be retrieved up to the axioms of traced comonoid categories. We also show how the framework of double pushout rewriting (DPO) can be adapted for traced comonoid categories by characterising the valid pushout complements for rewriting in our setting. We conclude by presenting a case study in the form of recent work on an equational theory for sequential circuits: circuits built from primitive logic gates with delay and feedback. The graph rewriting framework allows for the definition of an operational semantics for sequential circuits.

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arXiv2025

Equivalence Hypergraphs: DPO Rewriting for Monoidal E-Graphs

The technique of \emph{equality saturation}, which equips graphs with an equivalence relation, has proven effective for program optimisation. We give a categorical semantics to these structures, called \emph{e-graphs}, in terms of Cartesian categories enriched over the category of semilattices. This approach generalises to monoidal categories, which opens the door to new applications of e-graph techniques, from algebraic to monoidal theories. Finally, we present a sound and complete combinatorial representation of morphisms in such a category, based on a generalisation of hypergraphs which we call \emph{e-hypergraphs}. They have the usual advantage that many of their structural equations are absorbed into a general notion of isomorphism. This new principled approach to e-graphs enables double-pushout (DPO) rewriting for these structures, which constitutes the main contribution of this paper.

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arXiv2025

A robust graph-based approach to observational equivalence

We propose a new step-wise approach to proving observational equivalence, and in particular reasoning about fragility of observational equivalence. Our approach is based on what we call local reasoning. The local reasoning exploits the graphical concept of neighbourhood, and it extracts a new, formal, concept of robustness as a key sufficient condition of observational equivalence. Moreover, our proof methodology is capable of proving a generalised notion of observational equivalence. The generalised notion can be quantified over syntactically restricted contexts instead of all contexts, and also quantitatively constrained in terms of the number of reduction steps. The operational machinery we use is given by a hypergraph-rewriting abstract machine inspired by Girard's Geometry of Interaction. The behaviour of language features, including function abstraction and application, is provided by hypergraph-rewriting rules. We demonstrate our proof methodology using the call-by-value lambda-calculus equipped with (higher-order) state.

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arXiv2021

Functorial String Diagrams for Reverse-Mode Automatic Differentiation

We enhance the calculus of string diagrams for monoidal categories with hierarchical features in order to capture closed monoidal (and cartesian closed) structure. Using this new syntax we formulate an automatic differentiation algorithm for (applied) simply typed lambda calculus in the style of [Pearlmutter and Siskind 2008] and we prove for the first time its soundness. To give an efficient yet principled implementation of the AD algorithm we define a sound and complete representation of hierarchical string diagrams as a class of hierarchical hypergraphs we call hypernets.

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arXiv2021

Transparent Synchronous Dataflow

Dataflow programming is a popular and convenient programming paradigm in systems modelling, optimisation, and machine learning. It has a number of advantages, for instance the lacks of control flow allows computation to be carried out in parallel as well as in distributed machines. More recently the idea of dataflow graphs has also been brought into the design of various deep learning frameworks. They facilitate an easy and efficient implementation of automatic differentiation, which is the heart of modern deep learning paradigm. [abstract abridged]

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arXiv2021

Operational Semantics with Hierarchical Abstract Syntax Graphs

This is a motivating tutorial introduction to a semantic analysis of programming languages using a graphical language as the representation of terms, and graph rewriting as a representation of reduction rules. We show how the graphical language automatically incorporates desirable features, such as alpha-equivalence and how it can describe pure computation, imperative store, and control features in a uniform framework. The graph semantics combines some of the best features of structural operational semantics and abstract machines, while offering powerful new methods for reasoning about contextual equivalence. All technical details are available in an extended technical report by Muroya and the author and in Muroya's doctoral dissertation.

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arXiv2020

A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant Agda.

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arXiv2019

The Dynamic Geometry of Interaction Machine: A Token-Guided Graph Rewriter

In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. Aiming at a unified framework that would enable the study of this trade-off, we introduce an abstract machine, inspired by Girard's Geometry of Interaction (GoI), a machine combining token passing and graph rewriting. We show soundness and completeness of our abstract machine, called the \emph{Dynamic GoI Machine} (DGoIM), with respect to three evaluations: call-by-need, left-to-right call-by-value, and right-to-left call-by-value. Analysing time cost of its execution classifies the machine as ``efficient'' in Accattoli's taxonomy of abstract machines.

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arXiv2019

The far side of the cube

Game-semantic models usually start from the core model of the prototypical language PCF, which is characterised by a range of combinatorial constraints on the shape of plays. Relaxing each such constraint usually corresponds to the introduction of a new language operation, a feature of game semantics commonly known as the `Abramsky Cube'. In this presentation we relax all such combinatorial constraints, resulting in the most general game model, in which all the other game models live. This is perhaps the simplest set up in which to understand game semantics, so it should serve as a portal to the other, more complex, game models in the literature. It might also be interesting in its own right, as an extremal instance of the game-semantic paradigm.

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arXiv2018

Efficient Implementation of Evaluation Strategies via Token-Guided Graph Rewriting

In implementing evaluation strategies of the lambda-calculus, both correctness and efficiency of implementation are valid concerns. While the notion of correctness is determined by the evaluation strategy, regarding efficiency there is a larger design space that can be explored, in particular the trade-off between space versus time efficiency. We contributed to the study of this trade-off by the introduction of an abstract machine for call-by-need, inspired by Girard's Geometry of Interaction, a machine combining token passing and graph rewriting. This work presents an extension of the machine, to additionally accommodate left-to-right and right-to-left call-by-value strategies. We show soundness and completeness of the extended machine with respect to each of the call-by-need and two call-by-value strategies. Analysing time cost of its execution classifies the machine as "efficient" in Accattoli's taxonomy of abstract machines.

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arXiv2017

On the Learnability of Programming Language Semantics

Game semantics is a powerful method of semantic analysis for programming languages. It gives mathematically accurate models ("fully abstract") for a wide variety of programming languages. Game semantic models are combinatorial characterisations of all possible interactions between a term and its syntactic context. Because such interactions can be concretely represented as sets of sequences, it is possible to ask whether they can be learned from examples. Concretely, we are using long short-term memory neural nets (LSTM), a technique which proved effective in learning natural languages for automatic translation and text synthesis, to learn game-semantic models of sequential and concurrent versions of Idealised Algol (IA), which are algorithmically complex yet can be concisely described. We will measure how accurate the learned models are as a function of the degree of the term and the number of free variables involved. Finally, we will show how to use the learned model to perform latent semantic analysis between concurrent and sequential Idealised Algol.

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arXiv2017

Abductive functional programming, a semantic approach

We propose a call-by-value lambda calculus extended with a new construct inspired by abductive inference and motivated by the programming idioms of machine learning. Although syntactically simple the abductive construct has a complex and subtle operational semantics which we express using a style based on the Geometry of Interaction. We show that the calculus is sound, in the sense that well typed programs terminate normally. We also give a visual implementation of the semantics which relies on additional garbage collection rules, which we also prove sound.

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arXiv2017

The Dynamic Geometry of Interaction Machine: A Call-by-need Graph Rewriter

Girard's Geometry of Interaction (GoI), a semantics designed for linear logic proofs, has been also successfully applied to programming language semantics. One way is to use abstract machines that pass a token on a fixed graph along a path indicated by the GoI. These token-passing abstract machines are space efficient, because they handle duplicated computation by repeating the same moves of a token on the fixed graph. Although they can be adapted to obtain sound models with regard to the equational theories of various evaluation strategies for the lambda calculus, it can be at the expense of significant time costs. In this paper we show a token-passing abstract machine that can implement evaluation strategies for the lambda calculus, with certified time efficiency. Our abstract machine, called the Dynamic GoI Machine (DGoIM), rewrites the graph to avoid replicating computation, using the token to find the redexes. The flexibility of interleaving token transitions and graph rewriting allows the DGoIM to balance the trade-off of space and time costs. This paper shows that the DGoIM can implement call-by-need evaluation for the lambda calculus by using a strategy of interleaving token passing with as much graph rewriting as possible. Our quantitative analysis confirms that the DGoIM with this strategy of interleaving the two kinds of possible operations on graphs can be classified as "efficient" following Accattoli's taxonomy of abstract machines.

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arXiv2017

Diagrammatic Semantics for Digital Circuits

We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about abstract circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications.

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arXiv2013

From bounded affine types to automatic timing analysis

Bounded linear types have proved to be useful for automated resource analysis and control in functional programming languages. In this paper we introduce an affine bounded linear typing discipline on a general notion of resource which can be modeled in a semiring. For this type system we provide both a general type-inference procedure, parameterized by the decision procedure of the semiring equational theory, and a (coherent) categorical semantics. This is a very useful type-theoretic and denotational framework for many applications to resource-sensitive compilation, and it represents a generalization of several existing type systems. As a non-trivial instance, motivated by our ongoing work on hardware compilation, we present a complex new application to calculating and controlling timing of execution in a (recursion-free) higher-order functional programming language with local store.

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arXiv2013

Abstract machines for game semantics, revisited

We define new abstract machines for game semantics which correspond to networks of conventional computers, and can be used as an intermediate representation for compilation targeting distributed systems. This is achieved in two steps. First we introduce the HRAM, a Heap and Register Abstract Machine, an abstraction of a conventional computer, which can be structured into HRAM nets, an abstract point-to-point network model. HRAMs are multi-threaded and subsume communication by tokens (cf. IAM) or jumps. Game Abstract Machines (GAM), are HRAMs with additional structure at the interface level, but no special operational capabilities. We show that GAMs cannot be naively composed, but composition must be mediated using appropriate HRAM combinators. HRAMs are flexible enough to allow the representation of game models for languages with state (non-innocent games) or concurrency (non-alternating games). We illustrate the potential of this technique by implementing a toy distributed compiler for ICA, a higher-order programming language with shared state concurrency, thus significantly extending our previous distributed PCF compiler. We show that compilation is sound and memory-safe, i.e. no (distributed or local) garbage collection is necessary.

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