On interpretations of bounded arithmetic and bounded set theory

In a recent paper, Kaye and Wong proved the following result, which they considered to belong to the folklore of mathematical logic. THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom of infinity negated are bi-interpretable: that is, they are mutually interpretable with interpretations that are inverse to each other. In this note, I describe a theory of sets that stands in the same relation to the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of arithmetic in set theory. Instead, I am forced to produce a different interpretation.Comment: 12 pages; section on omega-models removed due to error; references added and typos correcte

Unifying Functional Interpretations: Past and Future

This article surveys work done in the last six years on the unification of various functional interpretations including G\"odel's dialectica interpretation, its Diller-Nahm variant, Kreisel modified realizability, Stein's family of functional interpretations, functional interpretations "with truth", and bounded functional interpretations. Our goal in the present paper is twofold: (1) to look back and single out the main lessons learnt so far, and (2) to look forward and list several open questions and possible directions for further research.Comment: 18 page

Formalizing Termination Proofs under Polynomial Quasi-interpretations

Usual termination proofs for a functional program require to check all the possible reduction paths. Due to an exponential gap between the height and size of such the reduction tree, no naive formalization of termination proofs yields a connection to the polynomial complexity of the given program. We solve this problem employing the notion of minimal function graph, a set of pairs of a term and its normal form, which is defined as the least fixed point of a monotone operator. We show that termination proofs for programs reducing under lexicographic path orders (LPOs for short) and polynomially quasi-interpretable can be optimally performed in a weak fragment of Peano arithmetic. This yields an alternative proof of the fact that every function computed by an LPO-terminating, polynomially quasi-interpretable program is computable in polynomial space. The formalization is indeed optimal since every polynomial-space computable function can be computed by such a program. The crucial observation is that inductive definitions of minimal function graphs under LPO-terminating programs can be approximated with transfinite induction along LPOs.Comment: In Proceedings FICS 2015, arXiv:1509.0282

Interpretations of Presburger Arithmetic in Itself

Presburger arithmetic PrA is the true theory of natural numbers with addition. We study interpretations of PrA in itself. We prove that all one-dimensional self-interpretations are definably isomorphic to the identity self-interpretation. In order to prove the results we show that all linear orders that are interpretable in (N,+) are scattered orders with the finite Hausdorff rank and that the ranks are bounded in terms of the dimension of the respective interpretations. From our result about self-interpretations of PrA it follows that PrA isn't one-dimensionally interpretable in any of its finite subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201

A herbrandized functional interpretation of classical first-order logic

We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio

On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures.Comment: 23 page