21 research outputs found
Ramsey's Theorem for Pairs and Colors as a Sub-Classical Principle of Arithmetic
The purpose is to study the strength of Ramsey's Theorem for pairs restricted
to recursive assignments of -many colors, with respect to Intuitionistic
Heyting Arithmetic. We prove that for every natural number , Ramsey's
Theorem for pairs and recursive assignments of colors is equivalent to the
Limited Lesser Principle of Omniscience for formulas over Heyting
Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is
equivalent to: for every recursively enumerable infinite -ary tree there is
some and some branch with infinitely many children of index .Comment: 17 page
An Intuitionistic Analysis of Size-change Termination
In 2001 Lee, Jones and Ben-Amram introduced the notion of size-change termination (SCT) for first order functional programs, a sufficient condition for termination. They proved that a program is size-change terminating if and only if it has a certain property which can be statically verified from the recursive definition of the program. Their proof of the size-change termination theorem used Ramsey\u27s Theorem for pairs, which is a purely classical result. In 2012 Vytiniotis, Coquand and Wahlsteldt intuitionistically proved a classical variant of the size-change termination theorem by using the Almost-Full Theorem instead of Ramsey\u27s Theorem for pairs. In this paper we provide an intuitionistic proof of another classical variant of the SCT theorem: our goal is to provide a statement and a proof very similar to the original ones. This can be done by using the H-closure Theorem, which differs from Ramsey\u27s Theorem for pairs only by a contrapositive step. As a side result we obtain another proof of the characterization of the functions computed by a tail-recursive SCT program, by relating the SCT Theorem with the Termination Theorem by Podelski and Rybalchenko. Finally, by investigating the relationship between them, we provide a property in the "language" of size-change termination which is equivalent to Podelski and Rybalchenko\u27s termination
Generic Large Cardinals and Systems of Filters
We introduce the notion of -system of filters, generalizing the
standard definitions of both extenders and towers of normal ideals. This
provides a framework to develop the theory of extenders and towers in a more
general and concise way. In this framework we investigate the topic of
definability of generic large cardinals properties.Comment: 36 page
Ramsey Theorem for Pairs As a Classical Principle in Intuitionistic Arithmetic
We produce a first order proof of a famous combinatorial result, Ramsey Theorem for pairs and in two colors. Our goal is to find the minimal classical principle that implies the "miniature" version of Ramsey we may express in Heyting Arithmetic HA. We are going to prove that Ramsey Theorem for pairs with recursive assignments of two colors is equivalent in HA to the sub-classical principle Sigma-0-3-LLPO. One possible application of our result could be to use classical realization to give constructive proofs of some combinatorial corollaries of Ramsey; another, a formalization of Ramsey in HA, using a proof assistant. In order to compare Ramsey Theorem with first order classical principles, we express it as a schema in the first order language of arithmetic, instead of using quantification over sets and functions as it is more usual: all sets we deal with are explicitly defined as arithmetical predicates. In particular, we formally define the homogeneous set which is the witness of Ramsey theorem as a Delta-0-3-arithmetical predicate
Reverse mathematical bounds for the Termination Theorem
In 2004 Podelski and Rybalchenko expressed the termination of transition-based programs as a property of well-founded relations. The classical proof by Podelski and Rybalchenko requires Ramsey's Theorem for pairs which is a purely classical result, therefore extracting bounds from the original proof is non-trivial task. Our goal is to investigate the termination analysis from the point of view of Reverse Mathematics. By studying the strength of Podelski and Rybalchenko's Termination Theorem we can extract some information about termination bounds
A DIRECT PROOF OF SCHWICHTENBERG'S BAR RECURSION CLOSURE THEOREM
In 1979 Schwichtenberg showed that the System definable
functionals are closed under a rule-like version Spector's bar recursion of
lowest type levels and . More precisely, if the functional which
controls the stopping condition of Spector's bar recursor is
-definable, then the corresponding bar recursion of type levels
and is already -definable. Schwichtenberg's original proof,
however, relies on a detour through Tait's infinitary terms and the
correspondence between ordinal recursion for and
primitive recursion over finite types. This detour makes it hard to calculate
on given concrete system input, what the corresponding system
output would look like. In this paper we present an alternative
(more direct) proof based on an explicit construction which we prove correct
via a suitably defined logical relation. We show through an example how this
gives a straightforward mechanism for converting bar recursive definitions into
-definitions under the conditions of Schwichtenberg's theorem.
Finally, with the explicit construction we can also easily state a sharper
result: if is in the fragment then terms built from
for this particular are definable in the
fragment .Comment: 12 page