On notions of computability-theoretic reduction between Π21 principles

Abstract

Abstract. Several notions of computability theoretic reducibility between Π 1 2 principles have been studied. This paper contributes to the program of analyzing the behavior of versions of Ramsey's Theorem and related principles under these notions. Among other results, we show that for each n 3, there is an instance of RT n 2 all of whose solutions have PA degree over ∅ (n−2) , and use this to show that König's Lemma lies strictly between RT 2 2 and RT 3 2 under one of these notions. We also answer two questions raised by Dorais, Dzhafarov, Hirst, Mileti, and Shafer [ta] on comparing versions of Ramsey's Theorem and of the Thin Set Theorem with the same exponent but different numbers of colors. Still on the topic of the effect of the number of colors on the computable aspects of Ramsey theoretic properties, we show that for each m 2, there is an (m + 1)-coloring c of N such that every m-coloring of N has an infinite homogeneous set that does not compute any infinite homogeneous set for c, and connect this result with the notion of infinite information reducibility introduced by Dzhafarov and Igusa [ta]. Next, we introduce and study a new notion that provides a uniform version of the idea of implication with respect to ω-models of RCA 0 , and related notions that allow us to count how many applications of a principle P are needed to reduce another principle to P . Finally, we fill in a gap in the proof of Theorem 12.2 in Cholak, Jockusch, and Slaman [2001].

On the Uniform Computational Content of Ramsey’s Theorem

“…We will be able to use our Theorem 3.24 on products to answer this question in the affirmative (see Theorem 4.22). Independently, similar results were obtained by Hirschfeldt and Jockusch [23] and Patey [32]. Altogether, the diagram in Figure 1 displays how Ramsey's theorem for different cardinalities and colors is situated in the Weihrauch lattice.…”

“…We mention that it was noted by Hirschfeldt and Jockusch [23,Figure 6] that the following corollary follows from [24,Theorem 2.3].…”

“…On the one hand, this algorithm ensures that the tree S is decided for larger and larger blocks of size n, and on the other hand, the construction guarantees that the resulting tree S is finitely branching: on each level i of T only finitely many different values can occur and at some stage n of the construction all words w of length i in T have been considered and corresponding words u w,v have been added to S. The fact that we always choose the lexicographically smallest v ensures that no new strings of length i are added to S after stage n. Finally, the construction also guarantees pr 1 ([S]) = [T ] as promised. Now Theorem 5.13 and Corollary 3.30 yield the following result, which was independently proved in [23,Corollary 2.3] by a very different method.…”

Coarse Reducibility and Algorithmic Randomness

“…We will show in the next section that, as one might expect, the converse fails. The development of the theory of notions of robust information coding and related concepts have led to interactions with computability theory (as in Jockusch and Schupp [13]; Downey, Jockusch, and Schupp [4]; Downey, Jockusch, Mc-Nicholl, and Schupp [5]; and Hirschfeldt, Jockusch, McNicholl, and Schupp [10]), reverse mathematics (as in Dzhafarov and Igusa [7] and Hirschfeldt and Jockusch [9]), and algorithmic randomness (as in Astor [1]).…”

“…Work on such notions has mainly focused on their uniform versions. (One exception is a result on nonuniform ii-reducibility in Hirschfeldt and Jockusch [13].) However, nonuniform versions of these reducibilities also seem to be of interest.…”