Calibrating the complexity of combinatorics: reverse mathematics and Weihrauch degrees of some principles related to Ramsey’s theorem

In this thesis, we study the proof-theoretical and computational strength of some combinatorial principles related to Ramsey's theorem: this will be accomplished chiefly by analyzing these principles from the points of view of reverse mathematics and Weihrauch complexity. We start by studying a combinatorial principle concerning graphs, introduced by Bill Rival and Ivan Sands as a form of ``inside-outside'' Ramsey's theorem: we will determine its reverse mathematical strength and present the result characterizing its Weihrauch degree. Moreover, we will study a natural restriction of this principle, proving that it is equivalent to Ramsey's theorem. We will then move to a related result, this time concerning countable partial orders, again introduced by Rival and Sands: we will give a thorough reverse mathematical investigation of the strength of this theorem and of its original proof. Moreover, we will be able to generalize it, and this generalization will itself be presented in the reverse mathematical perspective. After this, we will focus on two forms of Ramsey's theorem that can be considered asymmetric. First, we will focus on a restriction of Ramsey's theorem to instances whose solutions have a predetermined color, studying it in reverse mathematics and from the point of view of the complexity of the solutions in a computability theoretic sense. Next, we move to a classical result about partition ordinals, which will undergo the same type of analysis. Finally, we will present some results concerning a recently introduced operator on the Weihrauch degrees, namely the first-order part operator: after presenting an alternative characterization of it, we will embark on the study the result of its applications to jumps of Weak Kőnig's Lemma

The logical strength of Büchi's decidability theorem

We study the strength of axioms needed to prove various results related to automata on infinite words and Büchi's theorem on the decidability of the MSO theory of (N, less_or_equal). We prove that the following are equivalent over the weak second-order arithmetic theory RCA: 1. Büchi's complementation theorem for nondeterministic automata on infinite words, 2. the decidability of the depth-n fragment of the MSO theory of (N, less_or_equal), for each n greater than 5, 3. the induction scheme for Sigma^0_2 formulae of arithmetic. Moreover, each of (1)-(3) is equivalent to the additive version of Ramsey's Theorem for pairs, often used in proofs of (1); each of (1)-(3) implies McNaughton's determinisation theorem for automata on infinite words; and each of (1)-(3) implies the "bounded-width" version of König's Lemma, often used in proofs of McNaughton's theorem

An axiomatic approach to sustainable development

The paper proposes two axioms that capture the idea of sustainable development and derives the welfare criterion that they imply. The axioms require that neither the present nor the future should play a dictatorial role. Theorem 1 shows there exist sustainable preferences, which satisfy these axioms. They exhibit sensitivity to the present and to the long-run future, and specify trade-offs between them. It examines other welfare criteria which are generally utilized: discounted utility, lim inf. long run averages, overtaking and catching-up criteria, Ramsey's criterion, Rawlsian rules, and the criterion of satisfaction of basic needs, and finds that none satisfies the axioms for sustainability. Theorem 2 gives a characterization of all continuous independent sustainable preferences. Theorem 3 shows that in general sustainable growth paths cannot be approximated by paths which approximate discounted optima. Proposition 1 shows that paths which maximize the present value under a standard price system may fail to reach optimal sustainable welfare levels, and Example 4 that the two criteria can give rise to different value systems.sustainable development; economic development; welfare

Coloring trees in reverse mathematics

The tree theorem for pairs ($\mathsf{TT}^2_2$), first introduced by Chubb, Hirst, and McNicholl, asserts that given a finite coloring of pairs of comparable nodes in the full binary tree $2^{<\omega}$, there is a set of nodes isomorphic to $2^{<\omega}$ which is homogeneous for the coloring. This is a generalization of the more familiar Ramsey's theorem for pairs ($\mathsf{RT}^2_2$), which has been studied extensively in computability theory and reverse mathematics. We answer a longstanding open question about the strength of $\mathsf{TT}^2_2$, by showing that this principle does not imply the arithmetic comprehension axiom ($\mathsf{ACA}_0$) over the base system, recursive comprehension axiom ($\mathsf{RCA}_0$), of second-order arithmetic. In addition, we give a new and self-contained proof of a recent result of Patey that $\mathsf{TT}^2_2$ is strictly stronger than $\mathsf{RT}^2_2$. Combined, these results establish $\mathsf{TT}^2_2$ as the first known example of a natural combinatorial principle to occupy the interval strictly between $\mathsf{ACA}_0$ and $\mathsf{RT}^2_2$. The proof of this fact uses an extension of the bushy tree forcing method, and develops new techniques for dealing with combinatorial statements formulated on trees, rather than on $\omega$.Comment: 25 page

Making proofs without Modus Ponens: An introduction to the combinatorics and complexity of cut elimination

This paper is intended to provide an introduction to cut elimination which is accessible to a broad mathematical audience. Gentzen's cut elimination theorem is not as well known as it deserves to be, and it is tied to a lot of interesting mathematical structure. In particular we try to indicate some dynamical and combinatorial aspects of cut elimination, as well as its connections to complexity theory. We discuss two concrete examples where one can see the structure of short proofs with cuts, one concerning feasible numbers and the other concerning "bounded mean oscillation" from real analysis

An algorithmic approach for multi-color Ramsey graphs

The classical Ramsey number R(r1,r2,...,rm) is defined to be the smallest integer n such that no matter how the edges of Kn are colored with the m colors, 1, 2, 3, . . . ,m, there exists some color i such that there is a complete subgraph of size ri, all of whose edges are of color i. The problem of determining Ramsey numbers is known to be very difficult and is usually split into two problems, finding upper and lower bounds. Lower bounds can be obtained by the construction of a, so called, Ramsey graph. There are many different methods to construct Ramsey graphs that establish lower bounds. In this thesis mathematical and computational methods are exploited to construct Ramsey graphs. It was shown that the problem of checking that a graph coloring gives a Ramsey graph is NP-complete. Hence it is almost impossible to find a polynomial time algorithm to construct Ramsey graphs by searching and checking. Consequently, a method such as backtracking with good pruning techniques should be used. Algebraic methods were developed to enable such a backtrack search to be feasible when symmetry is assumed. With the algorithm developed in this thesis, two new lower bounds were established: R(3,3,5)≥45 and R(3,4,4)≥55. Other best known lower bounds were matched, such as R(3,3,4)≥30. The Ramsey graphs giving these lower bounds were analyzed and their full symmetry groups were determined. In particular it was shown that there are unique cyclic graphs up to isomorphism giving R(3,3,4)≥30 and R(3,4,4)≥55, and 13 non-isomorphic cyclic graphs giving R(3,3,5)≥45

Towards an epistemic theory of probability.

The main concern of this thesis is to develop an epistemic conception of probability. In chapter one we look at Ramsey's work. In addition to his claim that the axioms of probability ace laws of consistency for partial beliefs, we focus attention on his view that the reasonableness of our probability statements does not consist merely in such coherence, but is to be assessed through the vindication of the habits which give rise to them. In chapter two we examine de Finetti's account, and compare it with Ramsey's. One significant point of divergence is de Finetti's claim that coherence is the only valid form of appraisal for probability statements. His arguments for this position depend heavily on the implementation of a Bayesian model for belief change; we argue that such an approach fails to give a satisfactory account of the relation between probabilities and objective facts. In chapter three we stake out the ground for oar own positive proposals - for an account which is non-objective in so far as it does not require the postulation of probabilistic facts, but non-subjective in the sense that probability statements are open to objective forms of appraisal. we suggest that a certain class of probability statements are best interpreted as recommendations of partial belief; these being measurable by the betting quotients that one judges to be fair. Moreover, we argue that these probability statements are open to three main forms of appraisal (each quantifiable through the use of proper scoring rules), namely: (i) Coherence (ii) Calibration (iii) Refinement. The latter two forms of appraisal are applicable both in an ex ante sense (relative to the information known by the forecaster) and an ex post one (relative to the results of the events forecast). In chapters four and five we consider certain problems which confront theories of partial belief; in particular, (1) difficulties surrounding the justification of the rule to maximise one's information, and (2) problems with the ascription of probabilities to mathematical propositions. Both of these issues seem resolvable; the first through the principle of maximising expected utility (SEU), and the second either by amending the axioms of probability, or by making use of the notion that probabilities are appraisable via scoring rules. There do remain, however, various difficulties with SEU, in particular with respect to its application in real-life situations. These are discussed, but no final conclusion reached, except that an epistemic theory such as ours is not undermined by the inapplicability of SEU in certain situations

Taxing Human Capital Efficiently when Qualified Labour is Mobile

The paper studies the effect that skilled labour mobility has on efficient education policy. The model is one of two periods in which a representative taxpayer decides on labour, education, and saving. The government can only use linear tax and subsidy instruments. It is shown that the mobility of skilled labour well constrains government’s choice of policy instruments. The mobility does not however affect second best education policy in allocational terms. In particular, education should be effectively subsidized if, and only if, the elasticity of the earnings function is increasing in education. This rule applies regardless of whether labour is mobile or immobile.mobile labour, second-best efficient taxation, linear instruments, residence vs. source principle

Partition Theorems for Spaces of Variable Words

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135422/1/plms0449.pd

The strength of the tree theorem for pairs in reverse mathematics

International audienceNo natural principle is currently known to be strictly between the arithmetic comprehension axiom (ACA0) and Ramsey's theorem for pairs (RT 2 2) in reverse mathematics. The tree theorem for pairs (TT 2 2) is however a good candidate. The tree theorem states that for every finite coloring over tuples of comparable nodes in the full binary tree, there is a monochromatic subtree isomorphic to the full tree. The principle TT 2 2 is known to lie between ACA0 and RT 2 2 over RCA0, but its exact strength remains open. In this paper, we prove that RT 2 2 together with weak König's lemma (WKL0) does not imply TT 2 2 , thereby answering a question of Montálban. This separation is a case in point of the method of Lerman, Solomon and Towsner for designing a computability-theoretic property which discriminates between two statements in reverse mathematics. We therefore put the emphasis on the different steps leading to this separation in order to serve as a tutorial for separating principles in reverse mathematics