Decidability and essential undecidability

1957 ◽  
Vol 22 (1) ◽  
pp. 39-54 ◽  
Author(s):  
Hilary Putnam
Keyword(s):  
Simple Proof ◽  
Arithmetic Functions ◽  
Successor Function ◽  
Open Problems ◽  
Decidable Theory ◽  
Undecidable Theory ◽  
The Common

There are a number of open problems involving the concepts of decidability and essential undecidability. This paper will present solutions to some of these problems. Specifically:(1) Can a decidable theory have an essentially undecidable, axiomatizable extension (with the same constants)?(2) Are all the complete extensions of an undecidable theory ever decidable?We shall show that the answer to both questions is in the affirmative. In answering question (1), the decidable theory for which an essentially undecidable axiomatizable extension will be constructed is the theory of the successor function and a single one-place predicate. It will also be shown that the decidability of this theory is a “best possible” result in the following direction: the theory of either of the common diadic arithmetic functions and a one-place predicate; i.e., of addition and a one-place predicate, or of multiplication and a one-place predicate, is undecidable.Before establishing the main result, it is convenient to give a simple proof that a decidable theory can have an axiomatizable (simply) undecidable extension. This is, of course, an immediate consequence of the main result; but the proof is simple and illustrates the methods that we are going to use in this paper.

scholarly journals Introduction

2018 ◽  
Vol 27 (4) ◽  
pp. 441-441
Author(s):  
PAUL BALISTER ◽  
BÉLA BOLLOBÁS ◽  
IMRE LEADER ◽  
ROB MORRIS ◽  
OLIVER RIORDAN
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

The consistency of arithmetic

2021 ◽  
pp. 268-311
Author(s):  
Paolo Mancosu ◽  
Sergio Galvan ◽  
Richard Zach
Keyword(s):  
Simple Proof ◽  
Sequent Calculus ◽  
Cut Elimination ◽  
Successor Function ◽  
Cut Rule ◽  
Original Proof ◽  
First Order ◽  
Successive Elimination ◽  
Pure Logic ◽  
Classical Arithmetic

This chapter opens the part of the book that deals with ordinal proof theory. Here the systems of interest are not purely logical ones, but rather formalized versions of mathematical theories, and in particular the first-order version of classical arithmetic built on top of the sequent calculus. Classical arithmetic goes beyond pure logic in that it contains a number of specific axioms for, among other symbols, 0 and the successor function. In particular, it contains the rule of induction, which is the essential rule characterizing the natural numbers. Proving a cut-elimination theorem for this system is hopeless, but something analogous to the cut-elimination theorem can be obtained. Indeed, one can show that every proof of a sequent containing only atomic formulas can be transformed into a proof that only applies the cut rule to atomic formulas. Such proofs, which do not make use of the induction rule and which only concern sequents consisting of atomic formulas, are called simple. It is shown that simple proofs cannot be proofs of the empty sequent, i.e., of a contradiction. The process of transforming the original proof into a simple proof is quite involved and requires the successive elimination, among other things, of “complex” cuts and applications of the rules of induction. The chapter describes in some detail how this transformation works, working through a number of illustrative examples. However, the transformation on its own does not guarantee that the process will eventually terminate in a simple proof.

scholarly journals Introduction

2010 ◽  
Vol 19 (5-6) ◽  
pp. 641-641
Author(s):  
Noga Alon ◽  
Béla Bollobás
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

Two theories with axioms built by means of pleonasms

1957 ◽  
Vol 22 (1) ◽  
pp. 36-38 ◽  
Author(s):  
Andrzej Ehrenfeucht
Keyword(s):  
Finite Number ◽  
Logical Constants ◽  
Recursive Functions ◽  
Undecidable Theory ◽  
Binary Predicate ◽  
The Common ◽  
Image Position ◽  
Infinite Power ◽  
Recursive Set

This paper contains examples T1 and T2 of theories which answer the following questions:(1) Does there exist an essentially undecidable theory with a finite number of non-logical constants which contains a decidable, finitely axiomatizable subtheory?(2) Does there exist an undecidable theory categorical in an infinite power which has a recursive set of axioms? (Cf. [2] and [3].)The theory T1 represents a modification of a theory described by Myhill [7]. The common feature of theories T1 and T2 is that in both of them pleonasms are essential in the construction of the axioms.Let T1 be a theory with identity = which contains one binary predicate R(x, y) and is based on the axioms A1, A2, A3, B1, B2, B3, B4, Cnm which follow.A1: x = x. A2: x = y ⊃ y = x. A3: x = y ∧ y = z ⊃ x = z.(Axioms of identity.)B1: R(x, x). B2: R(x, y) ⊃ R(y, x). B3: R (x, y) ∧ R(y, z) ⊃ R(x, z).(Axioms of equivalence.)B4: x = y ⊃ [R(z, x) ≡ R(z,y)].Let φn be the formulawhich express that there is an abstraction class of the relation R which has exactly n elements.Let f(n) and g(n) be two recursive functions which enumerate two recursively inseparable sets [5], and call these sets X1 and X2.We now specify the axioms Cmm.It is obvious that the set composed of the formulas A1−A3, B1−B4, Cnm (n,m = 1,2, …) is recursive.The theory T1 is essentially undecidable; for if there were a complete and decidable extension T′1 (of it, then the recursive sets Z = {n: φn is provable in T′1} and Z′ = {n: ∼φn is provable in T′1} would separate the sets X1 and X2.

scholarly journals Introduction

2015 ◽  
Vol 24 (4) ◽  
pp. 584-584
Author(s):  
PAUL BALISTER
Keyword(s):  
Computer Science ◽  
Theoretical Computer Science ◽  
Special Issue ◽  
Open Problems ◽  
Theoretical Computer ◽  
Discrete Probability ◽  
The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 14th to 20th April 2013. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

DECIDABILITY FOR THEORIES OF MODULES OVER VALUATION DOMAINS

2015 ◽  
Vol 80 (2) ◽  
pp. 684-711 ◽  
Author(s):  
LORNA GREGORY
Keyword(s):  
Residue Field ◽  
Krull Dimension ◽  
Valuation Domain ◽  
Decidable Theory ◽  
Valuation Domains ◽  
Undecidable Theory

AbstractExtending work of Puninski, Puninskaya and Toffalori in [5], we show that if V is an effectively given valuation domain then the theory of all V-modules is decidable if and only if there exists an algorithm which, given a, b ε V, answers whether a ε rad(bV). This was conjectured in [5] for valuation domains with dense value group, where it was proved for valuation domains with dense archimedean value group. The only ingredient missing from [5] to extend the result to valuation domains with dense value group or infinite residue field is an algorithm which decides inclusion for finite unions of Ziegler open sets. We go on to give an example of a valuation domain with infinite Krull dimension, which has decidable theory of modules with respect to one effective presentation and undecidable theory of modules with respect to another. We show that for this to occur infinite Krull dimension is necessary.

scholarly journals About the Algebraic Yuzvinski Formula

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Simone Virili
Keyword(s):  
Vector Space ◽  
Topological Entropy ◽  
Characteristic Polynomial ◽  
Mahler Measure ◽  
Algebraic Proof ◽  
Open Problems ◽  
Algebraic Entropy ◽  
Rational Vector ◽  
The Common ◽  
Rational Vector Space

AbstractThe Algebraic Yuzvinski Formula expresses the algebraic entropy of an endomorphism of a finitedimensional rational vector space as the Mahler measure of its characteristic polynomial. In a recent paper, we have proved this formula, independently fromits counterpart – the Yuzvinski Formula – for the topological entropy proved by Yuzvinski in 1968. In this paper we first compare the proof of the Algebraic Yuzvinski Formula with a proof of the Yuzvinski Formula given by Lind and Ward in 1988, underlying the common ideas and the differences in the main steps. Then we describe several known applications of the Algebraic Yuzvinski Formula, and some related open problems are discussed. Finally,we give a new and purely algebraic proof of the Algebraic Yuzvinski Formula for the intrinsic algebraic entropy.

Catalogue reduction techniques

1978 ◽  
Vol 48 ◽  
pp. 389-390 ◽  
Author(s):  
Chr. de Vegt
Keyword(s):  
Adjustment Process ◽  
Proper Motions ◽  
Measuring Process ◽  
Aerial Photogrammetry ◽  
Reduction Techniques ◽  
Observational Technique ◽  
The Common ◽  
Astrometric Data

AbstractReduction techniques as applied to astrometric data material tend to split up traditionally into at least two different classes according to the observational technique used, namely transit circle observations and photographic observations. Although it is not realized fully in practice at present, the application of a blockadjustment technique for all kind of catalogue reductions is suggested. The term blockadjustment shall denote in this context the common adjustment of the principal unknowns which are the positions, proper motions and certain reduction parameters modelling the systematic properties of the observational process. Especially for old epoch catalogue data we frequently meet the situation that no independent detailed information on the telescope properties and other instrumental parameters, describing for example the measuring process, is available from special calibration observations or measurements; therefore the adjustment process should be highly self-calibrating, that means: all necessary information has to be extracted from the catalogue data themselves. Successful applications of this concept have been made already in the field of aerial photogrammetry.

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