The continuity of monadic stream functions

Brouwer’s continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Mart́ín Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment)

The continuity of monadic stream functions

Brouwer’s continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Mart́ín Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment)

From coinductive proofs to exact real arithmetic: theory and applications

Based on a new coinductive characterization of continuous functions we extract certified programs for exact real number computation from constructive proofs. The extracted programs construct and combine exact real number algorithms with respect to the binary signed digit representation of real numbers. The data type corresponding to the coinductive definition of continuous functions consists of finitely branching non-wellfounded trees describing when the algorithm writes and reads digits. We discuss several examples including the extraction of programs for polynomials up to degree two and the definite integral of continuous maps

Proofs of randomized algorithms in Coq

International audienceRandomized algorithms are widely used for finding efficiently approximated solutions to complex problems, for instance primality testing and for obtaining good average behavior. Proving properties of such algorithms requires subtle reasoning both on algorithmic and probabilistic aspects of programs. Thus, providing tools for the mechanization of reasoning is an important issue. This paper presents a new method for proving properties of randomized algorithms in a proof assistant based on higher-order logic. It is based on the monadic interpretation of randomized programs as probabilistic distributions. It does not require the definition of an operational semantics for the language nor the development of a complex formalization of measure theory. Instead it uses functional and algebraic properties of unit interval. Using this model, we show the validity of general rules for estimating the probability for a randomized algorithm to satisfy specified properties. This approach addresses only discrete distributions and gives rules for analysing general recursive functions. We apply this theory to the formal proof of a program implementing a Bernoulli distribution from a coin flip and to the (partial) termination of several programs. All the theories and results presented in this paper have been fully formalized and proved in the Coq proof assistant

London in space and time: Peter Ackroyd and Will Self

Copyright @ 2013 the author. This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 Unported License.This paper explores the treatment of London by two authors who are profoundly influenced by the concept of the power of place and the nature of urban space. The works of Peter Ackroyd, whose writings embody, according to Onega (1997, p. 208) “[a] yearning for mythical closure” where London is “a mystic centre of power” – spiritual, transhistorical and cultural – are considered alongside those of Will Self, who explores the city’s psychogeography as primarily a political, economic and cultural artefact. The paper draws on original interviews undertaken by the author with Ackroyd and Self. Both authors’ works are available for literary study during the 16-19 phase in the UK, and this paper explores how personal delineations of the urban environment are shaped by space and language. It goes on to consider how authors’ and students’ personal understandings of space and place can be used as pedagogical and theoretical lenses to “read” the city in the 16-19 literature classroom

Towards a Convenient Category of Topological Domains

We propose a category of topological spaces that promises to be convenient for the purposes of domain theory as a mathematical theory for modelling computation. Our notion of convenience presupposes the usual properties of domain theory, e.g. modelling the basic type constructors, fixed points, recursive types, etc. In addition, we seek to model parametric polymorphism, and also to provide a flexible toolkit for modelling computational effects as free algebras for algebraic theories. Our convenient category is obtained as an application of recent work on the remarkable closure conditions of the category of quotients of countably-based topological spaces. Its convenience is a consequence of a connection with realizability models

A Convenient Category of Domains

We motivate and define a category of "topological domains", whose objects are certain topological spaces, generalising the usual $omega$-continuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains combining such properties

Husserl, the Monad and Immortality

In an Appendix to his Analyses Concerning Passive and Active Synthesis dating from the early 1920s, Husserl makes the startling assertion that, unlike the mundane ego, the transcendental ego is immortal. The present paper argues that this claim is an ineluctable consequence of Husserl’s relentless pursuit of the ever deeper levels of time-constituting consciousness and, at the same time, of his increasing reliance on Leibniz’s model of monads as the true unifiers of all things, including minds. There are many structural and substantive parallels between Leibniz’s monadic scheme and Husserl’s later views on the primal ego, and these points of convergence are laid out step by step in this paper. For both theorists, the monad is a self-contained system of being, one “without windows”; a monad’s experiences unfold in harmonious concatenations; a monad is a mirror of its proximate environs and comprises multiple perspectives; the unconscious is a repository of potential activation; and, most importantly of all, a monad knows no birth and death and hence is immortal. In his very last years, Husserl proposed a third ego level, below (or beyond) the mundane ego and transcendental ego - the primal ego. It is neither psychical nor physical; it permits the transcendental ego to carry out its constitutive activities, including the mundane ego’s birth and death in time; it is always in a process of becoming, and so it can never be in a state of only “having-been”, that is, dead: and hence the primal ego’s enduring cannot itself ever come to an end. Indo-Pacific Journal of Phenomenology, Volume 7, Edition 2 September 200