Limited Principle Of Omniscience Research Articles

Stashing And Parallelization Pentagons

Parallelization is an algebraic operation that lifts problems to sequences in a natural way. Given a sequence as an instance of the parallelized problem, another sequence is a solution of this problem if every component is instance-wise a solution of the original problem. In the Weihrauch lattice parallelization is a closure operator. Here we introduce a dual operation that we call stashing and that also lifts problems to sequences, but such that only some component has to be an instance-wise solution. In this case the solution is stashed away in the sequence. This operation, if properly defined, induces an interior operator in the Weihrauch lattice. We also study the action of the monoid induced by stashing and parallelization on the Weihrauch lattice, and we prove that it leads to at most five distinct degrees, which (in the maximal case) are always organized in pentagons. We also introduce another closely related interior operator in the Weihrauch lattice that replaces solutions of problems by upper Turing cones that are strong enough to compute solutions. It turns out that on parallelizable degrees this interior operator corresponds to stashing. This implies that, somewhat surprisingly, all problems which are simultaneously parallelizable and stashable have computability-theoretic characterizations. Finally, we apply all these results in order to study the recently introduced discontinuity problem, which appears as the bottom of a number of natural stashing-parallelization pentagons. The discontinuity problem is not only the stashing of several variants of the lesser limited principle of omniscience, but it also parallelizes to the non-computability problem. This supports the slogan that "non-computability is the parallelization of discontinuity".

Principles of bar induction and continuity on Baire space

Brouwer-operations, also known as inductively defined neighbourhood functions, provide a good notion of continuity on Baire space which naturally extends that of uniform continuity on Cantor space. In this paper, we introduce a continuity principle for Baire space which says that every pointwise continuous function from Baire space to the set of natural numbers is induced by a Brouwer-operation. Working in Bishop constructive mathematics, we show that the above principle is equivalent to a version of bar induction whose strength is between that of the monotone bar induction and the decidable bar induction. We also show that the monotone bar induction and the decidable bar induction can be characterised by similar principles of continuity. Moreover, we show that the $\Pi^{0}_{1}$ bar induction in general implies LLPO (the lesser limited principle of omniscience). This, together with a fact that the $\Sigma^{0}_{1}$ bar induction implies LPO (the limited principle of omniscience), shows that an intuitionistically acceptable form of bar induction requires the bar to be monotone.

A continuity principle, a version of Baire's theorem and a boundedness principle

AbstractWe deal with a restricted form WC-N′ of the weak continuity principle, a version BT′ of Baire's theorem, and a boundedness principle BD-N. We show, in the spirit of constructive reverse mathematics, that WC-N′, BT′ + ¬LPO and BD-N + ¬LPO are equivalent in a constructive system, where LPO is the limited principle of omniscience.

The Gibbard–Satterthwaite theorem of social choice theory in an infinite society and LPO (limited principle of omniscience)

This paper is an attempt to examine the main theorems of social choice theory from the viewpoint of constructive mathematics. We examine the Gibbard–Satterthwaite theorem [A.F. Gibbard, Manipulation of voting schemes: a general result, Econometrica 41 (1973) 587–601; M.A. Satterthwaite, Strategyproofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory 10 (1975) 187–217] in a society with an infinite number of individuals (infinite society). We will show that the theorem that any coalitionally strategy-proof social choice function may have a dictator or has no dictator in an infinite society is equivalent to LPO (limited principle of omniscience). Therefore, it is non-constructive. A dictator of a social choice function is an individual such that if he strictly prefers an alternative (denoted by x) to another alternative (denoted by y), then the social choice function chooses an alternative other than y. Coalitional strategy-proofness is an extension of the ordinary strategy-proofness. It requires non-manipulability for coalitions of individuals as well as for a single individual.

Constructive Mathematics in Theory and Programming Practice

The first part of the paper introduces the varieties of modern constructive mathematics, concentrating on Bishop’s constructive mathematics (BISH). It gives a sketch of both Myhill’s axiomatic system for BISH and a constructive axiomatic development of the real line R. The second part of the paper focusses on the relation between constructive mathematics and programming, with emphasis on Martin-Lof’s theory of types as a formal system for BISH. 1 What is Constructive Mathematics? The story of modern constructive mathematics begins with the publication, in 1907, of L.E.J. Brouwer’s doctoral dissertation Over de Grondslagen der Wiskunde [18], in which he gave the first exposition of his philosophy of intuitionism (a general philosophy, not merely one for mathematics). According to Brouwer, mathematics is a creation of the human mind, and precedes logic: the logic we use in mathematics grows from mathematical practice, and is not some a priori given before mathematical activity can be undertaken. It is not difficult to see how, with this view of mathematics as a strictly creative activity, Brouwer came to the view that the phrase “there exists” should be interpreted strictly and uniquely as “there can be constructed” or, in more modern parlance, “we can compute”. In turn, this interpretation of existence led Brouwer to reject the unbridled use of the Law of Excluded Middle (LEM), P ∨¬P, in mathematical arguments. For example, consider the following statement, the Limited Principle of Omniscience (LPO): ∀a ∈ {0, 1} (a = 0 ∨ a 6= 0) . (1) Here, N = {0, 1, 2, . . .} is the set of natural numbers, {0, 1} is the set of all

Markov's principle, isols and Dedekind finite sets

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:1. ∀A(A Є ⊿ ↔ CP(A)).2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.

Markov's principle, isols and Dedekind finite sets

Markov's principle is more than a convenience in constructive arithmetic and analysis; it is absolutely essential to significant areas of constructive cardinal arithmetic. In turn, logical relations among intuitively appealing principles of constructive cardinal arithmetic parallel relations between MPS and other “problematic axioms” for constructive mathematics, such as the limited principle of omniscience. Finally, simple closure properties on the Dedekind finite sets provide ready examples of statements which are strictly weaker than Markov's principle and yet are independent of extensions of IZF.MPS, Markov's principle with variables over sets, is equivalent to each of these elementary properties of ⊿, the class of Dedekind finite sets:1. ∀A(A Є ⊿ ↔ CP(A)).2. [*]A Є ⊿ ↔ ∀B (A + B is infinite ↔ B is infinite).3. [**]A Є ⊿ ↔ ∀B((A + 1) x B is infinite ↔ B is infinite).CP is Tarski's cancellation property. Consequently, MPS is tantamount, in constructive mathematics, to the standard classical characterizations of ⊿ in terms of cardinality.[*] and [**] imply that ⊿ is closed under addition and multiplication. Closure under addition is, in turn, constructively equivalent to the closure of ⊿ under all combinatorial functions and to the fact that ⊿ is closed under each strict combinatorial function individually. Generally speaking, a function on P(ω) is combinatorial whenever it preserves finiteness, respects cardinality and has a “moduluslike” associate function. It is strict when it is strictly increasing relative to the subset ordering, [cf. §6 for precise definitions.] From this, we see that MPS is foundational for cardinal arithmetic on the constructive Dedekind finite sets.