Programs viewed as Skolem functions

2005 ◽  
pp. 125-134
Author(s):  
Ralf Steinbrüggen
Keyword(s):  
Skolem Functions

scholarly journals Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic

2012 ◽  
Vol 97 ◽  
pp. 1-18
Author(s):  
Federico Aschieri ◽  
Margherita Zorzi
Keyword(s):  
Peano Arithmetic ◽  
Skolem Functions

scholarly journals Exponential-constructible functions in P-minimal structures

2019 ◽  
Vol 20 (02) ◽  
pp. 2050005
Author(s):  
Saskia Chambille ◽  
Pablo Cubides Kovacsics ◽  
Eva Leenknegt
Keyword(s):  
Skolem Functions ◽  
Constructible Functions ◽  
Stability Results

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of [Formula: see text]-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages.

Σ1-compactness in languages stronger than

1978 ◽  
Vol 43 (3) ◽  
pp. 508-520 ◽  
Author(s):  
Nigel Cutland
Keyword(s):  
Reflection Principle ◽  
Necessary Condition ◽  
Closed Set ◽  
Skolem Functions ◽  
And Function

One of the main results of Barwise [2] (see also [7, Chapter VIII]) showed that the s – reflection principle for a set A is equivalent to Σ1-compactness of . Here A is any transitive p.r. closed set, and is the infinitary language on A which allows conjunction and disjunction over arbitrary sets Φ Є A, and finite quantification.In this paper we consider languages , where B is a Δ0 subset of A, which is like but we allow quantifiers ∀x and ∃x where x is any set of variables indexed by an element of B. A treatment similar to that of [2] for establishes a sufficient, and in some cases necessary, condition for to be Σ1-compact. The use of infinitary Skolem functions is intrinsic to the method, so to avoid a separate development of the rudiments of the Skolem language we actually define to have b-ary relation and function symbols for every b Є B.

Sets of theorems with short proofs

1974 ◽  
Vol 39 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Daniel Richardson
Keyword(s):  
Proof Complexity ◽  
Predicate Calculus ◽  
Definable Set ◽  
Semantic Tableaux ◽  
Skolem Functions ◽  
Definition Of

R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x1, …, xn) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a1, …, an): A(a1, …, an) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x1, …, xn) and a number k so that (a1 …, an) ∈ S if and only if A(a1 …, an) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.

scholarly journals Term extraction and Ramsey's theorem for pairs

2012 ◽  
Vol 77 (3) ◽  
pp. 853-895 ◽  
Author(s):  
Alexander P. Kreuzer ◽  
Ulrich Kohlenbach
Keyword(s):  
Uniform Method ◽  
Second Step ◽  
Functional Interpretation ◽  
Ramsey's Theorem ◽  
Term Extraction ◽  
Ramsey’S Theorem ◽  
Skolem Functions ◽  
Primitive Recursive

AbstractIn this paper we study with proof-theoretic methods the function(al)s provably recursive relative to Ramsey's theorem for pairs and the cohesive principle (COH).Our main result on COH is that the type 2 functional provably recursive fromare primitive recursive. This also provides a uniform method to extract bounds from proofs that use these principles. As a consequence we obtain a new proof of the fact thatis-conservative over PRA.Recent work of the first author showed thatis equivalent to a weak variant of the Bolzano-Weierstraß principle. This makes it possible to use our results to analyze not only combinatorial but also analytical proofs.For Ramsey's theorem for pairs and two colorswe obtain the upper bounded that the type 2 functional provable recursive relative toare inT1. This is the fragment of Gödel's systemTcontaining only type 1 recursion—roughly speaking it consists of functions of Ackermann type. With this we also obtain a uniform method for the extraction ofT1-bounds from proofs that use. Moreover, this yields a new proof of the fact thatis-conservative over.The results are obtained in two steps: in the first step a term including Skolem functions for the above principles is extracted from a given proof. This is done using Gödel's functional interpretation. After this the term is normalized, such that only specific instances of the Skolem functions are used. In the second step this term is interpreted using-comprehension. The comprehension is then eliminated in favor of induction using either elimination of monotone Skolem functions (for COH) or Howard's ordinal analysis of bar recursion (for).

Almost Combinatorial Skolem Functions

1970 ◽  
Vol 35 (1) ◽  
pp. 65 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Skolem Functions

scholarly journals Elimination of Skolem functions for monotone formulas in analysis

1998 ◽  
Vol 37 (5-6) ◽  
pp. 363-390 ◽  
Author(s):  
Ulrich Kohlenbach
Keyword(s):  
Skolem Functions

Skolem Functions and Herbrand Universes in a Tree Generalization of First Order Logic

2006 ◽  
Author(s):  
Roger Villemaire ◽  
Sylvain Halle ◽  
Rudy Deca ◽  
Omar Cherkaoui
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Skolem Functions

Kueker's conjecture for stable theories

1989 ◽  
Vol 54 (1) ◽  
pp. 207-220 ◽  
Author(s):  
Ehud Hrushovski
Keyword(s):  
Independent Interest ◽  
Linear Ordering ◽  
Stable Case ◽  
Skolem Functions

AbstractKueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.

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