Formalization of the MRDP Theorem in the Mizar System

Summary This article is the final step of our attempts to formalize the negative solution of Hilbert’s tenth problem. In our approach, we work with the Pell’s Equation defined in [2]. We analyzed this equation in the general case to show its solvability as well as the cardinality and shape of all possible solutions. Then we focus on a special case of the equation, which has the form x2 − (a2 − 1)y2 = 1 [8] and its solutions considered as two sequences $\left\{ {{x_i}(a)} \right\}_{i = 0}^\infty ,\left\{ {{y_i}(a)} \right\}_{i = 0}^\infty$ . We showed in [1] that the n-th element of these sequences can be obtained from lists of several basic Diophantine relations as linear equations, finite products, congruences and inequalities, or more precisely that the equation x = yi(a) is Diophantine. Following the post-Matiyasevich results we show that the equality determined by the value of the power function y = xz is Diophantine, and analogously property in cases of the binomial coe cient, factorial and several product [9]. In this article, we combine analyzed so far Diophantine relation using conjunctions, alternatives as well as substitution to prove the bounded quantifier theorem. Based on this theorem we prove MDPR-theorem that every recursively enumerable set is Diophantine, where recursively enumerable sets have been defined by the Martin Davis normal form. The formalization by means of Mizar system [5], [7], [4] follows [10], Z. Adamowicz, P. Zbierski [3] as well as M. Davis [6].

On Some Properties of Positive and Strongly Positive Arithmetical Sets

The notions of positive and strongly positive arithmetical sets are defined as in [1]-[4] (see, for example, [2], p. 33). It is proved (Theorem 1) that any arithmetical set is positive if and only if it can be defined by an arithmetical formula containing only logical operations ∃, &,∨ and the elementary subformulas having the forms 𝑥𝑥=0 or 𝑥𝑥=𝑦𝑦+1, where 𝑥𝑥and 𝑦𝑦 are variables.Corollary:the logical description of the class of positive sets is obtained from the logical description of the class of strongly positive sets replacing the list of operations &,∨ by the list ∃, &,∨. It is proved (Theorem 2) that for any one-dimensional recursively enumerable set 𝑀𝑀 there exists 6- dimensional strongly positive set 𝐻𝐻 such that 𝑥𝑥 ∈𝑀𝑀 holds if and only if (1, 2𝑥𝑥, 0, 0, 1, 0)∈𝐻𝐻+, where 𝐻𝐻+ is the transitive closure of 𝐻𝐻.

Non-deterministic structures of computation

Divergence and non-determinism play a fundamental role in the theory of computation, and their combined effect on computational equality deserves further study. By looking at the issue from the point of view of both computation and interaction, we are led to a canonical equality for non-deterministic computation, revealing its rich algebraic structure. We study this structure in three ways. First, we construct a complete equational system for finite-state non-deterministic computation. The challenge with such a system is to find an equational alternative to fixpoint inductionà laMilner. We establish a negative result in the form of the non-existence of a finite equational system for the canonical equality of non-deterministic computation to support our approach. We then investigate infinite-state non-deterministic computation in the light of definability and show that every recursively enumerable set is generated by an unobservable process. Finally, we prove that, as far as computation is concerned, the effect produced jointly by divergence and non-determinism is model independent for a large class of process models.We use C-graphs, which are interesting in their own right, as abstract representations of the computational objects throughout the paper.

Variants of P Colonies with Very Simple Cell Structure

We study two very simple variants of P colonies: systems with only one object inside the cells, and systems with insertion-deletion programs, so called P colonies with senders and consumers. We show that both of these extremely simple types of systems are able to compute any recursively enumerable set of vectors of non-negative integers.

CELL/SYMBOL COMPLEXITY OF TISSUE P SYSTEMS WITH SYMPORT/ANTIPORT RULES

We consider tissue P systems with symport/antiport rules and investigate their computational power when using only a (very) small number of symbols and cells. Even when using only one symbol, we need at most six (seven when allowing only one channel between a cell and the environment) cells to generate any recursively enumerable set of natural numbers. On the other hand, with only one cell we can only generate regular sets when using one channel with the environment, whereas one cell with two channels between the cell and the environment obtains computational completeness with five symbols. Between these extreme cases of one symbol and one cell, respectively, there seems to be a trade-off between the number of cells and the number of symbols. For example, for the case of tissue P systems with two channels between a cell and the environment we show that computational completeness can be obtained with two cells and three symbols as well as with three cells and two symbols, respectively. Moreover, we also show that some variants of tissue P systems characterize the families of finite or regular sets of natural numbers.

Recent Advances in Σ-definability over Continuous Data Types

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Sigma-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the reals by the structure of hereditarily finite sets. In this setting it is crucial to consider the real numbers without equality since the equality test is undecidable over the reals. We prove Engeler's Lemma for Sigma-definability over the reals without the equality test which relates Sigma-definability with definability in the constructive infinitary language L_{omega_1 omega}. Thus, a relation over the real numbers is Sigma-definable if and only if it is definable by a disjunction of a recursively enumerable set of quantifier free formulas. This result reveals computational aspects of Sigma-definability and also gives topological characterisation of Sigma-definable relations over the reals without the equality test. We also illustrate how computability over the real numbers can be expressed in the language of Sigma-formulas.

On Some Properties of 𝑟-Maximal Sets and 𝑄1–𝑁-Reducibility

It is shown that if 𝑀1, 𝑀2 are 𝑟-maximal sets and 𝑀1 ≡ 𝑄1–𝑁𝑀2, then 𝑀1 ≡ 𝑚𝑀2. In addition, we prove that there exists a simultaneously 𝑄1–𝑁- and 𝑊-complete recursively enumerable set which is not 𝑠𝑄-complete.

A General Schema for Constructing One-Point Bases in the Lambda Calculus

In this paper, we present a schema for constructing one-point bases for recursively enumerable sets of lambda terms. The novelty of the approach is that we make no assumptions about the terms for which the one-point basis is constructed: They need not be combinators and they may contain constants and free variables. The significance of the construction is twofold: In the context of the lambda calculus, it characterises one-point bases as ways of ``packaging'' sets of terms into a single term; And in the context of realistic programming languages, it implies that we can define a single procedure that generates any given recursively enumerable set of procedures, constants and free variables in a given programming language.

Friedberg splittings in Σ30 quotient lattices of

Let {We}e∈ω be a standard enumeration of the recursively enumerable (r.e.) subsets of ω = {0,1,2,…}. The lattice of recursively enumerable sets, , is the structure ({We}e∈ω,∪,∩). ≡ is a congruence relation on if ≡ is an equivalence relation on and if for all U, U′ ∈ and V, V′ ∈ , if U ≡ U′ and V ≡ V′, then U ∪ V ≡ U′ ∪ V′ and U ∩ V ≡ U′ ∩ V′. [U] = {V ∈ | V ≡ U} is the equivalence class of U. If ≡ is a congruence relation on , the elements of the quotient lattice / ≡ are the equivalence classes of ≡. [U] ∪ [V] is defined as [U ∪ V], and [U] ∩ [V] is defined as [U ∩ V]. We say that a congruence relation ≡ on is if {(i, j)| Wi ≡ Wj} is . Define =* by putting Wi, =* Wj if and only if (Wi − Wj)∪ (Wj − Wi) is finite. Then =* is a congruence relation. If D is any set, then we can define a congruence relation by putting Wi Wj if and only if Wi ∩ D =* Wj ∩D. By Hammond [2], a congruence relation ≡ ⊇ =* is if and only if ≡ is equal to for some set D.The Friedberg splitting theorem [1] asserts that if A is any recursively enumerable set, then there exist disjoint recursively enumerable sets A0 and A1 such that A = A0∪ A1 and such that for any recursively enumerable set B

Working below a high recursively enumerable degree

In recent work, Cooper [3, 1990] has extended results of Jockusch and Shore [6, 1984] to show that the Turing jump is definable in the structure given by the Turing degrees and the ordering of Turing reducibility. In his definition of x′ from x, Cooper identifies an order-theoretic property shared by all of the degrees that are recursively enumerable in x and above x. He then shows that x′ is the least upper bound of all the degrees with this property. Thus, the jump of x is identified by comparing the recursively enumerable degrees with other degrees which are not recursively enumerable. Of course, once the jump operator is known to be definable, the relation of jump equivalence x′ = y′ is also known to be a definable relation on x and y. If we consider how much of the global theory of the Turing degrees is sufficient for Cooper's methods, it is immediately clear that his methods can be implemented to show that the jump operator and its weakening to the relation of jump equivalence are definable in any ideal closed under the Turing jump. However, his methods do not localize to , the degrees, or to the recursively enumerable degrees.This paper fits, as do Shore and Slaman [16, 1990] and [17, to appear], within the general project to develop an understanding of the relationship between the local degree-theoretic properties of a recursively enumerable set A and its jump class. For an analysis of the possibility of defining jump equivalence in , consult Shore [15, to appear] who shows that the relation x(3) = y(3) is definable. In this paper, we will restrict our attention to definitions expressed completely in ℛ (Note: All sets and degrees discussed for the remainder of this paper will be recursively enumerable.) Ultimately, one would like to find some degree-theoretic properties definable in terms of the ordering of Turing reducibility and quantifiers over the recursively enumerable degrees that would define the relation of jump equivalence or define one or more of the jump classes Hn = {w∣ wn = 0n+1} or Ln = {w ∣ wn = 0n}. Such a result could very likely then be used as a springboard to other general definability results for the recursively enumerable degrees. It would be especially interesting to know whether every recursively enumerable degree is definable and whether every arithmetical degree-invariant property of the recursively enumerable sets is definable in .