arXiv subjects

Fabio Zanasi

Publications and source records attributed to Fabio Zanasi.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Complete Axiomatisation of Equivalence for Discrete Probabilistic Programming

We introduce a sound and complete equational theory capturing equivalence of discrete probabilistic programs, that is, programs extended with primitives for Bernoulli distributions and conditioning, to model distributions over finite sets of events. To do so, we translate these programs into a graphical syntax of probabilistic circuits, formalised as string diagrams, the two-dimensional syntax of symmetric monoidal categories. We then prove a first completeness result for the equational theory of the conditioning-free fragment of our syntax. Finally, we extend this result to a complete equational theory for the entire language. Note these developments are also of interest for the development of probability theory in Markov categories: our first result gives a presentation by generators and equations of the category of Markov kernels, restricted to objects that are powers of the two-elements set.

cs.LO↗

arXiv2024

Learning Closed Signal Flow Graphs

We develop a learning algorithm for closed signal flow graphs - a graphical model of signal transducers. The algorithm relies on the correspondence between closed signal flow graphs and weighted finite automata on a singleton alphabet. We demonstrate that this procedure results in a genuine reduction of complexity: our algorithm fares better than existing learning algorithms for weighted automata restricted to the case of a singleton alphabet.

cs.LO↗

arXiv2024

Deep Learning with Parametric Lenses

We propose a categorical semantics for machine learning algorithms in terms of lenses, parametric maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as MSE and Softmax cross-entropy, and different architectures, shedding new light on their similarities and differences. Furthermore, our approach to learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realised in the discrete setting of Boolean and polynomial circuits. We demonstrate the practical significance of our framework with an implementation in Python.

cs.LG↗

arXiv2024

A Categorical Approach to DIBI Models

The logic of Dependence and Independence Bunched Implications (DIBI) is a logic to reason about conditional independence (CI); for instance, DIBI formulas can characterise CI in probability distributions and relational databases, using the probabilistic and relational DIBI models, respectively. Despite the similarity of the probabilistic and relational models, a uniform, more abstract account remains unsolved. The laborious case-by-case verification of the frame conditions required for constructing new models also calls for such a treatment. In this paper, we develop an abstract framework for systematically constructing DIBI models, using category theory as the unifying mathematical language. In particular, we use string diagrams -- a graphical presentation of monoidal categories -- to give a uniform definition of the parallel composition and subkernel relation in DIBI models. Our approach not only generalises known models, but also yields new models of interest and reduces properties of DIBI models to structures in the underlying categories. Furthermore, our categorical framework enables a logical notion of CI, in terms of the satisfaction of specific DIBI formulas. We compare it with string diagrammatic approaches to CI and show that it is an extension of string diagrammatic CI under reasonable conditions.

cs.LO↗

arXiv2023

An Introduction to String Diagrams for Computer Scientists

This document is an elementary introduction to string diagrams. It takes a computer science perspective: rather than using category theory as a starting point, we build on intuitions from formal language theory, treating string diagrams as a syntax with its semantics. After the basic theory, pointers are provided to contemporary applications of string diagrams in various fields of science.

cs.LO↗

arXiv2023

A Categorical Model for Retrosynthetic Reaction Analysis

We introduce a mathematical framework for retrosynthetic analysis, an important research method in synthetic chemistry. Our approach represents molecules and their interaction using string diagrams in layered props - a recently introduced categorical model for partial explanations in scientific reasoning. Such principled approach allows one to model features currently not available in automated retrosynthesis tools, such as chirality, reaction environment and protection-deprotection steps.

cs.LO↗

arXiv2023

String Diagrams for Layered Explanations

We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems biology, electrical circuit theory, and concurrency. We demonstrate how three explanatory models in these seemingly diverse areas can be all understood uniformly via a graphical calculus of layered props. Layered props allow for a compact visual presentation of the same phenomenon at different levels of precision, as well as the translation between these levels. Notably, our approach allows for partial explanations, that is, for translating just one part of a diagram while keeping the rest of the diagram untouched. Furthermore, our approach paves the way for formal reasoning about counterfactual models in systems biology.

cs.LO↗

arXiv2023

Data-Parallel Algorithms for String Diagrams

We give parallel algorithms for string diagrams represented as structured cospans of ACSets. Specifically, we give linear (sequential) and logarithmic (parallel) time algorithms for composition, tensor product, construction of diagrams from arbitrary $Σ$-terms, and application of functors to diagrams. Our datastructure can represent morphisms of both the free symmetric monoidal category over an arbitrary signature as well as those with a chosen Special Frobenius structure. We show how this additional (hypergraph) structure can be used to map diagrams to diagrams of optics. This leads to a case study in which we define an algorithm for efficiently computing symbolic representations of gradient-based learners based on reverse derivatives. The work we present here is intended to be useful as a general purpose datastructure. Implementation requires only integer arrays and well-known algorithms, and is data-parallel by constuction. We therefore expect it to be applicable to a wide variety of settings, including embedded and parallel hardware and low-level languages.

math.CT↗

arXiv2023

A Complete Diagrammatic Calculus for Boolean Satisfiability

We propose a calculus of string diagrams to reason about satisfiability of Boolean formulas, and prove it to be sound and complete. We then showcase our calculus in a few case studies. First, we consider SAT-solving. Second, we consider Horn clauses, which leads us to a new decision method for propositional logic programs equivalence under Herbrand model semantics.

cs.LO↗

arXiv2023

A Finite Axiomatisation of Finite-State Automata Using String Diagrams

We develop a fully diagrammatic approach to finite-state automata, based on reinterpreting their usual state-transition graphical representation as a two-dimensional syntax of string diagrams. In this setting, we are able to provide a complete equational theory for language equivalence, with two notable features. First, the proposed axiomatisation is finite. Second, the Kleene star is a derived concept, as it can be decomposed into more primitive algebraic blocks.

cs.LO↗

arXiv2022

The Cost of Compositionality: A High-Performance Implementation of String Diagram Composition

String diagrams are an increasingly popular algebraic language for the analysis of graphical models of computations across different research fields. Whereas string diagrams have been thoroughly studied as semantic structures, much less attention has been given to their algorithmic properties, and efficient implementations of diagrammatic reasoning are almost an unexplored subject. This work intends to be a contribution in such a direction. We introduce a data structure representing string diagrams in terms of adjacency matrices. This encoding has the key advantage of providing simple and efficient algorithms for composition and tensor product of diagrams. We demonstrate its effectiveness by showing that the complexity of the two operations is linear in the size of string diagrams. Also, as our approach is based on basic linear algebraic operations, we can take advantage of heavily optimised implementations, which we use to measure performances of string diagrammatic operations via several benchmarks.

math.CT↗

arXiv2022

String Diagram Rewrite Theory II: Rewriting with Symmetric Monoidal Structure

Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thoughts as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

cs.LO↗

arXiv2022

Categories of Differentiable Polynomial Circuits for Machine Learning

Reverse derivative categories (RDCs) have recently been shown to be a suitable semantic framework for studying machine learning algorithms. Whereas emphasis has been put on training methodologies, less attention has been devoted to particular \emph{model classes}: the concrete categories whose morphisms represent machine learning models. In this paper we study presentations by generators and equations of classes of RDCs. In particular, we propose \emph{polynomial circuits} as a suitable machine learning model. We give an axiomatisation for these circuits and prove a functional completeness result. Finally, we discuss the use of polynomial circuits over specific semirings to perform machine learning with discrete values.

cs.LG↗

arXiv2022

String Diagram Rewrite Theory III: Confluence with and without Frobenius

In this paper we address the problem of proving confluence for string diagram rewriting, which was previously shown to be characterised combinatorically as double-pushout rewriting with interfaces (DPOI) on (labelled) hypergraphs. For standard DPO rewriting without interfaces, confluence for terminating rewrite systems is, in general, undecidable. Nevertheless, we show here that confluence for DPOI, and hence string diagram rewriting, is decidable. We apply this result to give effective procedures for deciding local confluence of symmetric monoidal theories with and without Frobenius structure by critical pair analysis. For the latter, we introduce the new notion of path joinability for critical pairs, which enables finitely many joins of a critical pair to be lifted to an arbitrary context in spite of the strong non-local constraints placed on rewriting in a generic symmetric monoidal theory.

cs.LO↗

arXiv2022

String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

String diagrams are a powerful and intuitive graphical syntax, originated in the study of symmetric monoidal categories. In the last few years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In many such proposals, the transformations of the described systems are modelled as rewrite rules of diagrams. These developments demand a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses additional challenges. This work systematises and expands a series of recent conference papers laying down such foundation. As first step, we focus on the case of rewrite systems for string diagrammatic theories which feature a Frobenius algebra. This situation ubiquitously appear in various approaches: for instance, in the algebraic semantics of linear dynamical systems, Frobenius structures model the wiring of circuits; in categorical quantum mechanics, they model interacting quantum observables. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures, in terms of double-pushout hypergraph rewriting. Furthermore, we prove this interpretation to be sound and complete. In the last part, we also see that the approach can be generalised to model rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

cs.LO↗

arXiv2021

Equivalence checking for weak bi-Kleene algebra

Pomset automata are an operational model of weak bi-Kleene algebra, which describes programs that can fork an execution into parallel threads, upon completion of which execution can join to resume as a single thread. We characterize a fragment of pomset automata that admits a decision procedure for language equivalence. Furthermore, we prove that this fragment corresponds precisely to series-rational expressions, i.e., rational expressions with an additional operator for bounded parallelism. As a consequence, we obtain a new proof that equivalence of series-rational expressions is decidable.

cs.FL↗

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.

cs.PL↗

arXiv2021

Categorical Foundations of Gradient-Based Learning

We propose a categorical semantics of gradient-based machine learning algorithms in terms of lenses, parametrised maps, and reverse derivative categories. This foundation provides a powerful explanatory and unifying framework: it encompasses a variety of gradient descent algorithms such as ADAM, AdaGrad, and Nesterov momentum, as well as a variety of loss functions such as as MSE and Softmax cross-entropy, shedding new light on their similarities and differences. Our approach to gradient-based learning has examples generalising beyond the familiar continuous domains (modelled in categories of smooth maps) and can be realized in the discrete setting of boolean circuits. Finally, we demonstrate the practical significance of our framework with an implementation in Python.

cs.LG↗