arXiv subjects
Publications and source records attributed to Keita Yokoyama.
The class of quasi-Polish spaces admits several equivalent representations, including UF spaces, NP spaces, $\mathbfΠ_2^0$ subspaces of $\mathcal{P}(\mathbb{N})$, and sober spaces of countably presented frames. In this paper, we formalize these structures within second-order arithmetic and conduct a systematic reverse mathematical analysis of the transitions between them.
The thin set theorem $\mathsf{RT}_{<\infty,\ell}^{n}$ asserts that for every natural number $k$, each coloring $c\colon[\mathbb{N}]^n \to \{0,1,\dots,k-1\}$ admits an infinite set $H$ such that $|c([H]^n)| \le \ell$. Within the framework of the reverse mathematics of second-order arithmetic, $\mathsf{RT}_{<\infty,\ell}^{n}$ implies the $Σ_{n+1}^{0}$-bounding principle ($\mathsf{B}Σ_{n+1}^{0}$) over $\mathsf{RCA}_0$ for all natural numbers $n, \ell \ge 1$.
This paper investigates the logical strength of two theorems in modal propositional logic - the Hennessy-Milner theorem and the van Benthem characterization theorem - within the framework of second-order arithmetic. We demonstrate that the Hennessy-Milner theorem is equivalent to $\mathrm{ACA}_0$ over $\mathrm{RCA}_0$. For the van Benthem characterization theorem, we introduce three variants: the semantic, syntactic, and hybrid forms. We show that the semantic form is provable in $\mathrm{RCA}_0$, the syntactic form is provable in $\mathrm{PRA}$, and the hybrid form is equivalent to the weak completeness theorem for first-order logic over $\mathrm{RCA}_0$.
In this note, we give some generalizations of Gödel's second incompleteness theorem and study their surroundings. We revisit it from two perspectives. One perspective is the relationship between the definable complexity of a theory and unprovability of its soundness. We clarify the relationship between this perspective and induction axioms. We also determine the logical strength of Craig's trick, which is important for studying the definability of a theory, from the point of view of reverse mathematics. The other perspective is semantic incompleteness. The second incompleteness theorem may be seen as the unprovability of the existence of a model. It is known that `model' is replaced with `$ω$-model' or `$β_n$-model'. We give a new and unified proof of the $ω$-model and $β_n$-model versions of the incompleteness theorem.
In this article, we prove that Ramsey's theorem for pairs and two colors is $Π^1_1$-conservative over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \mathsf{WF}(ε_0)$ and over~$\mathsf{RCA}_0 + \mathsf{B}Σ^0_2 + \bigcup_n \mathsf{WF}(ω^ω_n)$. These results improve theorems from Chong, Slaman and Yang and Kołodziejczyk and Yokoyama and belong to a long line of research towards the characterization of the first-order part of Ramsey's theorem for pairs.
In this article, we prove that Ramsey's theorem for pairs and two colors is a $\forall Π^0_4$ conservative extension of $\mathsf{RCA}_0 + \mathsf{B}Σ^0_2$, where a $\forall Π^0_4$ formula consists of a universal quantifier over sets followed by a $Π^0_4$ formula. The proof is an improvement of a result by Patey and Yokoyama and a step towards the resolution of the longstanding question of the first-order part of Ramsey's theorem for pairs.
A classical theorem of Lusin states that all analytic sets are Lebesgue-measurable. In this article we established the reverse mathematical strength of Lusin's theorem, which depends on how precisely it is formalized. By doing so, we answer to a question of Simpson. Our main proof is motivated towards proving a specific version of that result, namely that analytic sets are Lesbesgue-regular, which requires the equality of the outer and inner measures of the set in question. We prove this statement to be equivalent to $Σ^{1}_{1}$-$\mathrm{IND}$ over $\mathrm{ATR}_{0}$. The full statement of the theorem, that is the one implying the existence of the measure as a real number, is equivalent to $Π^{1}_{1}$-$\mathrm{CA}_{0}$, again provably over $\mathrm{ATR}_{0}$. In our main proof, we draw inspiration from Solovay's construction of a model of Zermelo-Fraenkel set theory where every set is Lebesgue measurable. In our case the argument requires the use of class forcing over a family of standard and non-standard models of a very weak set theory obtained through the method of pseudohierarchies.
We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}Σ^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}Σ^0_{3}}$. The $\ll^2$-basis theorem is a variant of the low basis theorem, which has recently received focus in the context of the first-order part of Ramsey type theorems. For the construction, we use Mathias forcing in an effectively coded $ω$-model of $\mathsf{WKL_0}$ to ensure sufficient computability under the system with weaker induction. Using a similar method, we also show the $\ll^2$-basis theorem for $\mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$, a version of Erdős-Moser principle, within $\mathrm{I}Σ^0_{2}$. These results provide simpler proofs of known results on the $Π^1_1$-conservativities of $\mathrm{RT}^2, \mathrm{RT}^2_2$ and $\mathrm{EM}_{<\infty}$ as corollaries.
Current acoustic radiation force (ARF) based methods for quantifying tissue elasticity primarily rely on shear wave propagation. However, their spatial resolution is limited by the need for spatial averaging, and their accuracy is affected by shear wave guidance, out of plane reflections, and geometric dispersion, which reduce their applicability in mechanically complex tissues. This study introduces a novel technique called Double Profile Intersection (DoPIo) ultrasound, which enables pointwise estimation of shear elastic modulus within the region of ARF excitation by leveraging the scatterer shearing rate. This rate is inferred by tracking ARF induced displacement using two tracking beams with different lateral widths. The wider beam captures scatterers located outside the ARF excitation region that begin to displace as shearing propagates. The time at which the two resulting displacement profiles intersect is mapped to shear elastic modulus using an empirically derived model based on finite element simulations. In silico, DoPIo estimated shear elastic modulus with a median error of -0.02 kPa and a median absolute deviation of 1.98 kPa in elastic materials up to 35 kPa. Experimental validation in vitro and ex vivo demonstrated that DoPIo reliably distinguished softer regions from stiffer ones, and its modulus estimates remained consistent across varying ARF push amplitudes, provided sufficient displacement estimation signal to noise ratio. DoPIo offers a feasible approach for high resolution, on axis shear elasticity estimation and holds promise as a quantitative biomarker that is independent of ARF amplitude.
This paper investigates the logical strength of completeness theorems for modal propositional logic within second-order arithmetic. We demonstrate that the weak completeness theorem for modal propositional logic is provable in $\mathrm{RCA}_0$, and that, over $\mathrm{RCA}_0$, $\mathrm{ACA}_0$ is equivalent to the strong completeness theorem for modal propositional logic using canonical models. We also consider a simpler version of the strong completeness theorem without referring to canonical models and show that it is equivalent to $\mathrm{WKL}_0$ over $\mathrm{RCA}_0$.
In this paper, we give characterizations of the set of $Π^1_{e}$-consequences, $Σ^1_{e}$-consequences and $\mathsf{B}(Π^1_{e})$-consequences of the axiomatic system of the strong dependent choice for $Σ^1_i$ formulas $Σ^1_i$-$\mathsf{SDC}_0$ for $i > 0$ and $e < i+2$. Here, $\mathsf{B}(Γ)$ denotes the set generated by $\land,\lor,\lnot$ starting from $Γ$.
In this paper, we introduce a hierarchy dividing the set $\{σ\in Π^1_2 : Π^1_1$-$\mathsf{CA}_0 \vdash σ\}$. Then, we give some characterizations of this set using weaker variants of some principles equivalent to $Π^1_1$-$\mathsf{CA}_0$: leftmost path principle, Ramsey's theorem for $Σ^0_n$ classes of $[\mathbb{N}]^{\mathbb{N}}$ and determinacy for $(Σ^0_1)_n$ classes of $\mathbb{N}^{\mathbb{N}}$.
We investigate some Weihrauch problems between $\mathsf{ATR}_2$ and $\mathsf{C}_{ω^ω}$ . We show that the fixed point theorem for monotone operators on the Cantor space (a weaker version of the Knaster-Tarski theorem) is not Weihrauch reducible to $\mathsf{ATR}_2$. Furthermore, we introduce the $ω$-model reflection $\mathsf{ATR}_2^{\mathrm{rfn}}$ of $\mathsf{ATR} $ and show that it is an upper bound for problems provable from the axiomatic system $\mathrm{ATR}_0$ which are of the form $\forall X(θ(X) \to \exists Y η(X, Y ))$ with arithmetical formulas $θ, η$. We also show that Weihrauch degrees of relativized least fixed point theorem for monotone operators on the Cantor space forms a linear hierarchy between $\mathsf{ATR}^{\mathrm{rfn}}$ and $\mathsf{C}_{ω^ω} $.
It is known that several variations of the axiom of determinacy play important roles in the study of reverse mathematics, and the relation between the hierarchy of determinacy and comprehension are revealed by Tanaka, Nemoto, Montalbán, Shore, and others. We prove variations of a result by Kołodziejczyk and Michalewski relating determinacy of arbitrary boolean combinations of $Σ^0_2$ sets and reflection in second-order arithmetic. Specifically, we prove that: over $\mathsf{ACA}_0$, $Π^1_2$-$\mathsf{Ref}(\mathsf{ACA}_0)$ is equivalent to $\forall n.(Σ^0_1)_n$-$\mathsf{Det}^*_0$; $Π^1_3$-$\mathsf{Ref}(Π^1_1$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(Σ^0_1)_n$-$\mathsf{Det}$; and $Π^1_3$-$\mathsf{Ref}(Π^1_2$-$\mathsf{CA}_0)$ is equivalent to $\forall n.(Σ^0_2)_n$-$\mathsf{Det}$. We also restate results by Montalbán and Shore to show that $Π^1_3$-$\mathsf{Ref}(\mathsf{Z}_2)$ is equivalent to $\forall n.(Σ^0_3)_n$-$\mathsf{Det}$ over $\mathsf{ACA}_0$.
We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in $\mathsf{RCA}_0$. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between $\mathsf{ATR}_0$ and $Π^1_1\mbox{-}\mathsf{CA}_0$. We also exhibit several weakenings of Caristi's theorem that are equivalent to $\mathsf{WKL}_0$ and to $\mathsf{ACA}_0$.
We introduce the notion of the \emph{first-order part} of a problem in the Weihrauch degrees. Informally, the first-order part of a problem $\mathsf{P}$ is the strongest problem with codomaixn $ω$ that is Weihrauch reducible to $\mathsf{P}$. We show that the first-order part is always well-defined, examine some of the basic properties of this notion, and characterize the first-order parts of several well-known problems from the literature.
We study the parameterized complexity of the problem to decide whether a given natural number $n$ satisfies a given $Δ_0$-formula $φ(x)$; the parameter is the size of $φ$. This parameterization focusses attention on instances where $n$ is large compared to the size of $φ$. We show unconditionally that this problem does not belong to the parameterized analogue of $\mathsf{AC}^0$. From this we derive that certain natural upper bounds on the complexity of our parameterized problem imply certain separations of classical complexity classes. This connection is obtained via an analysis of a parameterized halting problem. Some of these upper bounds follow assuming that $IΔ_0$ proves the MRDP theorem in a certain weak sense.
We prove that if $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are countable models of the theory $\mathrm{WKL}^*_0$ such that $\mathrm{I}Σ_1(A)$ fails for some $A \in \mathcal{X} \cap \mathcal{Y}$, then $(M,\mathcal{X})$ and $(M,\mathcal{Y})$ are isomorphic. As a consequence, the analytic hierarchy collapses to $Δ^1_1$ provably in $\mathrm{WKL}^*_0 + \neg\mathrm{I}Σ^0_1$, and $\mathrm{WKL}$ is the strongest $Π^1_2$ statement that is $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \neg\mathrm{I}Σ^0_1$. Applying our results to the $Δ^0_n$-definable sets in models of $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ that also satisfy an appropriate relativization of Weak König's Lemma, we prove that for each $n \ge 1$, the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n + \neg\mathrm{I}Σ^0_n$ is c.e. In contrast, we prove that the set of $Π^1_2$ sentences that are $Π^1_1$-conservative over $\mathrm{RCA}^*_0 + \mathrm{B}Σ^0_n$ is $Π_2$-complete. This answers a question of Towsner. We also show that $\mathrm{RCA}_0 + \mathrm{RT}^2_2$ is $Π^1_1$-conservative over $\mathrm{B}Σ^0_2$ if and only if it is conservative over $\mathrm{B}Σ^0_2$ with respect to $\forall Π^0_5$ sentences.