Epsilon substitution for transfinite induction

We apply Mints technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.     

Order-enriched categorical models of the classical sequent calculus

It is well-known that weakening and contraction cause naïve categorical models of the classical sequent calculus to collapse to Boolean lattices. Starting from a convenient formulation of the well-known categorical semantics of linear classical sequent proofs, we give models of weakening and contraction that do not collapse. Cut-reduction is interpreted by a partial order between morphisms. Our models make no commitment to any translation of classical logic into intuitionistic logic and distinguish non-deterministic choices of cut-elimination. We show soundness and completeness via initial models built from proof nets, and describe models built from sets and relations.     

Unprovability threshold for the planar graph minor theorem

This note is part of the implementation of a programme in foundations of mathematics to find exact threshold versions of all mathematical unprovability results known so far, a programme initiated by Weiermann. Here we find the exact versions of unprovability of the finite graph minor theorem with growth rate condition restricted to planar graphs, connected planar graphs and graphs embeddable into a given surface, assuming an unproved conjecture (*): ‘there is a number a>0 such that for all k≥3, and all n≥1, the proportion of connected graphs among unlabelled planar graphs of size n omitting the k-element circle as minor is greater than a’. Let γ be the unlabelled planar growth constant (27.2269≤γ<30.061). Let P(c) be the following first-order arithmetical statement with real parameter c: “for every K there is N such that whenever G₁,G₂,…,G_N are unlabelled planar graphs with |G_i|<K+c⋅log₂i then for some i<j≤N, G_i is isomorphic to a minor of G_j”. Then

1.

for every , P(c) is provable in IΔ₀+exp;

2.

for every , P(c) is unprovable in .

We also give proofs of some upper and lower bounds for unprovability thresholds in the general case of the finite graph minor theorem.     

Kripke models for classical logic

We introduce a notion of the Kripke model for classical logic for which we constructively prove the soundness and cut-free completeness. We discuss the novelty of the notion and its potential applications.     

On the constructive Dedekind reals

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class.     

Epsilon substitution method for elementary analysis

We formulate epsilon substitution method for elementary analysisEA (second order arithmetic with comprehension for arithmetical formulas with predicate parameters). Two proofs of its termination are presented. One uses embedding into ramified system of level one and cutelimination for this system. The second proof uses non-effective continuity argument.     

Epsilon substitution method for [Π0 1, Π0 1]-FIX

We formulate epsilon substitution method for a theory [Π⁰₁, Π⁰₁]-FIX for two steps non-monotonic Π⁰₁ inductive definitions. Then we give a termination proof of the H-processes based on Ackermann [1].     

A local limit theorem for random walks conditioned to stay positive

We consider a real random walk S_n=X₁+...+X_n attracted (without centering) to the normal law: this means that for a suitable norming sequence a_n we have the weak convergence S_n/a_n⇒ϕ(x)dx, ϕ(x) being the standard normal density. A local refinement of this convergence is provided by Gnedenko's and Stone's Local Limit Theorems, in the lattice and nonlattice case respectively. Now let denote the event (S₁>0,...,S_n>0) and let S_n⁺ denote the random variable S_n conditioned on : it is known that S_n⁺/a_n ↠ ϕ⁺(x) dx, where ϕ⁺(x):=x exp (−x²/2)1_(x≥0). What we establish in this paper is an equivalent of Gnedenko's and Stone's Local Limit Theorems for this weak convergence. We also consider the particular case when X₁ has an absolutely continuous law: in this case the uniform convergence of the density of S_n⁺/a_n towards ϕ⁺(x) holds under a standard additional hypothesis, in analogy to the classical case. We finally discuss an application of our main results to the asymptotic behavior of the joint renewal measure of the ladder variables process. Unlike the classical proofs of the LLT, we make no use of characteristic functions: our techniques are rather taken from the so–called Fluctuation Theory for random walks.     

Almost strictly totally positive matrices

A determinantal identity, frequently used in the study of totally positive matrices, is extended, and then used to re-prove the well-known univariate knot insertion formula for B-splines. Also we introduce a class of matrices, intermediate between totally positive and strictly totally positive matrices. The determinantal identity is used to show any minor of such matrices is positive if and only if its diagonal entries are positive. Among others, this class of matrices includes B-splines collocation matrices and Hurwitz matrices.This author acknowledges a sabbatical stay at IBM T.J. Watson Research Center in 1990, which was supported by a DGICYT grant from Spain.     

Cut-free formulations for a quantified logic of here and there

A predicate extension SQHT⁼ of the logic of here-and-there was introduced by V. Lifschitz, D. Pearce, and A. Valverde to characterize strong equivalence of logic programs with variables and equality with respect to stable models. The semantics for this logic is determined by intuitionistic Kripke models with two worlds (here and there) with constant individual domain and decidable equality. Our sequent formulation has special rules for implication and for pushing negation inside formulas. The soundness proof allows us to establish that SQHT⁼ is a conservative extension of the logic of weak excluded middle with respect to sequents without positive occurrences of implication. The completeness proof uses a non-closed branch of a proof search tree. The interplay between rules for pushing negation inside and truth in the “there” (non-root) world of the resulting Kripke model can be of independent interest. We prove that existence is definable in terms of remaining connectives.     

Generalizing realizability and Heyting models for constructive set theory

This article presents a common generalization of the two main methods for obtaining class models of constructive set theory. Heyting models are a generalization of the Boolean models for classical set theory which are a variant of forcing, while realizability is a decidedly constructive method that has first been developed for number theory by Kleene and was later very fruitfully adapted to constructive set theory. In order to achieve the generalization, a new kind of structure (applicative topologies) is introduced, which contains both elements of formal topology and applicative structures. This approach not only deepens the understanding of class models and leads to more efficiency in proofs about these kinds of models, but also makes it possible to prove new results about the two special cases that were not known before and to construct new models.     

A new rating system for duplicate bridge

We propose a new way to rate individual duplicate bridge players, which we believe is superior to the masterpoint system currently used by the American Contract Bridge League. This method measures only a player’s current skill level, and not how long or how frequently he has played. It is based on simple ideas from the theory of statistics and from linear algebra, and should be easy to implement.One particular issue which can occur within any system proposing to rate individual players using results earned by partnerships is what we call the “nonuniqueness problem”. This refers to the occasional inability for data to distinguish who is the “good player” and who is the “bad player” within particular partnerships. We prove that under our system this problem disappears if either (a) a certain “partnership graph” has no bipartite components, or if (b) every player is required to participate in at least one individual game.Finally, we present some data from a bridge club in Reno, NV. They show that even if (a) and (b) do not hold, our system will provide (unique) ratings for most players.     

Proof internalization in generalized Frege systems for classical logic

We present a general method for inserting proofs in Frege systems for classical logic that produces systems that can internalize their own proofs.     

Proof-theoretic conservations of weak weak intuitionistic constructive set theories

The paper aims to provide precise proof theoretic characterizations of Myhill–Friedman-style “weak” constructive extensional set theories and Aczel–Rathjen analogous constructive set theories both enriched by Mostowski-style collapsing axioms and/or related anti-foundation axioms. The main results include full intuitionistic conservations over the corresponding purely arithmetical formalisms that are well known in the reverse mathematics – which strengthens analogous results obtained by the author in the 80s. The present research was inspired by the more recent Sato-style “weak weak” classical extensional set theories whose proof theoretic strengths are shown to strongly exceed the ones of the intuitionistic counterparts in the presence of the collapsing axioms.     

Wellfoundedness proofs by means of non-monotonic inductive definitions II: First order operators

In this paper, we give two proofs of the wellfoundedness of a recursive notation system for Π_N-reflecting ordinals. One is based on distinguished classes, and the other is based on -inductive definitions.     

Classical proof forestry

Classical proof forests are a proof formalism for first-order classical logic based on Herbrand’s Theorem and backtracking games in the style of Coquand. First described by Miller in a cut-free setting as an economical representation of first-order and higher-order classical proof, defining features of the forests are a strict focus on witnessing terms for quantifiers and the absence of inessential structure, or ‘bureaucracy’.This paper presents classical proof forests as a graphical proof formalism and investigates the possibility of composing forests by cut-elimination. Cut-reduction steps take the form of a local rewrite relation that arises from the structure of the forests in a natural way. Yet reductions, which are significantly different from those of the sequent calculus, are combinatorially intricate and do not exclude the possibility of infinite reduction traces, of which an example is given.Cut-elimination, in the form of a weak normalisation theorem, is obtained using a modified version of the rewrite relation inspired by the game-theoretic interpretation of the forests. It is conjectured that the modified reduction relation is, in fact, strongly normalising.     

Interpolation property for bicartesian closed categories

Summary We show that proofs in the intuitionistic propositional logic factor through interpolants-in this way we prove a stronger interpolation property than the usual one which gives only the existence of interpolants.Translating that to categorical terms, we show that Pushouts (bipushouts) of bicartesian closed categories have the interpolation property (Theorem 3.2).     

A semantical proof of De Jongh's theorem

Summary In 1969, De Jongh proved the maximality of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all true ₂ ⁰ -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA.