The Chapter will provide a brief and informal introduction to diagrammatic reasoning (DR) and network modelling (NM) using string diagrams, which can be shown to possess the same degree of rigor as symbolic algebra, while achieving greater abbreviative power (and pedagogical insight) than more conventional techniques of diagram-chasing. This review of the research literature will set the context for a detailed examination of two case-studies of semantic technologies which have been applied to the management of emergency services and search-and-rescue operations. The next section of the Chapter will consider the implications of contemporary and closely related developments in software engineering for disaster management. Conclusions will follow.
This Chapter is concerned with developments in applied mathematics and theoretical computing that can provide a formal and technical support for practices of disaster management. To this end it will draw on recent developments in applied category theory , which inform semantic technologies. In the interests of brevity, it will be obliged to eschew formal exposition of these techniques, but to this end, comprehensive references will be provided. The justification for what might at first seem to be an unduly narrow focus, is that applied category theory facilitates translation between different mathematical, computational and scientific domains.
For its part, Semantic Technology (ST) can be loosely conceived as an approach treating the World-Wide-Web as a “giant global graph”, so that valuable and timely information can be extracted from it using rich structured-query languages and extended description logics. These query languages must be congruent with pertinent (organizational, application, and database) ontologies so that the extracted information can be converted into intelligence. Significantly, database instances can extend beyond relational or graph databases, to include Boolean matrices, relational data embedded within the category of linear relations, and that pertaining to systems of differential equations in finite vector space, or even quantum tensor networks within a finite Hilbert space.
More specifically, this chapter will introduce the formalism of string diagrams, which were initially derived from the work of the mathematical physicists, Roger Penrose (1971) and Richard Feynman (1948). However, this diagrammatic approach has since been extended and re-interpreted by category theorists such as Andre Joyal and Roy Street (1988, 1991). For example, Feynman diagrams can be viewed as morphisms in the category Hilb of Hilbert spaces and bounded linear operators (Westrich, 2006, fn. 3: 8), while Baez and Lauda (2009) interpret them as “a notation for intertwining operators between positive-energy representations of the Poincaré group”. Penrose diagrams can be viewed as a representation of operations within a tensor category.
Joyal and Street have demonstrated that when these string diagrams are manipulated in accordance with certain axioms—the latter taking the form of a set of equivalence relations established between related pairs of diagrams—the movements from one diagram to another can be shown to reproduce the algebraic steps of a non-diagrammatic proof. Furthermore, they can be shown to possess a greater degree of abbreviative power. This renders an approach using string diagrams extremely useful for teaching, experimentation, and exposition.
In addition to these conceptual and pedagogical advantages, however, there are additional implementation advantages associated with string diagrams including: (i) those of compositionality and layering (e.g. in Willems’s 2007 behavioural approach to systems theory, complex systems can be construed as the composites of smaller and simpler building blocks, which are then linked together in accordance with certain coherence conditions); (ii) a capacity for direct translation into functional programming (and thus, into propositions within a linear or resource-using logic); and, (iii) the potential for the subsequent application of software design and verification tools. It should be appreciated that these formal attributes will become increasingly important as the correlative features of what some have described as the digital economy.
This chapter will consider the specific role of string diagrams in the development and deployment of semantic technologies, which in turn have been developed for applications of relevance to disaster management practices. Techniques based on string diagrams have been developed to encompass a wide variety of dynamic systems and application domains, such as Petri nets, the π-calculus, and Bigraphs (Milner, 2009), Bayesian networks (Kissinger & Uijlen, 2017), thermodynamic networks (Baez and Pollard, 2017), and quantum tensor networks (Biamonte & Bergholm, 2017), as well as reaction-diffusion systems (Baez and Biamonte, 2012). Furthermore, they have the capacity to encompass graphical forms of linear algebra (Sobociński, Blog), universal algebras (Baez, 2006), and signal flow graphs (Bonchi, Sobociński and Zanasi (2014, 2015), along with computational logics based on linear logic and graph rewriting (on this see Mellies, 2018; and Fong and Spivak, 2018, for additional references).
1. Applied Category Theory
Category theory and topos theory have taken over large swathes in the field of formal or theoretical computation, because categories serve to link together the structures found in algebraic topology, and with the logical connectives and inferences to be found in formal logic, as well as with recursive processes and other operations in computation. The following diagram taken from Baez and Stay (2011), highlights this capability.
John Bell (1988: 236) succinctly explains why it is that category theory also possesses enormous ormous powers of generalization:
A category may be said to bear the same relation to abstract algebra as does the latter to elementary algebra. Elementary algebra results from the replacement of constant quantities (i.e. numbers) by variables, keeping the operations on these quantities fixed. Abstract algebra, in its turn, carries this a stage further by allowing the operations to vary while ensuring that the resulting mathematical structures (groups, rings, etc) remain of a prescribed kind. Finally, category theory allows even the kind of structure to vary: it is concerned with structure in general.
Category theory can also be interpreted as a universal approach to the analysis of process, across various domains including: (a) mathematic practice (theorem proving); (b) physical systems (their evolution and measurement); (c) computing (data types and programs); (d) chemistry (chemicals and reactions); (e) finance (currencies and various transactions); (f) engineering (flows of materials and production).
This way of thinking about processes now serves as a unifying interdisciplinary framework that researchers within business and the social sciences have also taken up. Alternative approaches to those predicated on optimizing behaviour on the part of individual economic agents include the work evolutionary economists and those in the business world who are obliged to work with computational systems designed for the operational management of commercial systems. However, these techniques are also grounded in conceptions of process
Another way of thinking about dynamic processes is in terms of circuit diagrams, which can represent displacement, flow, momentum and effort—phenomenon modelled by the Hamiltonians and Lagrangians of Classical Mechanics. It can be appreciated that key features of economic systems are also amenable to diagrammatic representations of this kind, including asset pricing based on notion of arbitrage, a concept initially formalized by Augustin Cournot in 1838. Cournot’s analysis arbitrage conditions is grounded in Kirchoff voltage law (Ellerman, 1984). The analogs of displacement, flow, momentum and effort are depicted below for a wide range of disciplines.
Applied Category Theory: in the US, contemporary developments in applied category theory (ACT) have been spurred along and supported by a raft of EU, DARPA and ONR Grants. A key resource on ACT is Fong and Spivak’s (2018) downloadable text on compositionality. This publication explores the relationship between wiring diagrams or string diagrams and a wide variety of mathematical and categorical constructs, including as a means for representing symmetric monoidal preorders, signal flow graphs, along with functorial translation between signal flow graphs and matrices and other aspects of functorial semantics, graphical linear algebra, hypergraph categories and operads, applied to electric circuits and network compositionality. Topos theory is introduced to characterise the logic of system behaviour on the basis of indexed sets, glueings, and sheaf conditions for every open cover.
2. Diagrammatic Reasoning
Authors such as Sáenz-Ludlow and Kadunz (2015), Shin (1995), Sowa (2000), and Stjernfelt (2007), who have published research on knowledge representation and diagrammatic approaches to reasoning, tend to work within a philosophical trajectory that stretches from F. W. Schelling and C. S. Peirce, through to E. Husserl and A. N. Whitehead, then on to M. Merleau-Ponty and T. Adorno. Where Kant and Hegel privileged symbolic reasoning over the iconic or diagrammatic, Peirce, Whitehead, and Merleau-Ponty followed the lead of Schelling for whom ‘aesthetics trumps epistemology’! It is, in fact, this shared philosophical allegiance that not only links diagrammatic research to the semantic (or embodied) cognition movement (Stjernfeld himself refers to the embodied cognition theorists Eleanor Rosch, George Lakoff, Mark Johnson, Leonard Talmy, Mark Turner, and Gilles Fauconnier), but also to those researchers who have focused on issues of educational equity in the teaching of mathematics and computer science, including Ethnomathematics and critical work on ‘Orientalism’ specialized to emphasize a purported division between the ‘West and the Rest’ in regard to mathematical and computational thought and practice.
As such, insights from this research carry over to questions of ethnic ‘marginalization’ or ‘positioning’ in the mathematical sciences (see the papers reproduced in Forgasz and Rivera, eds., 2012 and Herbel-Eisenmann et al., 2012). In a nutshell, diagrammatic reasoning is sensitive to both context and positioning and, thus, is closely allied to this critical axis of mathematics education.
The following illustration of the elements and flows associated with diagrammatic forms of reasoning comes from Michael Hoffman’s (2011) explication of the concept first outlined by the American philosopher and logician, Charles Sanders Peirce.
The above Figure depicts three stages in the process of diagrammatic reasoning: (i) constructing a diagram as a consistent representation of key relations; (ii) analysing a problem on the basis of this representation; and (iii) experimenting with the diagram and then observing the results. Consistency is ensured in two ways. First, the researcher or research team develop an ontology specifying elements of the problem and the relations holding between these elements, along with pertinent rules of operation. Second, language is specified in terms of both syntactical and semantic properties. Furthermore, in association with this language, a rigorous axiomatic system is specified, which both constrains and enables any pertinent diagrammatic transformations.
3a. Case-Study One:
A 2010 paper by SAP Professors, Paulheim and Probst reviews an application of STs to the management and coordination of emergency services in the Darmstadt region of Germany. The aim of the following diagram, reproduced from their work, is to highlight the fact that, from a computational perspective, the integrative effort of STs can apply to different organizational levels: that of the common user interface, shared business logics and that of data sources.
In their software engineering application, the upper-level ontology DOLCE is deployed to link a core domain ontology together with a user-interface interaction ontology. In turn, each of these ontologies draws on inputs from an ontology on deployment regulations and various application ontologies. Improved search capabilities across this hierarchy of computational ontologies, are achieved through the adoption of the ONTOBROKER and F-Logic systems.
3b. Case-study Two:
An important contribution to the field of network modelling has come from the DARPA-funded CASCADE Project (Complex Adaptive System Composition and Design Environment), which has invested in long-term research into the “system-of-systems” perspective (see John Baez’s extended discussion of this project on his Azimuth blog). This research has been influenced by Willems’s (2007) behavioural approach to systems, which in turn, is based on the notion that large and complex systems can be built up from simple building blocks.
Baez et al. (2020) introduce ‘network models’ to encode different ways of combining networks both through overlaying one model on top of another and by setting each model side by side. In this way, complex networks can be constructed using simple networks as components. Vertices in the network represent fixed or moving agents, while edges represent communication channels.
The components of their networks are constructed using coloured operads, which include vertices representing entities of various types and edges representing the relationships between these entities. Each network model gives rise to a typed operad with an associated canonical algebra, whose operations represent ways of assembling a more complex network from smaller parts. The various different ways to compose these operations characterize a more general notion of an operation, which must be complemented by ways of permuting the arguments of an operation a process yielding a permutation group of inputs and outputs).
In research conducted under the auspices of the CASCADE Project, Baez, Foley, Moeller, and Pollard (2020) have worked out how to combine two formalisms. First, there are Petri nets, commonoly used as an alternative to process algebras as a foralism for business process management. The vertices in a Petri net represent collections of different types of entities (species) with morphisms between them used to describe processes (transitions) that can be carried out by combining various sets of entities (conceived as resources or inputs into a transition node or process of production) together to make new sets of entities (concived as outputs or vertices are positioned after the relevant transition node). The stocks of each type of entity that is available is enumerated as a ‘marking’ specific to each type or colour together with the set of outputs that can be produced by activated the said transition.
Second, there are network models, which describe processes that a given collection of agents (say, cars, boats, people, planes in a search-and-rescue operation) can carry out. However, in this kind of network, while each type of object or vertex can move around within a delineated space, they are not allowed to turn into other types of agent or object.
In these networks, morphisms are functors (generalised functions) which describe everything that can be done with a specific collection of agents. The following Figure depicts this kind of operational network in an informal manner, where icons represent helicopters, boats, victims floating in the sea, and transmission towers with communication thresholds.
By combining Petri nets with an underlying network model resource-using operations can be defined. For example, a helicopter may be able to drop supplies gathered from different depots and packaged into pallets, onto the deck of a sinking ship or to a remote village cut off by an earthquake or flood.
The formal mechanism for combining a network model with a Petri net relies on treating different type of entities as catalysts, in the sense that the relevant species are neither increased nor decreased in number by any given transition. The derived category is symmetric monoidal and possesses a tensor product (representing processes for each catalyst that occur side-by-side), a coproduct (or disjoint union of amounts of each catalyst present), and within each subcategory of a particular catalyst, an internal tensor product describes how one process can follow another while reusing the pertinent catalysts.
The following diagram taken from Baez et al. (2020), illustrates the overlaying process which enables more complex networks to be constructed from simpler components. The use of the Grothendieck construction in this research ensures that when two or more diagrams are overlayed there will be no ‘double-counting’ of edges and vertices. When components are ‘tensored’ each of the relevant blocks would be juxtaposed “side-by-side”.
Each network model is characterized by a “plug-and-play” feature based on an algebraic component called an operad. The operad serves as the construct for a canonical algebra, whose operations are ways of assembling a network of the given kind from smaller parts. This canonical algebra, in turn, accommodates a set of types, a set of operations, ways to compose these operations to arrive at more general operations, and ways to permute an operation’s arguments (i.e. via a permutation group), along with a set of relevant distance constraints (e.g. pertinent communication thresholds for each type of entity) .
One of Baez’s co-authors, John Foley, works for Metron, Inc., VA, a company which specializes in applying the advanced mathematics of network models to such phenomena as “search-and-rescue” operations, the detection of network incursions, and sports analytics. Their 2017 paper mentions a number of formalisms that have relevance to “search-and-rescue” applications, especially the ability to distinguish between different communication channels (different radio frequencies and capacities) and vertices (e.g. planes, boats, walkers, individuals in need of rescue etc.) and the capacity to impose distance constraints over those agents who may fall outside the reach of communication networks.
In related research paper, Schultz, Spivak, Vasilakopoulou, Wisnesky (2016) argue thay dynamical systems can be gainfully thought of as ‘machines’ with inputs and outputs, carrying some sort of signal that occurs through some notion of time”. Special cases of this general approach include discrete, continuous, and hybrid dynamical systems. The authors deploy lax functors out of monoidal categories, which provide them with a language of compositionality. As with Baez and his co-authors, Schultz et al. (2016) draw on an operadic construct so as to understand systems that result from an “arbitrary interconnection of component subsystems”. They also draw on the mathematics of sheaf theory, to flexibly capture the crucial notion of time. The resulting sheaf-theoretic perspective relates continuous- and discrete-time systems together via functors (a kind of generalized ‘function of functions’, which preserves structure). Their approach can also account for synchronized continuous time, in which each moment is assigned a specific phase within the unit interval.
4. Related Developments in Software Engineering
This section of the Chapter examines contemporary advances in software engineering that have implications for ‘system-of-sytems’ approaches to semantic technology. The work of the Statebox group at the University of Oxford and that of Evan Patterson, from Stanford University, who is also affiliated with researchers from the MIT company, Categorical Informatics, will be discussed to indicate where these new developments are likely to be moving in the near future. This will be supplemented by an informal overview of some recent innovations in functional programming, which have been informed by the notion of a derivative applied to an algorithmic step. These initiatives have the potential to transform software for machine-learning and the optimization of networks
The Statebox team based at Oxford University have developed a language for software engineering that uses diagrammatic representations of generalized Petri nets. In this context, transitions in the net are morphisms between data-flow objects represent terminating functional programming algorithms. In Statebox (integer and semi-integer) Petri nets are constructed with both positive and negative tokens to account for contracting. Negative tokens represent borrowing while positive tokens represent lending and, likewise, the taking of short and long positions in asset markets. This allows for the representation of smart contracts, conceived as separable nets. Nets are also endowed with interfaces that allow for channelled communications through user-defined addresses. Furthermore, guarded and timed nets, with side-effects (which are mapped to standard nets using the Grothendieck construction), offer greater expressive power in regard to the conditional behaviour affecting transitions (The Statebox Team, 2018).
Patterson (2017) begins his paper with a discussion of description logics (e.g. OWL, WC3), which he interprets as calculi for knowledge representation (KR). These logics, which are the actual substrates responsible for the World-Wide-Web (WWW), lie somewhere between propositional logic and first-order predicate logic possessing the capability to express the (∃,∧,T,=) fragment of first-order logic. Patterson highlights the trade-off that must be made between computational tractability and expressivity before introducing a third knowledge representation formalism that interpolates between description logic and ontology logs (see Spivak and Kent, 2012, for an the extensive description of ologs, which express key constructs from category theory, such as products and coproducts, pullbacks and pushforwards, and representations of recursive operations using diagrams labelled with concepts drawn from everyday conversation). Patterson (2017) calls this construct the relational ontology log, or relational olog, because it is based on, Rel, the category of sets and relations and, as such, draws on relational algebra, which is the (∃,∧, , T,⊥,=) fragment of first-order logic. He calls Spivak and Kent’s, 2012, version, a functional olog to avoid any confusion, because these are solely based on Set, the category of sets and functions. Relational ologs achieve their expressivity through categorical limits and colimits (products, pullbacks, pushforwards, and so forth
The advantages of Patterson’s framework are that functors allow instance data to be associated with a computational ontology in a mathematically precise way, by interpreting it as a relational or graph database, Boolean matrix, or category of linear relations. Moreover, relational ologs are, by default, typed, which he suggests can mitigate the maintainability challenges posed by the open world semantics of description logic.
String diagrams (often labelled Markov-Penrose diagramsby those working in the field of brain science imaging) are routinely deployed by data-scientists used to represent the structure of deep-learning convolution neural networks. However, string diagrams can also serve as a tool for representing the computational aspects of machine-learning.
For example, influenced by the program idioms of machine-learning, Ghica and Muroya (2017) have developed what they choose to call a ‘Dynamic Geometry of Interaction Machine’, which can be defined as a state transition system operating whose transitions not only account for ‘token passing’ but also for ‘graph rewriting’ (where the latter can be construed as a graph-based approach to the proving of mathematical hypotheses and theories). Their proposes system is supported by diagrammatic implementation based on the proof structures of the multiplicative and exponential fragment of linear logic (MELL). In Muroya, Cheung and Ghica (2017), this logical approach is complemented by a sound call-by-value lambda calculus inspired, in turn by Peircean notions of abductive inference. The resulting bimodal programming model operates in both: (a) direct mode, with new inputs provided, new outputs obtained; and, (b) learning mode, with special inputs applied for which outputs are known; to achieve optimal tuning of parameters to ensure desired outputs approach actual outputs. The authors contend that their holistic approach is superior to that of the TensorFlow software package developed for machine-learning, which they describe as a ‘shallow embedding’ of a domain specific language (DSL) into PYTHON” rather than a ‘stand-alone’ programming language.
Adopting a somewhat different approach, Cruttwell, Gallagher and MacAdam (2019) extend Plotkin’s differential programming framework, which is itself a generalization of differential neural computers, where arbitrary programs with control structures encode smooth functions also represented as programs. Within this generalized domain, the derivative can be directly applied to programs or to algorithmic steps and, furthermore, can be rendered entirely congruent with categorical approaches to Riemannian and Differential geometry such as Lawvere’s Synthetic Differential Geometry.
Cruttwell and his colleagues go on to observe that, when working in a simple neural network, back-propagation takes the derivative of the error function, then uses the chain rule to push errors backwards. They point out that, for convolution neural networks, the necessary procedure is less straightforward due to the presence of looping constructs.
In this context, the authors further note that attempts to work with the usual ‘if-then-else’ and ‘while’ commands can also be problematic. To overcome these problems associated with recursion, they deploy what have been called ‘join restriction tangent categories’, which express the requisite domain of definition and detect and achieve disjointness of domains, while expressing iteration using the join of disjoint domains (i.e. in technical terms, this is the trace of a coproduct in the idempotent splitting). The final mathematical construct they arrive at, is that of a differential join restriction category along with the associated join restriction functor which, they suggest, admits a coherent interpretation of differential programming.
It should be stressed that each of these category-theoretic initiatives to formalize the differential of an algorithmic step will become important in future efforts to develop improved, yet diagrammatically-based forms of software for machine learning that have greater capability and efficiency than existing software suites. The fact that both differential and integral categories can be provided with a coherent string diagram formalism (Lemay, 2017) provides a link back to the earlier discussion about the role of diagrammatic reasoning in semantic technologies.
It is clear that techniques of this kind could also be applied to a wide variety of network models (e.g. for the centralized and decentralized control of hybrid cyber-physical systems), where optimization routines may be required (including those for effective disaster management).
In conclusion, the innovations in software engineering described above, have obvious implications for those attempting to develop new semantic technologies for the effective management of emergency services and search-and-rescue operations in the aftermath of a major disaster. Hopefully, the material surveyed in this Chapter should serve to highlight the advantages of a category-theoretic approach to the issue at hand, along with the specific benefits of adopting an approach that is grounded in the pedagogical, computational, and formal representational power of string-diagrams, especially within a networked computational environment charactrised by Big Data, parallel processing, hybridity, and some degree of decentralized control.
While a Chapter of this kind cannot go into too much detail about the formalisms that have been discussed, it is to be hoped that enough pertinent references have been provided for those who would like to find out more about the mathematical detail. Of course, it is not always necessary to be a computer programmer both to understand and to effectively deploy powerful suites of purpose-built software. It is also to be hoped that diagrammatic reasoning may assist the interested reader in acquiring a deeper understanding of the requisite mathematical techniques.
Author: Professor Dr. James Juniper – Conjoint Academic, University of Newcastle; PhD in Economics, University of Adelaide
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