Decidable Predicates Research Articles

Infinite sets that Satisfy the Principle of Omniscience in any Variety of Constructive Mathematics

AbstractWe show that there are plenty of infinite sets that satisfy the omniscience principle, in a minimalistic setting for constructive mathematics that is compatible with classical mathematics. A first example of an omniscient set is the one-point compactification of the natural numbers, also known as the generic convergent sequence. We relate this to Grilliot's and Ishihara's Tricks. We generalize this example to many infinite subsets of the Cantor space. These subsets turn out to be ordinals in a constructive sense, with respect to the lexicographic order, satisfying both a well-foundedness condition with respect to decidable subsets, and transfinite induction restricted to decidable predicates. The use of simple types allows us to reach any ordinal below εQ, and richer type systems allow us to get higher.

On the contrapositive of countable choice

We show that in elementary analysis (EL) the contrapositive of countable choice is equivalent to double negation elimination for $${\Sigma_{2}^{0}}$$ -formulas. By also proving a recursive adaptation of this equivalence in Heyting arithmetic (HA), we give an instance of the conservativity of EL over HA with respect to recursive functions and predicates. As a complement, we prove in HA enriched with the (extended) Church thesis that every decidable predicate is recursive.

Decidability in Analysis

In this paper, we introduce the notion of almost decidable predicates of real variables. The notion comes from the concept of decidability of number theoretical predicates together with the idea of effective convergence in computable analysis. The weakness of traditional definitions of decidability is discussed. The definition of almost decidable predicates of real variables is given. It is proved that some commonly used predicates on Open image in new window such as Open image in new window between computable reals and recursive open/closed sets are almost decidable, which justifies our definition of decidability in Euclidean spaces. Additionally, the relation between almost decidability and computability of reals and recursiveness of subsets of Open image in new window is considered, which provides a bridge to include our works here to the earlier literature on computability in analysis.

Getting results from programs extracted from classical proofs

We present a new method to extract from a classical proof of ∀ x( I[ x]→∃ y( O[ y]∧ S[ x, y])) a program computing y from x. This method applies when O is a data type and S is a decidable predicate. Algorithms extracted this way are often far better than a stupid enumeration of all the possible outputs and this is verified on a nontrivial example: a proof of Dickson's lemma.

On the induction schema for decidable predicates

AbstractWe study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates,IΔ1. We show thatIΔ1is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.An open problem formulated by J. Paris (see [4, 5]) is whetherIΔ1proves the corresponding least element principle for decidable predicates,LΔ1(or, equivalently, the Σ1-collection principleBΣ1). We reduce this question to a purely computation-theoretic one.

Eventually infinite time Turing machine degrees: infinite time decidable reals

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

On barinduction of higher types for decidable predicates

1.1. The statement to be proved This paper is essentially devoted to the proof of a single statement. In order to state it we denote by BID a certain formal system phrased in the language of functionals of higher types and having the following properties: ( 1 ) BID is an extensicn of Heyting's arithmetic; (2) it contains a vari~ nt o~ GiSdel's Dialectica system T; (3) it contains all axioms of barinduction of higher types for decidable predicates; (4) it contains certain continuity axioms which express that every functional of type ((0/o)]o) is continuous; (5) it contains axioms which express a certain form of extensionality for functions of higher types. Let ZF be Zermelo-Fraenkel set theory without power-set ax ioJ . Let W(<) in this chapter be short for (~)(Ex) q a(x + 1 ) < ~t(x) (x of type 0). The statement $ which we are going to prove (Theorems 9, 10) is:

Computability Over Arbitrary Fields

In most attempts to make precise the concept of a computable function, or decidable predicate, over a field F, it is considered necessary that the elements of F should be in some sense effectively describable, and hence that F itself should be countable. This is the attitude taken in the study of computable fields (see Rabin1). Our proposed definition of computability over arbitrary fields is based on the Shepherdson - Sturgis2 concept of an unlimited register machine.