On the logical strengths of partial solutions to mathematical problems

Laurent Bienvenu 1 Ludovic Patey 2, 3 Paul Shafer 4
2 AGL - Algèbre, géométrie, logique
ICJ - Institut Camille Jordan [Villeurbanne]
3 PI.R2 - Design, study and implementation of languages for proofs and programs
Inria de Paris, CNRS - Centre National de la Recherche Scientifique, UPD7 - Université Paris Diderot - Paris 7, PPS - Preuves, Programmes et Systèmes
Abstract : We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood [9], we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to König's lemma, such as restrictions of König's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak König's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak König's lemma is robust in the sense of Montalbán [26]: it is equivalent to several perturbations. We also clarify the relationship between Ramsey-type weak König's lemma and algorithmic randomness by showing that Ramsey-type weak weak König's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak König's lemma. This answers a question of Flood.
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Laurent Bienvenu, Ludovic Patey, Paul Shafer. On the logical strengths of partial solutions to mathematical problems. Transactions of the London Mathematical Society, 2017, 4 (1), pp.30 - 71. ⟨10.1112/tlm3.12001⟩. ⟨hal-01888542⟩

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