10 research outputs found
Functional first order definability of LRTp
The language LRTp is a non-deterministic language for exact real number computation. It has been shown that all computable rst order relations in the sense of Brattka are denable in the language. If we restrict the language to single-valued total relations (e.g. functions), all polynomials are denable in the language. This paper is an expanded version of [12] in which we show that the non-deterministic version of the limit operator, which allows to dene all computable rst order relations, when restricted to single-valued total inputs, produces single-valued total outputs. This implies that not only the polynomials are denable in the language but also allcomputable rst order functions
Coinductive Formal Reasoning in Exact Real Arithmetic
In this article we present a method for formally proving the correctness of
the lazy algorithms for computing homographic and quadratic transformations --
of which field operations are special cases-- on a representation of real
numbers by coinductive streams. The algorithms work on coinductive stream of
M\"{o}bius maps and form the basis of the Edalat--Potts exact real arithmetic.
We use the machinery of the Coq proof assistant for the coinductive types to
present the formalisation. The formalised algorithms are only partially
productive, i.e., they do not output provably infinite streams for all possible
inputs. We show how to deal with this partiality in the presence of syntactic
restrictions posed by the constructive type theory of Coq. Furthermore we show
that the type theoretic techniques that we develop are compatible with the
semantics of the algorithms as continuous maps on real numbers. The resulting
Coq formalisation is available for public download.Comment: 40 page
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
. 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 . 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
Constructive Domains with Classical Witnesses
We develop a constructive theory of continuous domains from the perspective
of program extraction. Our goal that programs represent (provably correct)
computation without witnesses of correctness is achieved by formulating
correctness assertions classically. Technically, we start from a predomain base
and construct a completion. We then investigate continuity with respect to the
Scott topology, and present a construction of the function space. We then
discuss our main motivating example in detail, and instantiate our theory to
real numbers that we conceptualise as the total elements of the completion of
the predomain of rational intervals, and prove a representation theorem that
precisely delineates the class of representable continuous functions