Paul Shafer

• PhD
• Lecturer at University of Leeds

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C . Our main results are the following. The...

We analyze the axiomatic strength of the following theorem due to Rival and Sands in the style of reverse mathematics. "Every infinite partial order $P$ of finite width contains an infinite chain $C$ such that every element of $P$ is either comparable with no element of $C$ or with infinitely many elements of $C$." Our main results are the followin...

A quasi-order Q induces two natural quasi-orders on \({\mathcal{P}(Q)}\), but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-o...

An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a \cup c = b$. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and W...

We prove that several versions of the Tietze extension theorem for functions with moduli of uniform continuity are equivalent to WKL_0 over RCA_0. This confirms a conjecture of Giusto and Simpson that was also phrased as a question in Montalb\'an's "Open questions in reverse mathematics."

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 [‘Reverse mathematics and a Ramsey-type König's lemma’, J. Symb. Log. 77 (2012) 1272–1280], we say that a Ramsey-type variant o...

A Martin-Löf test is universal if it captures all non-Martin-Löf random sequences, and it is optimal if for every ML-test there is a such that . We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of ,...

A quasi-order $Q$ induces two natural quasi-orders on $P(Q)$, but if $Q$ is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq showed that moving from a well-quasi-order $Q$ to the quasi-orders on $P(Q)$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault...

We prove that neither the statement "there is a $k$ such that for every $f$ there is a $k$-bounded diagonally non-recursive function relative to $f$" nor the statement "for every $\ell$ there is a $k$ such that every finitely $\ell$-colorable graph is $k$-colorable" imply weak K\"onig's lemma over $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$. This answe...

A semi-measure is a generalization of a probability measure obtained by relaxing the additivity requirement to super-additivity. We introduce and study several randomness notions for left-c.e. semi-measures, a natural class of effectively approximable semi-measures induced by Turing functionals. Among the randomness notions we consider, the general...

The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic. In this setting, called reverse mathematics, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take pl...

This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section 2, we show that $\mathsf{WKL}_0$ is equivalent to the ability to extend $F$-automorphisms of field extensions to automorphisms of $\bar{F}$, the algebraic closure of $F$. Section 3 explores finitary conditions for embeddability. Nor...

We prove Menger’s theorem for countable graphs in ${{\Pi^1_1\tt{-CA}_0}}$ . Our proof in fact proves a stronger statement, which we call extended Menger’s theorem, that is equivalent to ${{\Pi^1_1\tt{-CA}_0}}$ over ${{\tt{RCA}_0}}$ .

Let Es denote the lattice of Medvedev degrees of non-empty Π10 subsets of 2ω, and let Ew denote the lattice of Muchnik degrees of non-empty Π10 subsets of 2ω. We prove that the first-order theory of Es as a partial order is recursively isomorphic to the first-order theory of true arithmetic. Our coding of arithmetic in Es also shows that the Σ30-th...

Using the techniques of reverse mathematics, we analyze the log-ical strength of statements similar to trichotomy and dichotomy for sequences of reals. Capitalizing on the connection between sequen-tial statements and constructivity, we use computable restrictions of the statements to indicate forms provable in formal systems of weak constructive a...

We prove that the first-order theory of the Medvedev degrees, the first-order theory of the Muchnik degrees, and the third-order theory of true arithmetic are pairwise recursively isomorphic (obtained independently by Lewis, Nies, and Sorbi [7]). We then restrict our attention to the degrees of closed sets and prove that the following theories are...

We characterize the join-irreducible Medvedev degrees as the degrees of complements of Turing ideals, thereby solving a problem posed by Sorbi. We use this characterization to prove that there are Medvedev degrees above the second-least degree that do not bound any join-irreducible degrees above this second-least degree. This solves a problem posed...

Experimental design and linear models.

In the mouse olfactory system, the role of the olfactory bulb in guiding olfactory sensory neuron (OSN) axons to their targets is poorly understood. What cell types within the bulb are necessary for targeting is unknown. What genes are important for this process is also unknown. Although projection neurons are not required, other cell-types within...

Appendix The EST2Prot webserver. The appendix takes the reader through the main web pages that make-up the EST2Prot webserver, with explanations of the output format of each page and snapshots that exemplify the type of information provided.

EST libraries are used in various biological studies, from microarray experiments to proteomic and genetic screens. These libraries usually contain many uncharacterized ESTs that are typically ignored since they cannot be mapped to known genes. Consequently, new discoveries are possibly overlooked. We describe a system (EST2Prot) that uses multiple...

Existing biological databases support a variety of queries such as keyword or definition search. However, they do not provide any measure of relevance for the instances reported, and result sets are usually sorted arbitrarily. We describe a system that builds upon the complex infrastructure of the Biozon database and applies methods similar to thos...