Retracing some paths in categorical semantics: From process-propositions-as-types to categorified reals and computers

The logical parallelism of propositional connectives and type constructors extends beyond the static realm of predicates, to the dynamic realm of processes. Understanding the logical parallelism of process propositions and dynamic types was one of the central problems of the semantics of computation, albeit not always clear or explicit. It sprung into clarity through the early work of Samson Abramsky, where the central ideas of denotational semantics and process calculus were brought together and analyzed by categorical tools, e.g. in the structure of interaction categories. While some logical structures borne of dynamics of computation immediately started to emerge, others had to wait, be it because the underlying logical principles (mainly those arising from coinduction) were not yet sufficiently well-understood, or simply because the research community was more interested in other semantical tasks. Looking back, it seems that the process logic uncovered by those early semantical efforts might still be starting to emerge and that the vast field of results that have been obtained in the meantime might be a valley on a tip of an iceberg.
In the present paper, I try to provide a logical overview of the gamut of interaction categories and to distinguish those that model computation from those that capture processes in general. The main coinductive constructions turn out to be of this latter kind, as illustrated towards the end of the paper by a compact category of all real numbers as processes, computable and uncomputable, with polarized bisimulations as morphisms. The addition of the reals arises as the biproduct, real vector spaces are the enriched bicompletions, and linear algebra arises from the enriched kan extensions. At the final step, I sketch a structure that characterizes the computable fragment of categorical semantics.


Quotients in monadic programming: Projective algebras are equivalent to coalgebras

with Peter-Michael Seidel

In monadic programming, datatypes are presented as free algebras, generated by data values, and by the algebraic operations and equations capturing some computational effects. These algebras are free in the sense that they satisfy just the equations imposed by their algebraic theory, and remain free of any additional equations. The consequence is that they do not admit quotient types. This is, of course, often inconvenient. Whenever a computation involves data with multiple representatives, and they need to be identified according to some equations that are not satisfied by all data, the monadic programmer has to leave the universe of free algebras, and resort to explicit destructors. We characterize the situation when these destructors are preserved under all operations, and the resulting quotients of free algebras are also their subalgebras. Such quotients are called *projective*. Although popular in universal algebra, projective algebras did not attract much attention in the monadic setting, where they turn out to have a surprising avatar: for any given monad, a suitable category of projective algebras is equivalent with the category of coalgebras for the comonad induced by any monad resolution. For a monadic programmer, this equivalence provides a convenient way to implement polymorphic quotients as coalgebras. The dual correspondence of injective coalgebras and all algebras leads to a different family of quotient types, which seems to have a different family of applications. Both equivalences also entail several general corollaries concerning monadicity and comonadicity.

Monoidal computer III: A coalgebraic view of computability and complexity

with Muzamil Yahia

Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a much needed high level language for theory of computation, flexible enough to allow abstracting away the low level implementation details when they are irrelevant, or taking them into account when they are genuinely needed. A salient feature of the approach through monoidal categories is the formal graphical language of string diagrams, which supports visual reasoning about programs and computations.

In the present paper, we provide a coalgebraic characterization of monoidal computer. It turns out that the availability of interpreters and specializers, that make a monoidal category into a monoidal computer, is equivalent with the existence of a *universal state space*, that carries a weakly final state machine for any pair of input and output types. Being able to program state machines in monoidal computers allows us to represent Turing machines, to capture their execution, count their steps, as well as, e.g., the memory cells that they use. The coalgebraic view of monoidal computer thus provides a convenient diagrammatic language for studying computability and complexity.


Testing randomness by matching pennies

In the game of Matching Pennies, Alice and Bob each hold a penny, and at every tick of the clock they simultaneously display the head or the tail sides of their coins. If they both display the same side, then Alice wins Bob’s penny; if they display different sides, then Bob wins Alice’s penny. To avoid giving the opponent a chance to win, both players seem to have nothing else to do but to randomly play heads and tails with equal frequencies. However, while not losing in this game is easy, not missing an opportunity to win is not. Randomizing your own moves can be made easy. Recognizing when the opponent’s moves are not random can be arbitrarily hard.

The notion of randomness is central in game theory, but it is usually taken for granted. The notion of outsmarting is not central in game theory, but it is central in the practice of gaming. We pursue the idea that these two notions can be usefully viewed as two sides of the same coin.


Smooth coalgebra: testing vector analysis

with Bertfried Fauser

Processes are often viewed as coalgebras, with the structure maps specifying the state transitions. In the simplest case, the state spaces are discrete, and the structure map simply takes each state to the next states. But the coalgebraic view is also quite effective for studying processes over structured state spaces, e.g. measurable, or continuous. In the present paper we consider coalgebras over manifolds. This means that the captured processes evolve over state spaces that are not just continuous, but also locally homeomorphic to Banach spaces, and thus carry a differential structure. Both dynamical systems and differential forms arise as coalgebras over such state spaces, for two different endofunctors over manifolds. A duality induced by these two endofunctors provides a formal underpinning for the informal geometric intuitions linking differential forms and dynamical systems in the various practical applications, e.g. in physics. This joint functorial reconstruction of tangent bundles and cotangent bundles uncovers the universal properties and a high level view of these fundamental structures, which are implemented rather intricately in their standard form. The succinct coalgebraic presentation provides unexpected insights even about the situations as familiar as Newton’s laws.


Tracing the Man in the Middle in Monoidal Categories

Man-in-the-Middle (MM) is not only a ubiquitous attack pattern in security, but also an important paradigm of network computation and economics. Recognizing ongoing MM-attacks is an important security task; modeling MM-interactions is an interesting task for semantics of computation. Traced monoidal categories are a natural framework for MM-modelling, as the trace structure provides a tool to hide what happens in the middle. An effective analysis of what has been traced out seems to require an additional property of traces, called normality. We describe a modest model of network computation, based on partially ordered multisets (pomsets), where basic network interactions arise from the monoidal trace structure, and a normal trace structure arises from an iterative, i.e. coalgebraic structure over terms and messages used in computation and communication. The correspondence is established using a convenient monadic description of normally traced monoidal categories.


A semantical approach to equilibria and rationality

Game theoretic equilibria are mathematical expressions of rationality. Rational agents are used to model not only humans and their software representatives, but also organisms, populations, species and genes, interacting with each other and with the environment. Rational behaviors are achieved not only through conscious reasoning, but also through spontaneous stabilization at equilibrium points.

Formal theories of rationality are usually guided by informal intuitions, which are acquired by observing some concrete economic, biological, or network processes. Treating such processes as instances of computation, we reconstruct and refine some basic notions of equilibrium and rationality from the some basic structures of computation.

It is, of course, well known that equilibria arise as fixed points; the point is that semantics of computation of fixed points seems to be providing novel methods, algebraic and coalgebraic, for reasoning about them.


Labelled Markov Processes as generalised stochastic relations

with Michael Mislove and James Worrell

Labelled Markov processes (LMPs) are labelled transition systems in which each transition has an associated probability. In this paper we present a universal LMP as the spectrum of a commutative C*-algebra consisting of formal linear combinations of labelled trees. This yields a simple trace-tree semantics for LMPs that is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with distinguished entry and exit points as stateful stochastic relations. This allows us to define a category LMP, with measurable spaces as objects and LMPs as morphisms. Our main result in this context is to provide a predicate-transformer duality for LMP that generalises Kozen’s duality for the category SRel of stochastic relations.


Testing semantics: Connecting processes and process logics

with Michael Mislove and James Worrell

We propose a methodology based on testing as a framework to capture the interactions of a machine represented in a denotational model and the data it manipulates. Using a connection that models ma- chines on the one hand, and the data they manipulate on the other, testing is used to capture the interactions of each with the objects on the other side: just as the data that are input into a machine can be viewed as tests that the machine can be subjected to, the machine can be viewed as a test that can be used to distinguish data. This approach is based on generalizing from duality theories that now are common in semantics to logical connections, which are simply contravariant adjunctions. In the process, it accomplishes much more than simply moving from one side of a duality to the other; it faithfully represents the interactions that embody what is happening as the computation proceeds.

Our basic philosophy is that tests can be used as a basis for modeling interactions, as well as processes and the data on which they operate. In more abstract terms, tests can be viewed as formulas of process logics, and testing semantics connects processes and process logics, and assigns computational meanings to both.


Duality for Labelled Markov Processes

with Michael Mislove, Joel Ouaknine and James Worrell

Labelled Markov processes (LMPs) are probabilistic labelled transition systems. In this paper we present a `universal’ LMP as the Stone-Gelfand-Naimark dual of a C*-algebra consisting of formal linear combinations of labelled trees. We characterize the state space of the universal LMP as the set of homomorphims from an ordered commutative monoid of labelled trees into the multiplicative unit interval. This yields a simple semantics for LMPs which is fully abstract with respect to probabilistic bisimilarity. We also consider LMPs with entry points and exit points in the framework of Elgot’s iterative theories. We define an iterative theory of LMPs by specifying its categorical dual: a category of commutative rings consisting of C*-algebras of trees and `shapely maps’ between them. We find that the basic operations for composing LMPs have simple definitions in the dual category.


The continuum as a final coalgebra

with Vaughan Pratt

We define the continuum in terms of the final coalgebras of simple functors: either the functor N×X , mapping the sets X into their products with the set of natural numbers, or the functor 1+N×X. This makes an attractive analogy with the definition of N itself as the initial algebra of the functor 1+X, which adds a singleton to each set X. We furthermore characterize Baire space and Cantor space in terms of these final coalgebras.

The presented coalgebraic approach leads to coinductive definitions of these infinitary concepts both in the category of sets, and in the category of posets. We analyze the structural differences that arise, and conclude with some paradoxical discrepancies between continuity and constructiveness in this setting.


Categories of processes enriched in final coalgebras

with Sava Krstic and John Launchbury

Simulations between processes can be understood in terms of coalgebra homomorphisms, with homomorphisms to the final coalgebra exactly identifying bisimilar processes. The elements of the final coalgebra are thus natural representatives of bisimilarity classes, and a denotational semantics of processes can be developed in a final-coalgebra-enriched category where arrows are processes, canonically represented.

In the present paper, we describe a general framework for building final-coalgebra-enriched categories. Every such category is constructed from a multivariant functor representing a notion of process, much like Moggi’s categories of computations arising from monads as notions of computation. The “notion of process” functors are intended to capture different flavors of processes as dynamically extended computations. These functors may involve a computational (co)monad, so that a process category in many cases contains an associated computational category as a retract. We further discuss categories of resumptions and of hyperfunctions, which are the main examples of prcess categories. Very informally, resumptions can be understood as computations extended in time, whereas hypercomputations are extended in space.


Process fusion

The technique of build fusion, also known as deforestation, removes intermediate results in a composition involving the “build” of an initial (inductive, finite) data structure, followed by its consumption. Here we show that it is analogously possible to do process fusion, removing intermediate final (coinductive, potentially infinite) data passing between a producer and a consumer.

The key observation leading to our results is the fact that the Curry-Howard isomorphism, relating types to propositions, programs to proofs, and sequential composition to cut, extends to the correspondence of fusion to cut elimination. this simple idea gives us logical interpretations of the basic methods of generic and transformational programming. In the present paper, we provide a logical analysis of the general form of build fusion over the inductive data types, regular or nested. The analysis is based on a novel logical interpretation of parametricity in terms of the paranatural transformations, introduced in the paper. We extend it to cover process fusion on coinductive data types.

The results obtained are truly generic, in the sense of applying to all coinductive (final) data types, including nested ones, and allow a far wider range of optimizations than previously possible. By the standard embedding of initial into final data types, it also applies to arbitrary intial-final mixtures (e.g., infinitely unfolding trees of finite lists).


On coalgebra of real numbers

with Vaughan Pratt

We define the continuum up to order isomorphism (and hence homeomorphism) as the final coalgebra of the functor capturing the ordinal product with omega. This makes an attractive analogy with the definition of the ordinal omega itself as the initial algebra of the functor prepending the unity, with both definitions made in the category of posets. The variants of these functors yield respectively Cantor space (surplus rationals), Baire space (no rationals), and again the continuum as their final coalgebras.


Calculus in coinductive form

with Martin Escardo

Coinduction is often seen as a way of implementing infinite objects. Since real numbers are typical infinite objects, it may not come as a surprise that calculus, when presented in a suitable way, is permeated by coinductive reasoning. What is surprising is that mathematical techniques, recently developed in the context of computer science, seem to be shedding a new light on some basic methods of calculus.

We introduce a coinductive formalization of elementary calculus that can be used as a tool for symbolic computation, and geared towards computer algebra and theorem proving. So far, we have covered ordinary differential and difference equations, Taylor series, Laplace transform and the basics of operator calculus.

Guarded induction on final coalgebras

We make an initial step towards categorical semantics of guarded induction. While ordinary induction is usually modelled in terms of least fixpoints and initial algebras, guarded induction is based on unique fixpoints of certain operations, called guarded, on finalcoalgebras. So far, such operations were treated syntactically. We analyse them categorically. Guarded induction appears as couched in coinduction.

The applications of the presented categorical analysis span across the gamut of the applications of coinduction, from modelling of computation to solving differential equations. A subsequent paper will provide an account of some domain theoretical aspects, which are presently left implicit.