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Richard A. Shore

Publications and source records attributed to Richard A. Shore.

8 recordsLinked to original sources
arXiv2023

Halin's Infinite Ray Theorems: Complexity and Reverse Mathematics: Version E

Halin [1965] proved that if a graph has $n$ many pairwise disjoint rays for each $n$ then it has infinitely many pairwise disjoint rays. We analyze the complexity of this and other similar results in terms of computable and proof theoretic complexity. The statement of Halin's theorem and the construction proving it seem very much like standard versions of compactness arguments such as König's Lemma. Those results, while not computable, are relatively simple. They only use arithmetic procedures or, equivalently, finitely many iterations of the Turing jump. We show that several Halin type theorems are much more complicated. They are among the theorems of hyperarithmetic analysis. Such theorems imply the ability to iterate the Turing jump along any computable well ordering. Several important logical principles in this class have been extensively studied beginning with work of Kreisel, H. Friedman, Steel and others in the 1960s and 1970s. Until now, only one purely mathematical example was known. Our work provides many more and so answers Question 30 of Montalbán's Open Questions in Reverse Mathematics [2011]. Some of these theorems including ones in Halin [1965] are also shown to have unusual proof theoretic strength as well.

math.LO↗
arXiv2019

The Strength of the Grätzer-Schmidt Theorem

The Grätzer-Schmidt theorem of lattice theory states that each algebraic lattice is isomorphic to the congruence lattice of an algebra. We study the reverse mathematics of this theorem. We also show that the set of indices of computable lattices that are complete is $Π^1_1$-complete; the set of indices of computable lattices that are algebraic is $Π^1_1$-complete; the set of compact elements of a computable lattice is $Π^{1}_{1}$ and can be $Π^1_1$-complete; and the set of compact elements of a distributive computable lattice is $Π^{0}_{3}$, and there is an algebraic distributive computable lattice such that the set of its compact elements is $Π^0_3$-complete.

math.LO↗
arXiv2014

Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $% \mathcal{D}_{h}$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $ω_{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $ω_{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$.

math.LO↗
arXiv2012

Computing Maximal Chains

In 1967 Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable wpos. Wolk's original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets.

math.LO↗
arXiv2011

Computably enumerable partial orders

We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the former is strictly stronger than the latter. We then show that every $\emptyset'$-computable structure (or even just of c.e.\ degree) has the same degree spectrum as some computably enumerable (co-c.e.)\ partial order, and hence that there is a c.e.\ (co-c.e.)\ partial order with spectrum equal to the set of nonzero degrees.

math.LO↗
arXiv2002

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

cs.LO↗
arXiv1999

Reasoning About Common Knowledge with Infinitely Many Agents

Complete axiomatizations and exponential-time decision procedures are provided for reasoning about knowledge and common knowledge when there are infinitely many agents. The results show that reasoning about knowledge and common knowledge with infinitely many agents is no harder than when there are finitely many agents, provided that we can check the cardinality of certain set differences G - G', where G and G' are sets of agents. Since our complexity results are independent of the cardinality of the sets G involved, they represent improvements over the previous results even with the sets of agents involved are finite. Moreover, our results make clear the extent to which issues of complexity and completeness depend on how the sets of agents involved are represented.

cs.LO↗