Martín Hötzel Escardó

PCF extended with real numbers

We extend the programming language PCF with a type for (total and partial) real numbers. By a partial real number we mean an element of a cpo of intervals, whose subspace of maximal elements (single-point intervals) is homeomorphic to the Euclidean real line. We show that partial real numbers can be considered as “continuous words”. Concatenation of continuous words corresponds to refinement of partial information. The usual basic operations cons, head and tail used to explicitly or recursively define functions on words generalize to partial real numbers. We use this fact to give an operational semantics to the above referred extension of PCF. We prove that the operational semantics is sound and complete with respect to the denotational semantics. A program of real number type evaluates to a head-normal form iff its value is different from b ; if its value is different from b then it successively evaluates to head-normal forms giving better and better partial results converging to its value. — Authors Abstract

Synthetic Topology

Synthetic topology as conceived in this monograph has three fundamental aspects: 1. to explain what has been done in classical topology in conceptual terms, 2. to provide one-line, enlightening proofs of the theorems that constitute the core of the theory, and 3. to smoothly export topological concepts and theorems to unintended situations, keeping the synthetic proofs unmodified. The unintended situation that we focus on is the theory of computation, in particular regarding programming languages from both operational (Part I) and denotational (Part III) points of view, with emphasis on sequential computation. We are aware of other applications of synthetic topology, e.g. to locales, convergence spaces, sequential spaces, equilogical spaces, and some sheaf and realizability toposes, but this will be reported elsewhere. Aspects 1 and 2 are the subject of Part II. However, it turns out that it is possible to tackle aspect 3 without previous reference to 1 or 2. In fact, we start by developing synthetic topology of programming-language data types in Part I, without assuming any background in classical topology and without introducing any. Part III combines ideas from Parts I and II, developing non-trivial computational applications. The main new result is a computational version of the Tychonoff theorem. We also review previously known applications and explain how topology and semantics interact in program-correctness proofs. Although computers are finite, infinity shows up in a number of important situations in the theory of computation, e.g. infinity in syntax : loops, recursion; infinity in time: non-terminating computations; infinity of data: stream computation and higher-type computation; infinity in precision: real-number computation; infinity through abstraction: probabilistic descriptions. The first few chapters of Part I explore how the fundamental topological notions of continuous map, open set, closed set, compact set, Hausdorff space, and discrete space reconcile the finite character of computers with the infinite nature of the entities one wishes to calculate with. One of the main contributions of this monograph is to explain the computational nature of the the notion of compactness. Roughly speaking, a set is compact if and only if, given any semidecidable property, one can semidecide whether it holds for all elements of the set in finite time. Surprisingly, there are infinite computationally compact sets, for example that of infinite streams of binary digits.

Properly injective spaces and function spaces

Abstract Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j : X → Y , Dana Scott asked whether the higher-order function [ X → D ] → [ Y → D ] which takes a continuous map f : X → D to its greatest continuous extension t f: Y → D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial one-point space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression “proper subspace embedding”, we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, these are known to be the continuous meet-semilattices. In the full subcategory of stably locally compact spaces these are again the continuous lattices. The above characterization of the injective spaces over finitary embeddings is an instance of a general result on injective objects in poset-enriched categories with the structure of a KZ-monad established in this paper, which we also apply to various full subcategories closed under the upper power space construction and to the upper and lower power locale monads. The above results also hold for the injective spaces over dense subspace embeddings (continuous Scott domains). Moreover, we show that every sober space has a smallest finitary dense sober subspace (its support ). The support always contains the subspace of maximal points, and in the stably locally compact case (which includes densely injective spaces) it is the subspace of maximal points iff that subspace is compact.

Calculus in coinductive form

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 parts of ordinary differential and difference equations, Taylor series, Laplace transform and the basics of the operator calculus.

Semantics of exact real arithmetic

In this paper, we incorporate a representation of the non-negative extended real numbers based on the composition of linear fractional transformations with non-negative integer coefficients into the Programming Language for Computable Functions (PCF) with products. We present two models for the extended language and show that they are computationally adequate with respect to the operational semantics.

Selection functions, bar recursion and backward induction

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

Integration in Real PCF

Real PCF is an extension of the programming language PCF with a data type for real numbers. Although a Real PCF definable real number cannot be computed in finitely many steps, it is possible to compute an arbitrarily small rational interval containing the real number in a sufficiently large number of steps. Based on a domain-theoretic approach to integration, we show how to define integration in Real PCF. We propose two approaches to integration in Real PCF. One consists in adding integration as primitive. The other consists in adding a primitive for maximization of functions and then recursively defining integration from maximization. In both cases we have an adequacy theorem for the corresponding extension of Real PCF. Moreover based on previous work on Real PCF definability, we show that Real PCF extended with the maximization operator is universal, which implies that it is also fully abstract.

Sequential games and optimal strategies

This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these higher type objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of this game are considered in a variety of areas such as fixed point theory, topology, game theory, higher type computability and proof theory. These examples are intended to illustrate how the fundamental construction of optimal strategies based on products of selection functions permeates several research areas.

The way-below relation of function spaces over semantic domains

Abstract For partially ordered sets that are continuous in the sense of D.S. Scott, the way-below relation is crucial. It expresses the approximation of an ideal element by its finite parts. We present explicit characterizations of the way-below relation on spaces of continuous functions from topological spaces into continuous posets. Although it is well known in which cases these function spaces are continuous posets, such characterizations were lacking until now.

The regular-locally-compact coreflection of a stably locally compact locale

Abstract The Scott continuous nuclei form a subframe of the frame of all nuclei. We refer to this subframe as the patch frame. We show that the patch construction exhibits (i) the category of regular locally compact locales and perfect maps as a coreflective subcategory of the category of stably locally compact locales and perfect maps, (ii) the category of compact regular locales and continuous maps as a coreflective subcategory of the category of stably compact locales and perfect maps, and (iii) the category of Stone locales and continuous maps as a coreflective subcategory of the category of spectral locales and perfect maps. (Here a stably locally compact locale is not necessarily compact, and a stably compact locale is a compact and stably locally compact locale.) We relate our patch construction to Banaschewski and Brummers construction of the dual equivalence of the category of stably compact locales and perfect maps with the category of compact regular biframes and biframe homomorphisms.