A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions. An intuitionistic theory IPVω is presented for reasoning about these functions. Interpretations for IPVω are developed both in the style of Kreisel's modified realizability and Gödel's Dialectica interpretation. Applications include alternative proofs for Buss's results concerning the classical first-order system S12 and its intuitionistic counterpart IS12 as well as proofs of some of Buss's conjectures concerning (...) IS12, and a proof that IS12 cannot prove that extended Frege systems are not polynomially bounded. (shrink)

This article presents a parametrized functional interpretation. Depending on the choice of two parameters one obtains well-known functional interpretations such as Gödel's Dialectica interpretation, Diller-Nahm's variant of the Dialectica interpretation, Kohlenbach's monotone interpretations, Kreisel's modified realizability, and Stein's family of functional interpretations. A functional interpretation consists of a formula interpretation and a soundness proof. I show that all these interpretations differ only on two design choices: first, on the number of counterexamples (...) for A which became witnesses for ¬A when defining the formula interpretation and, second, the inductive information about the witnesses of A which is considered in the proof of soundness. Sufficient conditions on the parameters are also given which ensure the soundness of the resulting functional interpretation. The relation between the parametrized interpretation and the recent bounded functional interpretation is also discussed. (shrink)

In the present paper we give a functional interpretation of Aczel's constructive set theories CZF − and CZF in systems T ∈ and T ∈ + of constructive set functionals of finite types. This interpretation is obtained by a translation × , a refinement of the ∧ -translation introduced by Diller and Nahm 49–66) which again is an extension of Gödel's Dialectica translation. The interpretation theorem gives characterizations of the definable set functions of CZF − and (...) CZF in terms of constructive set functionals. In a second part we introduce constructive set theories in all finite types. We expand the interpretation to these theories and give a characterization of the translation × . We further show that the simplest non-trivial axiom of extensionality is not interpretable by functionals of T ∈ and T ∈ + . We obtain this result by adapting Howard's notion of hereditarily majorizable functionals to set functionals. Subject of the last section is the translation ∨ that is defined in Burr for an interpretation of Kripke-Platek set theory with infinity. (shrink)

We present a new functional interpretation, based on a novel assignment of formulas. In contrast with Gödel’s functional “Dialectica” interpretation, the new interpretation does not care for precise witnesses of existential statements, but only for bounds for them. New principles are supported by our interpretation, including the FAN theorem, weak König’s lemma and the lesser limited principle of omniscience. Conspicuous among these principles are also refutations of some laws of classical logic. Notwithstanding, we end (...) up discussing some applications of the new interpretation to theories of classical arithmetic and analysis. (shrink)

We are concerned with showing how ‘labelled’ Natural Deduction presentation systems based on an extension of the so-called Curry-Howard functional interpretation can help us understand and generalise most of the deduction calculi designed to deal with the logical notion of existential quantification. We present the labelling mechanism for ‘’ using what we call ‘ɛ-terms’, which have the form of ‘a’) in a dual form to the ‘Ax.f’ terms of in the sense that the ‘witness’ is chosen at the (...) time of assertion/introduction.With the intention of demonstrating the generality of our framework, we provide a brief comparison with some other calculi of existentials, including the model-theoretic interpretations of Skolem and Herbrand, indicating the way in which some of the classical results on prenex normal forms can be extended to a wide range of logics. The essential ingredient is to look at the conceptual framework of labelled natural deduction as an attempted reconstruction of Frege's functional calculus where the devices of the Begriffsschrift work separately from, yet in harmony with, those of the Grundgesetze 1. (shrink)

We introduce constructive and classical systems for nonstandard arithmetic and show how variants of the functional interpretations due to Gödel and Shoenfield can be used to rewrite proofs performed in these systems into standard ones. These functional interpretations show in particular that our nonstandard systems are conservative extensions of E-HAω and E-PAω, strengthening earlier results by Moerdijk and Palmgren, and Avigad and Helzner. We will also indicate how our rewriting algorithm can be used for term extraction purposes. To (...) conclude the paper, we will point out some open problems and directions for future research, including some initial results on saturation principles. (shrink)

It is shown that functional interpretation can be used to show the existence property of intuitionistic number theory. On the basis of truth variants a comparison is then made between realisability and functional interpretation showing a structural difference between the two.

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce (...) a variant of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

In this article we study applications of the bounded functional interpretation to theories of feasible arithmetic and analysis. The main results show that the novel interpretation is sound for considerable generalizations of weak König’s Lemma, even in the presence of very weak induction. Moreover, when this is combined with Cook and Urquhart’s variant of the functional interpretation, one obtains effective versions of conservation results regarding weak König’s Lemma which have been so far only obtained non-constructively.

This book gives a detailed treatment of functional interpretations of arithmetic, analysis, and set theory. The subject goes back to Gödel's Dialectica interpretation of Heyting arithmetic which replaces nested quantification by higher type operations and thus reduces the consistency problem for arithmetic to the problem of computability of primitive recursive functionals of finite types. Regular functional interpretations, i.e. Dialectica and Diller-Nahm interpretation as well as Kreisel's modified realization, together with their Troelstra-style hybrids, are applied to constructive (...) as well as classical systems of arithmetic, analysis, and set theory. They yield relative consistency and conservativity results and closure under relevant rules of the theories in question as well as axiomatic characterizations of the functional translations. Prerequisites are: familiarity with classical and intuitionistic predicate logic, basics of computability theory, Gödel's incompleteness theorems. (shrink)

Logics for ‘generally’ are intended to express some vague notions, such as ‘generally’, ‘several’, ‘many’, ‘most’, etc., by means of the new generalized quantifier ∇ and to reason about assertions with ‘generally’ . We introduce the idea of functional interpretation for ‘generally’ and show that representative functions enable elimination of ∇ and reduce consequence to classical theories. Thus, one can use proof procedures and theorem provers for classical first-order logic to reason about assertions involving ‘generally’.

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity ( $\hbox{\sf KP} \omega$ ) given in [1] uses a choice functional (which is not a definable set function of ( $hbox{\sf KP} \omega$ ). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive recursive in $x\mapsto\omega$ . This yields the following characterization: The class of $\Sigma$ -definable set functions (...) of $\hbox{\sf KP} \omega$ coincides with the collection of set functionals of type 1 primitive recursive in $x\mapsto \omega$. (shrink)

Understanding the functional mechanisms underlying genetic signals associated with complex traits and common diseases, such as cancer, diabetes and Alzheimer's disease, is a formidable challenge. Many genetic signals discovered through genome‐wide association studies map to non‐protein coding sequences, where their molecular consequences are difficult to evaluate. This article summarizes concepts for the systematic interpretation of non‐coding genetic signals using genome annotation data sets in different cellular systems. We outline strategies for the global analysis of multiple association intervals and (...) the in‐depth molecular investigation of individual intervals. We highlight experimental techniques to validate candidate (potential causal) regulatory variants, with a focus on novel genome‐editing techniques including CRISPR/Cas9. These approaches are also applicable to low‐frequency and rare variants, which have become increasingly important in genomic studies of complex traits and diseases. There is a pressing need to translate genetic signals into biological mechanisms, leading to prognostic, diagnostic and therapeutic advances. (shrink)

In recent years, proof theoretic transformations that are based on extensions of monotone forms of Gödel’s famous functional interpretation have been used systematically to extract new content from proofs in abstract nonlinear analysis. This content consists both in effective quantitative bounds as well as in qualitative uniformity results. One of the main ineffective tools in abstract functional analysis is the use of sequential forms of weak compactness. As we recently verified, the sequential form of weak compactness for (...) bounded closed and convex subsets of an abstract Hilbert space can be carried out in suitable formal systems that are covered by existing metatheorems developed in the course of the proof mining program. In particular, it follows that the monotone functional interpretation of this weak compactness principle can be realized by a functional Ω∗ definable from bar recursion of lowest type. While a case study on the analysis of strong convergence results that are based on weak compactness indicates that the use of the latter seems to be eliminable, things apparently are different for weak convergence theorems such as the famous Baillon nonlinear ergodic theorem. For this theorem we recently extracted an explicit bound on a metastable version of this theorem that is primitive recursive relative to Ω∗.In the current paper we for the first time give the construction of Ω∗ . As a corollary to the fine analysis of the use of bar recursion in this construction we obtain that Ω∗ elevates arguments in Tn at most to resulting functionals in Tn+2 . In particular, one can conclude from this that our bound on Baillon’s theorem is at least definable in T4. (shrink)

The Dialectica-style functional interpretation of Kripke-Platek set theory with infinity (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbox{\sf KP} \omega$\end{document}) given in [1] uses a choice functional (which is not a definable set function of (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $hbox{\sf KP} \omega$\end{document}). By means of a Diller-Nahm-style interpretation (cf. [4]) it is possible to eliminate the choice functional and give an interpretation by set functionals primitive (...) recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $x\mapsto\omega$\end{document}. This yields the following characterization: The class of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma$\end{document}-definable set functions of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\hbox{\sf KP} \omega$\end{document} coincides with the collection of set functionals of type 1 primitive recursive in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $x\mapsto \omega$\end{document}. (shrink)

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name (...) a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧-interpretation theorem for constructive set theory in all finite types CZFω- shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω- under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω- and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. (shrink)

Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit ‘interpreting’ instances that make the implication valid. For proofs in constructive set theory CZF‐, it may not always be possible to find just one such instance, but it must suffice to explicitly name (...) a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧‐interpretation theorem for constructive set theory in all finite types CZFω‐ shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω‐ under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω‐ and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. (shrink)

Abstract.Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary first-order predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PL-proofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the β-normal-form of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in (...) the extended language, the terms are ordinary PL-terms. In contrast to approaches to Herbrand’s theorem based on cut elimination orɛ-elimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrand-term extraction tool. (shrink)

The aim of this article is to give an introduction to functional interpretations of set theory given by the authorin Burr (2000a). The first part starts with some general remarks on Gödel's functional interpretation with a focus on aspects related to problems that arise in the context of set theory. The second part gives an insight in the techniques needed to perform a functional interpretation of systems of set theory. However, the first part of this (...) article is not intended to be a complete survey of functional interpretations and here we recommend, for example, Avigad and Feferman (1998),Troelstra (1990) and Troelstra (1973). (shrink)

We introduce a new typed combinatory calculus with a type constructor that, to each type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}, associates the star type σ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma ^*$$\end{document} of the nonempty finite subsets of elements of type σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document}. 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. (shrink)

Gödel interpreted Heyting arithmetic HA in a “logic-free” fragment T 0 of his theory T of primitive recursive functionals of finite types by his famous Dialectica-translation D . This works because the logic of HA is extremely simple. If the logic of the interpreted system is different—in particular more complicated—, it forces us to look for different and more complicated functional translations. We discuss the arising logical problems for arithmetical and set theoretical systems from HA to CZF . We (...) want to test the thesis: While the functionals take care of the proof theoretic strength of the interpreted system, it is the functional translation that has to cope with the logical complexities of the system. (shrink)

We show that the types of the witnesses in the Herbrand functional interpretation can be simplified, avoiding the use of “sets of functionals” in the interpretation of implication and universal quantification. This is done by presenting an alternative formulation of the Herbrand functional interpretation, which we show to be equivalent to the original presentation. As a result of this investigation we also strengthen the monotonicity property of the original presentation, and prove a monotonicity property for (...) our alternative definition. (shrink)

We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application. we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case.

We adapt Streicher and Kohlenbach's proof of the factorization S = KD of the Shoenfield translation S in terms of Krivine's negative translation K and the Gödel functional interpretation D, obtaining a proof of the factorization U = KB of Ferreira's Shoenfield-like bounded functional interpretation U in terms of K and Ferreira and Oliva's bounded functional interpretation B.

We prove a result relating the author's monotone functional interpretation to the bounded functional interpretation due to Ferreira and Oliva. More precisely we show that largely a solution for the bounded interpretation also is a solution for the monotone functional interpretation although the latter uses the existence of an underlying precise witness. This makes it possible to focus on the extraction of bounds while using the conceptual benefit of having precise realizers at the (...) same time without having to construct them. (shrink)

The meaning of an experimental result depends on the experiment's conceptual backdrop, particularly its null hypothesis. This observation provides the basis for a functional interpretation of belief in the base rate fallacy. On this interpretation, if the base rate fallacy is to be labelled a “myth,” then it should be recognized that this label is not necessarily a disparaging one.

We show how to extract effective bounds Φ for $\bigwedge u^1 \bigwedge v \leq_\gamma tu \bigvee w^\eta G_0$ -sentences which depend on u only (i.e. $\bigwedge u \bigwedge v \leq_\gamma tu \bigvee w \leq_\eta \Phi uG_0$ ) from arithmetical proofs which use analytical assumptions of the form \begin{equation*}\tag{*}\bigwedge x^\delta\bigvee y \leq_\rho sx \bigwedge z^\tau F_0\end{equation*} (γ, δ, ρ, and τ are arbitrary finite types, η ≤ 2, G0 and F0 are quantifier-free, and s and t are closed terms). If τ (...) ≤ 2, (*) can be weakened to $\bigwedge x^\delta, z^\tau\bigvee y \leq_\rho sx \bigwedge \tilde{z} \leq_\tau z F_0$ . This is used to establish new conservation results about weak König's lemma. Applications to proofs in classical analysis, especially uniqueness proofs in approximation theory, will be given in subsequent papers. (shrink)

The Twin Paradox is not a new topic. What is new in this article is that it is an exercise toward interpretation that is functional, in the sense discovered by Lonergan in Method in Theology. The author being interpreted is P. Tipler; and the primary document is taken from his well known textbook. I try to lay out the basic argument in a way that reveals the operative insights, as well as the significant oversights. As it turns out, (...) it would seem that there is neither theoretical basis nor experimental evidence for the alleged “slowing of time” for the high speed traveler. (shrink)