Semantics of computation



Bicompletions of distance matrices -To Samson Abramsky on the occasion of his 60th birthday-

In the practice of information extraction, the input data are usually arranged into pattern matrices, and analyzed by the methods of linear algebra and statistics, such as principal component analysis. In some applications, the tacit assumptions of these methods lead to wrong results. The usual reason is that the matrix composition of linear algebra presents information as flowing in waves, whereas it sometimes flows in particles, which seek the shortest paths. This wave-particle duality in computation and information processing has been originally observed by Abramsky. In this paper we pursue a particle view of information, formalized in distance spaces, which generalize metric spaces, but are slightly less general than Lawvere's generalized metric spaces. In this framework, the task of extracting the 'principal components' from a given matrix of data boils down to a bicompletion, in the sense of enriched category theory. We describe the bicompletion construction for distance matrices. The practical goal that motivates this research is to develop a method to estimate the hardness of attack constructions in security.

Monoidal computer I: Basic computability by string diagrams

We present a new model of computation, described in terms of monoidal categories. It conforms the Church-Turing Thesis, and captures the same computable functions as the standard models. It provides a succinct categorical interface to most of them, free of their diverse implementation details, using the ideas and structures that in the meantime emerged from research in semantics of computation and programming. The salient feature of the language of monoidal categories is that it is supported by a sound and complete graphical formalism, string diagrams, which provide a concrete and intuitive interface for abstract reasoning about computation. The original motivation and the ultimate goal of this effort is to provide a convenient high level programming language for a theory of computational resources, such as one-way functions, and trapdoor functions, by adopting the methods for hiding the low level implementation details that emerged from practice. Only a modest first step towards this ambitious goal is presented.


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, test- ing 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 specifing 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.


Towards semantics of self-adaptive software

When people perform computations, they routinely monitor their results, and try to adapt and improve their algorithms when a need arises. The idea of self-adaptive software is to implement this common facility of human mind within the framework of the standard logical methods of software engineering. The ubiquitous practice of testing, debugging and improving programs at the design time should be automated, and established as a continuing run time routine.

Technically, the task thus requires combining functionalities of automated software development tools and of runtime environments. Such combinations lead not just to challenging engineering problems, but also to novel theoretical questions. Formal methods are needed, and the standard techniques do not suffice.

As a first contribution in this direction, we present a basic mathematical framework suitable for describing self-adaptive software at a high level of semantical abstraction. A static view leads to a structure akin to the Chu construction. An dynamic view is given by a coalgebraic presentation of adaptive transducers.

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).


Semantics of first order parametric specifications

Parametricity is one of the most effective ways to achieve compositionality and reuse in software development. Parametric specifications have been thoroughly analyzed in the algebraic setting and are by now a standard part of most software development toolkits. However, an effort towards classifying, specifying and refining algorithmic theories, rather than mere datatypes, quickly leads beyond the realm of algebra, and often to full first order theories. We extend standard semantics of parametric specifications to this more general setting.

The familiar semantic characterization of parametricity in the algebraic case is expressed in terms of the free functor, i.e. using the initial models. In the general case, the initial models may not exist, and the free functor is not available. Various syntactic, semantic and abstract definitions of parametricity have been offered, but their exact relationships are often unclear. Using the methods of categorical model theory, we establish the equivalence of two well known, yet so far unrelated definitions of parametricity, one syntactic, one semantic. Besides providing support for both underlying views, and a way for aligning the systems based on each of them, the offered general analysis and its formalism open up several avenues for future research and applications.


Specifying Interaction Categories --- with Samson Abramsky

We analyse two complementary methods for obtaining categorical models of process calculi. They allow adding new features to the captured notions of process and of type respectively. By alternating these two methods, all the familiar examples, as well as some new interaction categories, can be derived from basic monoidal categories.

Using the proposed constructions, interaction categories can be built and analysed without fixing a set of axioms for them. They are thus approached via specifications, just like algebras are approached via equations and relations, independantly of the intrinsic characterisation of varieties.

Categorical logic of names and abstraction in action calculi

Milner's action calculus implements abstraction in monoidal categories, so that the familiar lambda-calculi, together with the pi-calculus and Petri nets can be subsumed. Variables are generalised to names: only a restricted form of substitution is allowed.

In the present paper, the well-known categorical semantics of the lambda-calculus is generalised to the action calculus. A suitable functional completeness theorem for symmetric monoidal categories is proved: we determine the conditions under which the abstraction is definable. Algebraically, the distinction between the variables and the names boils down to the distinction between the transcendental and the algebraic elements. The former lead to the polynomial extensions, like e.g. the ring Z[x], the latter to the algebraic extensions like Z[i].

Building upon the work of P. Gardner, we introduce action categories, and show that they are related to the static action calculus exacly as cartesian closed categories are related to the lambda-calculus. Natural examples of this structure arise from allegories and cartesian bicategories. On the other hand, the free algebras for any commutative Moggi monad form an action category. The general correspondence of action calculi and Moggi monads will be worked out in a sequel to this work.


Convenient categories of processes and simulations II: Asynchronous cases

In the preceding paper, we have argued for a representation of processes taking into account computationally relevant morphisms. It has been shown that the category of synchronous processes, i.e. modulo strong bisimilarity, is isomorphic with the category of labelled irredundant trees.

In the present paper, we extend this representation to asynchronous processes, namely modulo the weak and the branching bisimulations and congruences. They are shown to correspond to interesting subcategories of the category of labelled irredundant trees. The form of the representatives in the case of the branching bisimilarity suggests possible connections with game theory. An abstract construction of a category of processes in a general setting is presented in the appendix.


Categorical logic of concurrency and interaction I: Synchronous processes

This is a report on a mathematician's effort to understand some concurrency theory. The starting point is a logical interpretation of Nielsen and Winskel's account of the basic models of concurrency. Upon the obtained logical structures, we build a calculus of relations which yields, when cut down by bisimulations, Abramsky's interaction category of synchronous processes. It seems that all interaction categories arise in this way. The obtained presentation uncovers some of their logical contents and perhaps sheds some more light on the original idea of processes as relations extended in time.

The sequel of this paper will address the issues of asynchronicity, preemption, noninterleaving and linear logic in the same setting.

Convenient categories of processes and simulations I: Synchronous case

Deep categorical analyses of various aspects of concurrency have been developed, but a uniform categorical treatment of the very first concepts seems to be hindered by the fact that the existing representations of processes as bisimilarity classes do not provide an account of computational morphisms.

In the present paper, we describe a category of processes modulo strong bisimulations, with the bisimilarity preserving simulations as morphisms, and show that it is equivalent to the category of labelled irredundant trees and the label preserving tree morphisms. The developed representation turns out to be applicable to weaker notions of process as well. They will be studied in the sequel.