Formalizing Hyperspaces for Extracting Efficient Exact Real Computation

We propose a framework for certified computation on hyperspaces by formalizing various higher-order data types and operations in a constructive dependent type theory. Our approach builds on our previous work on axiomatization of exact real computation where we formalize nondeterministic first-order partial computations over real and complex numbers. Based on the axiomatization, we first define open, closed, compact and overt subsets in an abstract topological way that allows short and elegant proofs with computational content coinciding with standard definitions in computable analysis. From these proofs we extract programs for testing inclusion, overlapping of sets, et cetera. To improve extracted programs, our framework specializes the Euclidean space ?^m making use of metric properties. To define interesting operations over hyperspaces of Euclidean space, we introduce a nondeterministic version of a continuity principle valid under the standard type-2 realizability interpretation. Instead of choosing one of the usual formulations, we define it in a way similar to an interval extension operator, which often is already available in exact real computation software. We prove that the operations on subsets preserve the encoding, and thereby define a small calculus to built new subsets from given ones, including limits of converging sequences with regards to the Hausdorff metric. From the proofs, we extract programs that generate drawings of subsets of ?^m with any given precision efficiently. As an application we provide a function that constructs fractals, such as the Sierpinski triangle, from iterated function systems using the limit operation, resulting in certified programs that errorlessly draw such fractals up to any desired resolution

Quantalic spectra of semirings

Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann–Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains 'differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory

Computable decision making on the reals and other spaces via partiality and nondeterminism

Though many safety-critical software systems use floating point to represent real-world input and output, programmers usually have idealized versions in mind that compute with real numbers. Significant deviations from the ideal can cause errors and jeopardize safety. Some programming systems implement exact real arithmetic, which resolves this matter but complicates others, such as decision making. In these systems, it is impossible to compute (total and deterministic) discrete decisions based on connected spaces such as $\mathbb{R}$. We present programming-language semantics based on constructive topology with variants allowing nondeterminism and/or partiality. Either nondeterminism or partiality suffices to allow computable decision making on connected spaces such as $\mathbb{R}$. We then introduce pattern matching on spaces, a language construct for creating programs on spaces, generalizing pattern matching in functional programming, where patterns need not represent decidable predicates and also may overlap or be inexhaustive, giving rise to nondeterminism or partiality, respectively. Nondeterminism and/or partiality also yield formal logics for constructing approximate decision procedures. We implemented these constructs in the Marshall language for exact real arithmetic.Comment: This is an extended version of a paper due to appear in the proceedings of the ACM/IEEE Symposium on Logic in Computer Science (LICS) in July 201

Apartness, sharp elements, and the Scott topology of domains

Working constructively, we study continuous directed complete posets (dcpos) and the Scott topology. Our two primary novelties are a notion of intrinsic apartness and a notion of sharp elements. Being apart is a positive formulation of being unequal, similar to how inhabitedness is a positive formulation of nonemptiness. To exemplify sharpness, we note that a lower real is sharp if and only if it is located. Our first main result is that for a large class of continuous dcpos, the Bridges-Vîţǎ apartness topology and the Scott topology coincide. Although we cannot expect a tight or cotransitive apartness on nontrivial dcpos, we prove that the intrinsic apartness is both tight and cotransitive when restricted to the sharp elements of a continuous dcpo. These include the strongly maximal elements, as studied by Smyth and Heckmann. We develop the theory of strongly maximal elements highlighting its connection to sharpness and the Lawson topology. Finally, we illustrate the intrinsic apartness, sharpness, and strong maximality by considering several natural examples of continuous dcpos: the Cantor and Baire domains, the partial Dedekind reals, the lower reals and, finally, an embedding of Cantor space into an exponential of lifted sets

Fibred contextual quantum physics

Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos. The contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints à la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle Σ → B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C — the commutative C*–algebras of a incommutative C*–algebras — and those of the spectral locale Σ as the couples (C, ψ), with ψ a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and Σ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps

Adelic Geometry via Topos Theory

Our starting point has to do with a key tension running through number theory: although all completions of the rationals Q should be treated symmetrically, this is complicated by fundamental disanalogies between the p-adics vs. the reals. Whereas prior work has typically been guided by classical point-set reasoning, this thesis explores various ways of pulling this problem away from the underlying set theory, revealing various surprises that are obscured by the classical perspective. Framing these investigations is the following test problem: construct and describe the topos of completions of Q (up to equivalence). Chapter 2 begins with the preliminaries: we set up the topos-theoretic framework of point-free topology, with a view towards highlighting the distinction between classical vs. geometric mathematics, before introducing the number-theoretic context. A key theme is that geometric mathematics possesses an intrinsic continuity, which forces us to think more carefully about the topological character of classical algebraic constructions. Chapter 3 represents the first step towards constructing the topos of completions. Here, we provide a pointfree account of real exponentiation and logarithms, which will allow us to define the equivalence of completions geometrically. Chapter 4 provides a geometric proof of Ostrowski's Theorem for both upper-valued abosolute values on Z as well as Dedekind-valued absolute values on Q, along with some key insights about the relationship between the multiplicative seminorms and upper reals. In a slightly more classical interlude, Chapter 5 extends these insights to obtain a surprising generalisation of a foundational result in Berkovich geometry. Namely, by replacing the use of classical rigid discs with formal balls, we obtain a classification of the points of Berkovich Spectra M(K{R^{-1}T}) via the language of filters [more precisely, what we call: R-good filters] even when the base field K is trivially-valued. Returning to geometricity, Chapter 6 builds upon Chapters 3 and 4 to investigate the space of places of Q via descent arguments. Here, we uncover an even deeper surprise. Although the non-Archimedean places correspond to singletons (as is classically expected), the Archimedean place corresponds to the subspace of upper reals in [0, 1], a sort of blurred unit interval. The chapter then analyses the topological differences between the non-Archimedean vs. Archimedean places. In particular, we discover that while the topos corresponding to Archimedean place witnesses non-trivial forking in the connected components of its sheaves, the topos corresponding to the non-Archimedean place eliminates all kinds of forking phenomena. We then conclude with some insights and observations, framed by the question: "How should the connected and the disconnected interact?

Locatedness and overt sublocales

Contains fulltext : 83527.pdf (preprint version ) (Open Access

Constructive topology of bishop spaces

The theory of Bishop spaces (TBS) is so far the least developed approach to constructive topology with points. Bishop introduced function spaces, here called Bishop spaces, in 1967, without really exploring them, and in 2012 Bridges revived the subject. In this Thesis we develop TBS. Instead of having a common space-structure on a set X and R, where R denotes the set of constructive reals, that determines a posteriori which functions of type X -> R are continuous with respect to it, within TBS we start from a given class of "continuous" functions of type X -> R that determines a posteriori a space-structure on X. A Bishop space is a pair (X, F), where X is an inhabited set and F, a Bishop topology, or simply a topology, is a subset of all functions of type X -> R that includes the constant maps and it is closed under addition, uniform limits and composition with the Bishop continuous functions of type R -> R. The main motivation behind the introduction of Bishop spaces is that function-based concepts are more suitable to constructive study than set-based ones. Although a Bishop topology of functions F on X is a set of functions, the set-theoretic character of TBS is not that central as it seems. The reason for this is Bishop's inductive concept of the least topology generated by a given subbase. The definitional clauses of a Bishop space, seen as inductive rules, induce the corresponding induction principle. Hence, starting with a constructively acceptable subbase the generated topology is a constructively graspable set of functions exactly because of the corresponding principle. The function-theoretic character of TBS is also evident in the characterization of morphisms between Bishop spaces. The development of constructive point-function topology in this Thesis takes two directions. The first is a purely topological one. We introduce and study, among other notions, the quotient, the pointwise exponential, the dual, the Hausdorff, the completely regular, the 2-compact, the pair-compact and the 2-connected Bishop spaces. We prove, among other results, a Stone-Cech theorem, the Embedding lemma, a generalized version of the Tychonoff embedding theorem for completely regular Bishop spaces, the Gelfand-Kolmogoroff theorem for fixed and completely regular Bishop spaces, a Stone-Weierstrass theorem for pseudo-compact Bishop spaces and a Stone-Weierstrass theorem for pair-compact Bishop spaces. Of special importance is the notion of 2-compactness, a constructive function-theoretic notion of compactness for which we show that it generalizes the notion of a compact metric space. In the last chapter we initiate the basic homotopy theory of Bishop spaces. The other direction in the development of TBS is related to the analogy between a Bishop topology F, which is a ring and a lattice, and the ring of real-valued continuous functions C(X) on a topological space X. This analogy permits a direct "communication" between TBS and the theory of rings of continuous functions, although due to the classical set-theoretic character of C(X) this does not mean a direct translation of the latter to the former. We study the zero sets of a Bishop space and we prove the Urysohn lemma for them. We also develop the basic theory of embeddings of Bishop spaces in parallel to the basic classical theory of embeddings of rings of continuous functions and we show constructively the Urysohn extension theorem for Bishop spaces. The constructive development of topology in this Thesis is within Bishop's informal system of constructive mathematics BISH, inductive definitions with rules of countably many premises included

Computable sets : located and overt locales

What is a computable set? One may call a bounded subset of the plane computable if it can be drawn at any resolution on a computer screen. Using the constructive approach to computability one naturally considers totally bounded subsets of the plane. We connect this notion with notions introduced in other frameworks. A subset of a totally bounded set is again totally bounded iff it is located. Locatedness is one of the fundamental notions in constructive mathematics. The existence of a positivity predicate on a locale, i.e. the locale being overt, or open, has proved to be fundamental in locale theory in a constructive, or topos theoretic, context. We show that the two notions are intimately connected. We propose a definition of located closed sublocale motivated by locatedness of subsets of metric spaces. A closed sublocale of a compact regular locale is located iff it is overt. Moreover, a closed subset of a complete metric space is Bishop compact — that is, totally bounded and complete — iff its localic completion is compact overt. For Baire space metric locatedness corresponds to having a decidable positivity predicate. Finally, we show that the points of the Vietoris locale of a compact regular locale are precisely its compact overt sublocales. We work constructively, predicatively and avoid the use of the axiom of countable choice. Consequently, all are results are valid in any predicative topos