arXiv subjects

Jade Master

Publications and source records attributed to Jade Master.

13 recordsLinked to original sources

arXiv2025

Colored Petri Nets are Monoidal Double Functors

We give a characterization of colored Petri nets as monoidal double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular, we show how morphisms of colored Petri nets may be understood as natural transformations. The displayed category construction explains how lax double functors are equivalent to functors with codomain their former domain. We use this result to characterize the unfolding of colored Petri nets in terms of free symmetric monoidal categories.

math.CT↗

arXiv2023

Relative fixed points of functors

We show how the relatively initial or relatively terminal fixed points for a well-behaved functor $F$ form a pair of adjoint functors between $F$-coalgebras and $F$-algebras. We use the language of locally presentable categories to find sufficient conditions for existence of this adjunction. We show that relative fixed points may be characterized as (co)equalizers of the free (co)monad on $F$. In particular, when $F$ is a polynomial functor on $\mathsf{Set}$ the relative fixed points are a quotient or subset of the free term algebra or the cofree term coalgebra. We give examples of the relative fixed points for polynomial functors and an example which is the Sierpinski carpet. Lastly, we prove a general preservation result for relative fixed points.

math.CT↗

arXiv2023

Proceedings Fifth International Conference on Applied Category Theory

The Fifth International Conference on Applied Category Theory took place at the University of Strathclyde in Glasgow, Scotland on 18-22 July 2022. This conference follows the previous meetings at Leiden (2018), Oxford (2019), MIT (2020, fully online), and Cambridge (2021). The conference comprised 59 contributed talks, a poster session, an industry showcase session, and a session where junior researchers who had attended the Adjoint School presented the results of their research at the school. Information regarding the conference may be found at (https://msp.cis.strath.ac.uk/act2022). The contributions to ACT2022 ranged from pure to applied and included contributions in a wide range of disciplines in science and engineering. ACT2022 included talks in linguistics, functional programming, classical mechanics, quantum physics, probability theory, electrical engineering, epidemiology, thermodynamics, engineering, and logic. ACT2022 was sponsored by Huawei, Protocol Labs, Cambridge Quantum, Conexus, Topos, and SICSA (Scottish Informatics and Computer Science Alliance). Submission to ACT2022 had three tracks: extended abstracts, software demonstrations, and proceedings. The extended abstract and software demonstration submissions had a page limit of 2 pages, and the proceedings track had a page limit of 14 pages. Only papers submitted to the proceedings track were considered for publication in this volume. In total, there were 97 submissions, of which 59 were accepted for presentation and 24 for publication in this volume. Publication of accepted submissions in the proceedings was determined by personal choice of the authors and not based on quality. Each submission received a review from three different members of the programming committee, and papers were selected based on discussion and consensus by these reviewers.

cs.LO↗

arXiv2023

Beyond Initial Algebras and Final Coalgebras

We provide a construction of the fixed points of functors which may not be inital algebras or final coalgebras. For an endofunctor F, this fixed point construction may be expressed as a pair of adjoint functors between F-coalgebras and F-algebras. We prove a version of the limit colimit coincidence theorem for these generalized fixed points.

math.CT↗

arXiv2022

How to Compose Shortest Paths

The composition problem for shortest paths asks the following: given shortest paths on weighted graphs M and N which share a common boundary, find the shortest paths on their union. This problem is a crucial step in any algorithm which uses the divide and conquer method to find shortest paths. This extended abstract details how this problem may be understood categorically. Finding shortest paths is represented by a functor and the composition problem asks to find the value of this functor on a pushout using the values of the functor on the components. Furthermore, we present an algorithm which solves the composition problem for shortest paths. When implemented in Python, this algorithm reduces the computation time for finding shortest paths by relying on precompilation.

cs.DM↗

arXiv2021

Composing Behaviors of Networks

This thesis aims to develop a compositional theory for the operational semantics of networks. The networks considered are described by either internal or enriched graphs. In the internal case we focus on $\mathsf{Q}$-nets, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$. $\mathsf{Q}$-nets include many known variants of Petri nets including pre-nets, integer nets, elementary net systems, and bounded nets. In the enriched case we focus on graphs enriched in a quantale $R$ regarded as matrices with entries in $R$. These $R$-matrices represent distance networks, Markov processes, capacity networks, non-deterministic finite automata, simple graphs, and more. The operational semantics of $\mathsf{Q}$-nets is constructed as an adjunction between $\mathsf{Q}$-nets and categories internal to the category of models of $\mathsf{Q}$. Similarly, the operational semantics of $R$-matrices is constructed as an adjunction between $R$-matrices and categories enriched in $R$. The left adjoint of this adjunction sends an $R$-matrix $M$ to the $R$-category $F_R(M)$ whose hom-objects are solutions of the algebraic path problem: a generalization of the shortest path problem to graphs weighted in $R$. For both $\mathsf{Q}$-nets and $R$-matrices we use the theory of structured cospans to study the compositionality of the above operational semantics. For each type of network we construct a double category whose morphisms are "open networks", i.e. networks with certain vertices designated as input or output. We introduce the black-boxing of an open network, a profunctor describing the externally observable behavior of an open network. We introduce a class of open networks called "functional open networks" for which black-boxing preserves composition.

math.CT↗

arXiv2021

Categories of Nets

We present a unified framework for Petri nets and various variants, such as pre-nets and Kock's whole-grain Petri nets. Our framework is based on a less well-studied notion that we call $\Sigma$-nets, which allow finer control over whether tokens are treated using the collective or individual token philosophy. We describe three forms of execution semantics in which pre-nets generate strict monoidal categories, $\Sigma$-nets (including whole-grain Petri nets) generate symmetric strict monoidal categories, and Petri nets generate commutative monoidal categories, all by left adjoint functors. We also construct adjunctions relating these categories of nets to each other, in particular showing that all kinds of net can be embedded in the unifying category of $\Sigma$-nets, in a way that commutes coherently with their execution semantics.

math.CT↗

arXiv2020

The Open Algebraic Path Problem

The algebraic path problem provides a general setting for shortest path algorithms in optimization and computer science. This work extends the algebraic path problem to networks equipped with input and output boundaries. We show that the algebraic path problem is functorial as a mapping from a double category whose horizontal composition is gluing of open networks. We introduce functional open matrices, for which the functoriality of the algebraic path problem has a more practical expression.

math.CT↗

arXiv2020

Why is Homology so Powerful?

My short answer to this question is that homology is powerful because it computes invariants of higher categories. In this article we show how this true by taking a leisurely tour of the connection between category theory and homological algebra.

math.AT↗

arXiv2019

String Diagrams for Assembly Planning

Assembly planning is a difficult problem for companies. Many disciplines such as design, planning, scheduling, and manufacturing execution need to be carefully engineered and coordinated to create successful product assembly plans. Recent research in the field of design for assembly has proposed new methodologies to design product structures in such a way that their assembly is easier. However, present assembly planning approaches lack the engineering tool support to capture all the constraints associated to assembly planning in a unified manner. This paper proposes CompositionalPlanning, a string diagram based framework for assembly planning. In the proposed framework, string diagrams and their compositional properties serve as the foundation for an engineering tool where CAD designs interact with planning and scheduling algorithms to automatically create high-quality assembly plans. These assembly plans are then executed in simulation to measure their performance and to visualize their key build characteristics. We demonstrate the versatility of this approach in the LEGO assembly domain. We developed two reference LEGO CAD models that are processed by CompositionalPlanning's algorithmic pipeline. We compare sequential and parallel assembly plans in a Minecraft simulation and show that the time-to-build performance can be optimized by our algorithms.

cs.RO↗

arXiv2019

Petri Nets Based on Lawvere Theories

We give a definition of $\mathsf{Q}$-net, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of $\mathsf{Q}$-nets. To justify our definition of $\mathsf{Q}$-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which $\mathsf{Q}$-nets present free models of $\mathsf{Q}$ in $\mathsf{Cat}$. This gives a functorial description of the operational semantics for an arbitrary category of $\mathsf{Q}$-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

math.CT↗

arXiv2018

Translating and Evolving: Towards a Model of Language Change in DisCoCat

The categorical compositional distributional (DisCoCat) model of meaning developed by Coecke et al. (2010) has been successful in modeling various aspects of meaning. However, it fails to model the fact that language can change. We give an approach to DisCoCat that allows us to represent language models and translations between them, enabling us to describe translations from one language to another, or changes within the same language. We unify the product space representation given in (Coecke et al., 2010) and the functorial description in (Kartsaklis et al., 2013), in a way that allows us to view a language as a catalogue of meanings. We formalize the notion of a lexicon in DisCoCat, and define a dictionary of meanings between two lexicons. All this is done within the framework of monoidal categories. We give examples of how to apply our methods, and give a concrete suggestion for compositional translation in corpora.

cs.CL↗

arXiv2018

Open Petri Nets

The reachability semantics for Petri nets can be studied using open Petri nets. For us an "open" Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri})$, which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. We describe two forms of semantics for open Petri nets using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$. The first, an operational semantics, gives for each open Petri net a category whose morphisms are the processes that this net can carry out. This is done in a compositional way, so that these categories can be computed on smaller subnets and then glued together. The second, a reachability semantics, simply says which markings of the outputs can be reached from a given marking of the inputs.

math.CT↗