Keita Yokoyama

Reverse mathematics and Peano categoricity

Abstract We investigate the reverse-mathematical status of several theorems to the effect that the natural number system is second-order categorical. One of our results is as follows. Define a system to be a triple A , i , f such that A is a set and i ∈ A and f : A → A . A subset X ⊆ A is said to be inductive if i ∈ X and ∀a ( a ∈ X ⇒ f ( a ) ∈ X ). The system A , i , f is said to be inductive if the only inductive subset of A is A itself. Define a Peano system to be an inductive system such that f is one-to-one and i ∉ the range of f. The standard example of a Peano system is N , 0 , S where N = { 0 , 1 , 2 , … , n , … } = the set of natural numbers and S : N → N is given by S ( n ) = n + 1 for all n ∈ N . Consider the statement that all Peano systems are isomorphic to N , 0 , S . We prove that this statement is logically equivalent to WKL 0 over RCA 0 ⁎ . From this and similar equivalences we draw some foundational/philosophical consequences.

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 Ramseys 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 Rybalchenkos Termination Theorem we can extract some information about termination bounds.

On the Ramseyan Factorization Theorem

We study, in the context of reverse mathematics, the strength of Ramseyan factorization theorem (\({\rm RF}^{s}_{k}\)), a Ramsey-type theorem used in automata theory. We prove that \({\rm RF}^s_k\) is equivalent to \({\rm RT}^2_2\) for all s,k ≥ 2, k ∈ ω over RCAo. We also consider a weak version of Ramseyan factorization theorem and prove that it is in between ADS and CAC.

On the strength of Ramsey's theorem without Σ1-induction

In this paper, we show that is a -conservative extension of BΣ1 + exp, thus it does not imply IΣ1.

Propagation of partial randomness

Let f be a computable function from finite sequences of 0s and 1s to real numbers. We prove that strong f-randomness implies strong f- randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Lof random rel- ative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial random- ness, including non-K-triviality and autocomplexity. We prove that f- randomness relative to a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree.

On principles between ∑1- and ∑2-induction, and monotone enumerations

We show that many principles of first-order arithmetic, previously only known to lie strictly between ∑1-induction and ∑2-induction, are equivalent to the well-foundedness of ωω. Among these principles are the iteration of partial functions (P∑1) of Hajek and Paris, the bounded monotone enumerations principle (non-iterated, BME1) by Chong, Slaman, and Yang, the relativized Paris–Harrington principle for pairs, and the totality of the relativized Ackermann–Peter function. With this we show that the well-foundedness of ωω is a far more widespread than usually suspected. Further, we investigate the k-iterated version of the bounded monotone iterations principle (BMEk), and show that it is equivalent to the well-foundedness of the (k + 1)-height ω-tower ω⋰ω.

The Strength of the SCT Criterion

We undertake the study of size-change analysis in the context of Reverse Mathematics. In particular, we prove that the SCT criterion [9, Theorem 4] is equivalent to \(\mathsf {I}\varSigma ^0_{2}\) over \(\mathsf {RCA_0}\).

A parameterized halting problem, the linear time hierarchy, and the MRDP theorem

The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy. On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.

Nonstandard second-order arithmetic and Riemannʼs mapping theorem

Abstract In this paper, we introduce systems of nonstandard second-order arithmetic which are conservative extensions of systems of second-order arithmetic. Within these systems, we do reverse mathematics for nonstandard analysis, and we can import techniques of nonstandard analysis into analysis in weak systems of second-order arithmetic. Then, we apply nonstandard techniques to a version of Riemannʼs mapping theorem, and show several different versions of Riemannʼs mapping theorem.

A Note on the Sequential Version of {\rm \Pi^1_2} Statements

In connection with uniform computability and intuitionistic provability, the strength of the sequential version of \({\rm \Pi^1_2}\) theorems has been investigated in reverse mathematics. In some examples, we illustrate that it occasionally depends on the way of formalizing the \({\rm \Pi^1_2}\) statement, so the investigation of sequential strength demands careful attention to the formalization. Moreover our results suggest the optimality of Dorais’s uniformization theorems.