In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called "uniform definability of types over finite sets" (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform (...) definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets. This settles a conjecture of La...

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets. This settles a conjecture of La...

We prove first-order definability of the prime subring inside polynomial rings, whose coefficient rings are reduced and indecomposable. This is achieved by means of a uniform formula in the language of rings with signature $$. In the characteristic zero case, the claim implies that the full theory is undecidable, for rings of the referred type. This extends a series of results by Raphael Robinson, holding for certain polynomial integral domains, to a more general class.

We define and investigate a uniformly locally o-minimal structure of the second kind in this paper. All uniformly locally o-minimal structures of the second kind have local monotonicity, which is a local version of monotonicity theorem of o-minimal structures. We also demonstrate a local definable cell decomposition theorem for definably complete uniformly locally o-minimal structures of the second kind. We define dimension of a definable set and investigate its basic properties when the given structure is a locally o-minimal structure which (...) admits local definable cell decomposition. (shrink)

Consider a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group. Let $f:X \rightarrow R^n$ be a definable map, where X is a definable set and R is the universe of the structure. We demonstrate the inequality $\dim ) \leq \dim $ in this paper. As a corollary, we get that the set of the points at which f is discontinuous is of dimension smaller than $\dim $. We also show that the (...) structure is definably Baire in the course of the proof of the inequality. (shrink)

This paper defines the form of prior knowledge that is required for sound inferences by analogy and single-instance generalizations, in both logical and probabilistic reasoning. In the logical case, the first order determination rule defined in Davies (1985) is shown to solve both the justification and non-redundancy problems for analogical inference. The statistical analogue of determination that is put forward is termed 'uniformity'. Based on the semantics of determination and uniformity, a third notion of "relevance" is defined, both logically and (...) probabilistically. The statistical relevance of one function in determining another is put forward as a way of defining the value of information: The statistical relevance of a function F to a function G is the absolute value of the change in one's information about the value of G afforded by specifying the value of F. This theory provides normative justifications for conclusions projected by analogy from one case to another, and for generalization from an instance to a rule. The soundness of such conclusions, in either the logical or the probabilistic case, can be identified with the extent to which the corresponding criteria (determination and uniformity) actually hold for the features being related. (shrink)

Allen, B., Arithmetizing Uniform NC, Annals of Pure and Applied Logic 53 1–50. We give a characterization of the complexity class Uniform NC as an algebra of functions on the natural numbers which is the closure of several basic functions under composition and a schema of recursion. We then define a fragment of bounded arithmetic, and, using our characterization of Uniform NC, show that this fragment is capable of proving the totality of all of the functions in (...) Uniform NC. Lastly, in the spirit of Buss, we show that any function which is definable by a ∑b1-formula in our theory is a function which is in Uniform NC. (shrink)

Can rational communication proceed when interlocutors are uncertain which contents utterances contribute to discourse? An influential negative answer to this question is embodied in the Stalnakerian principle of uniformity, which requires speakers to produce only utterances that express the same content in every possibility treated as live for the purposes of the conversation. The principle of uniformity enjoys considerable intuitive plausibility and, moreover, seems to follow from platitudes about assertion; nevertheless, it has recently proven controversial. In what follows, I defend (...) the principle by developing two arguments for it based on premises reflecting the central aims and assumptions of possibility-carving frameworks for modeling inquiry—that is, frameworks which describe the evolution of individuals’ attitudinal states in terms of set-theoretic operations defined over a domain of objects representing possibilities. (shrink)

Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. (...) The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed. (shrink)

We define an easily verifiable notion of an atomic formula having uniformly bounded arrays in a structure M. We prove that if T is a complete L-theory, then T is mutually algebraic if and only if there is some model M of T for which every atomic formula has uniformly bounded arrays. Moreover, an incomplete theory T is mutually algebraic if and only if every atomic formula has uniformly bounded arrays in every model M of T.

We prove uniformly computable versions of the Implicit Function Theorem in its differentiable and non-differentiable forms. We show that the resulting operators are not computable if information about some of the partial derivatives of the implicitly defining function is omitted. Finally, as a corollary, we obtain a uniformly computable Inverse Function Theorem, first proven by M. Ziegler.

Defining a class of sets to be uniform Δ02 if it is derived from a binary {0,1}{0,1}-valued function f≤TKf≤TK, we show that, for any C⊆DeC⊆De induced by such a class, there exists a high Δ02 degree c which is incomparable with every degree b ϵ Ce \ {0e, 0'e}. We show how this result can be applied to quite general subclasses of the Ershov Hierarchy and we also prove, as a direct corollary, that every nonzero low degree caps with (...) both a high and a nonzero low Δ02 degree. (shrink)

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between metric spaces, and the second declares that sequentially compact pseudometric spaces are \—meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.

We prove that a function definable with parameters in an o-minimal structure is bounded away from ∞ as its argument goes to ∞ by a function definable without parameters, and that this new function can be chosen independently of the parameters in the original function. This generalizes a result in [1]. Moreover, this remains true if the argument is taken to approach any element of the structure , and the function has limit any element of the structure.

Self-referential sentences have played a key role in Tarski's proof [9] of the non-definibility of arithmetic truth within arithmetic and Gödel's proof [2] of the incompleteness of Peano Arithmetic. In this article we consider some new methods of achieving self-reference in a uniform manner.

This book provides an antidote to the ungrounded, anecdotal components that define the contemporary conversation regarding policies of standardized dress in American K-12 districts and schools. The contributing authors draw upon years of educational teaching and administrative experience and research directed at objectively and empirically understanding the issue of school uniform policies at two focal levels of schooling: elementary and middle.

This note deals with the prepositional uniformity principlep-UP: p x N A (p, x) x N p A (p, x) ( species of all propositions) in intuitionistic mathematics.p-UP is implied by WC and KS. But there are interestingp-UP-cases which require weak KS resp. WC only. UP for number species follows fromp-UP by extended bar-induction (ranging over propositions) and suitable weak continuity. As corollaries we have the disjunction property and the existential definability w.r.t. concrete objects. Other consequences are: there is (...) no non-trivial countable partition of;id is the only injective function from to; there are no many-place injective prepositional functions; card () is incomparable with the cardinality of all metric spaces containing at least three elements. (shrink)

This paper discusses sufficient conditions for a definably complete expansion of a densely linearly ordered abelian group to have uniformly locally o‐minimal open cores of the first/second kind and strongly locally o‐minimal open core, respectively.

When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $\mathbb Q_{p}\langle X\rangle $, $\mathbb Z_{p}\langle X\rangle $, and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This (...) builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg.prepared by Madeline G. Barnicle.E-mail: [email protected]: https://escholarship.org/uc/item/6t02q9s4. (shrink)

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have a definable choice function (by a monadic formula with parameters)? A natural dichotomy arises where the trees that fall in the first class don't have a definable choice function and the trees in the second class have even a definable well ordering of their elements. This has a close connection to (...) the uniformization problem. (shrink)

A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds . By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine (...) codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful. (shrink)

We prove that every countable relation on the enumeration degrees, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}, is uniformly definable from parameters in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document}. Consequently, the first order theory of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\frak E}$\end{document} is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order (...) theory of the enumeration degrees of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\Sigma^0_2$\end{document} sets is not decidable. (shrink)

The uniform continuity theorem states that every pointwise continuous real‐valued function on the unit interval is uniformly continuous. In constructive mathematics, is strictly stronger than the decidable fan theorem, but Loeb [17] has shown that the two principles become equivalent by encoding continuous real‐valued functions as type‐one functions. However, the precise relation between such type‐one functions and continuous real‐valued functions (usually described as type‐two objects) has been unknown. In this paper, we introduce an appropriate notion of continuity for a (...) modulus of a continuous real‐valued function on [0, 1], and show that real‐valued functions with continuous moduli are exactly those functions induced by Loeb's codes. Our characterisation relies on two assumptions: (1) real numbers are represented by regular sequences (equivalently Cauchy sequences with explicitly given moduli); (2) the continuity of a modulus is defined with respect to the product metric on the regular sequences inherited from the Baire space. Our result implies that is equivalent to the statement that every pointwise continuous real‐valued function on [0, 1] with a continuous modulus is uniformly continuous. We also show that is equivalent to a similar principle for real‐valued functions on the Cantor space. These results extend Berger's [2] characterisation of for integer‐valued functions on and unify some characterisations of in terms of functions having continuous moduli. (shrink)

The biinterpretability conjecture for the r.e. degrees asks whether, for each sufficiently large k, the [Formula: see text] relations on the r.e. degrees are uniformly definable from parameters. We solve a weaker version: for each k ≥ 7, the [Formula: see text] relations bounded from below by a nonzero degree are uniformly definable. As applications, we show that Low 1 is parameter definable, and we provide methods that lead to a new example of a ∅-definable ideal. Moreover, we prove that (...) automorphisms restricted to intervals [d, 1], d ≠ 0, are [Formula: see text]. We also show that, for each c ≠ 0, can be interpreted in [0, c] without parameters. (shrink)

Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO. We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first-order define PLUS, that < and DIVIDES can first-order define TIMES, and that < and COPRIME can first-order define TIMES. The first result sharpens the equivalence FO (...) =FO to FO = FO, answering a question raised by Barrington et al. about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that FO is strictly less expressive than FO = FO = FO. In more colorful language, one could say that, for parallel computation, multiplication is harder than addition. (shrink)

The monadic second-order theory of trees allows quantification over elements and over arbitrary subsets. We classify the class of trees with respect to the question: does a tree T have definable Skolem functions? This continues [6] where the question was asked only with respect to choice functions. A natural subclass is defined and proved to be the class of trees with definable Skolem functions. Along the way we investigate the spectrum of definable well orderings of well ordered chains.

Popper functions allow one to take conditional probabilities as primitive instead of deriving them from unconditional probabilities via the ratio formula P=P/P. A major advantage of this approach is it allows one to condition on events of zero probability. I will show that under plausible symmetry conditions, Popper functions often fail to do what they were supposed to do. For instance, suppose we want to define the Popper function for an isometrically invariant case in two dimensions and hence require the (...) Popper function to be rotationally invariant and defined on pairs of sets from some algebra that contains at least all countable subsets. Then it turns out that the Popper function trivializes for all finite sets: P=1 for all A ) if B is finite. Likewise, Popper functions invariant under all sequence reflections can’t be defined in a way that models a bidirectionally infinite sequence of independent coin tosses. (shrink)

Let be a uniform space with its uniformity generated by a set of pseudo-metrics Γ. Let the symbol ≃ denote the usual infinitesimal relation on *X , and define a new infinitesimal relation ≈ on *X by writing x ≈ y whenever *ϱ ≃ *ϱ for each ϱ ∈ Γ and each p ∈ X . We call an S-space if the relations ≃ and ≈ coincide on fin. S -spaces are interesting because their nonstandard hulls have representations within (...) Nelson's internal set theory . This was shown in [1], where it was also observed that the class of uniform spaces that have invariant nonstandard hulls is contained in the class of S -spaces. The question of whether there are S -spaces that do not have invariant nonstandard hulls was left open in [1]. In this note we show that when the uniformity of an S -space is given by a single pseudometric, the space has invariant nonstandard hulls. (shrink)

Assume [Formula: see text]. Let [Formula: see text] be a [Formula: see text] equivalence relation coded in [Formula: see text]. [Formula: see text] has an ordinal definable equivalence class without any ordinal definable elements if and only if [Formula: see text] is unpinned. [Formula: see text] proves [Formula: see text]-class section uniformization when [Formula: see text] is a [Formula: see text] equivalence relation on [Formula: see text] which is pinned in every transitive model of [Formula: see text] containing the real (...) which codes [Formula: see text]: Suppose [Formula: see text] is a relation on [Formula: see text] such that each section [Formula: see text] is an [Formula: see text]-class, then there is a function [Formula: see text] such that for all [Formula: see text], [Formula: see text]. [Formula: see text] proves that [Formula: see text] is Jónsson whenever [Formula: see text] is an ordinal: For every function [Formula: see text], there is an [Formula: see text] with [Formula: see text] in bijection with [Formula: see text] and [Formula: see text]. (shrink)

The main purpose of this paper is to define and study a particular variety of Montague-Scott neighborhood semantics for modal propositional logic. We call this variety the first-order neighborhood semantics because it consists of the neighborhood frames whose neighborhood operations are, in a certain sense, first-order definable. The paper consists of two parts. In Part I we begin by presenting a family of modal systems. We recall the Montague-Scott semantics and apply it to some of our systems that have hitherto (...) be uncharacterized. Then, we define the notion of a first-order indefinite semantics, along with the more specific notion of a first-order uniform semantics, the latter containing as special cases the possible world semantics of Kripke. In Part II we prove consistency and completeness for a broad range of the systems considered, with respect to the first-order indefinite semantics, and for a selected list of systems, with respect to the first-order uniform semantics. The completeness proofs are algebraic in character and make essential use of the finite model property. A by-product of our investigations is a result relating provability in S-systems and provability in T-systems, which generalizes a known theorem relating provability in the systems S 2° and C 2. (shrink)

A uniform carbon tax with equal per capita dividends is usually advocated as a cost-effective way of reducing greenhouse gas (GHG) emissions without increasing, and in many cases even reducing, economic inequality, in particular because of the positive balance between the carbon taxes paid by the worse off and the carbon dividends they receive back. In this article, I argue that a uniform carbon tax reform is unjust regardless of how the revenue is used, because it does not (...) discourage the rich from indulging in wasteful emissions. This is at odds with a normative account of the distribution of the GHG budget that I define as emissions limitarianism, which is a climate derivative of economic limitarianism. Emissions limitarianism holds that, under low GHG budget conditions, emissions permits should not be used to achieve functionings that over-consume the GHG budget without producing significant welfare gains relative to the achievement of alternative functionings. Accordingly, I argue that a uniform carbon tax should be complemented by a limitarian carbon tax, which restricts the capabilities of the rich to achieve wasteful emission-generating functionings. The article adopts a methodological multi-level perspective on capabilities as a measure of welfare, distinguishing between basic, secondary and tertiary capabilities. (shrink)

Algebraic approach to study of classical and non-classical logical calculi was developed and systematically presented by Helena Rasiowa in [48], [47]. It is very fruitful in investigation of non-classical logics because it makes possible to study large families of logics in an uniform way. In such research one can replace logics with suitable classes of algebras and apply powerful machinery of universal algebra. In this paper we present an overview of results on interpolation and definability in modal and (...) positive logics,and also in extensions of Johansson's minimal logic. All these logics are strongly complete under algebraic semantics. It allows to combine syntactic methods with studying varieties of algebras and to flnd algebraic equivalents for interpolation and related properties. Moreover, we give exhaustive solution to interpolation and some related problems for many families of propositional logics and calculi. (shrink)

In [5] it has been shown that for first-order definability over the reals there exists an effective procedure which by a finite formula with equality defining an open set produces a finite formula without equality that defines the same set. In this paper we prove that there exists no such procedure for Σ-definability over the reals. We also show that there exists even no uniform effective transformation of the definitions of Σ-definable sets into new definitions of Σ-definable (...) sets in such a way that the results will define open sets, and if a definition defines an open set, then the result of this transformation will define the same set. These results highlight the important differences between Σ-definability with equality and Σ-definability without equality. (shrink)

Dewey defines knowledge as the outcome of competent inquiry. but knowledge is for dewey fundamentally predictive. this gives rise to a difficulty: should the course of nature change, a man might both know something (having carried out the relevant inquiry) and not know it (his relevant predictions being false). this difficulty is set out formally, and a solution is proposed in terms of dewey's concept of warranted assertibility.

In any model theoretic logic, Beth’s definability property together with Feferman-Vaught’s uniform reduction property for pairs imply recursive compactness, and the existence of models with infinitely many automorphisms for sentences having infinite models. The stronger Craig’s interpolation property plus the uniform reduction property for pairs yield a recursive version of Ehrenfeucht-Mostowski’s theorem. Adding compactness, we obtain the full version of this theorem. Various combinations of definability and uniform reduction relative to other logics yield corresponding results (...) on the existence of non-rigid models. (shrink)

We introduce a natural class of quantifiersTh containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic ofL ωω (Th) or countably compact regular sublogic ofL ∞ω (Th), properly extendingL ωω , satisfies the uniform reduction property for quotients. As a consequence, none of these logics satisfies eitherΔ-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any (...) sublogic ofL ∞ω (Th) in which Chang's quantifier or some cardinality quantifierQ α, with α≧1, is definable. (shrink)

We estimate the derivation lengths of functionals in Gödel's system \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a (...) collapsing f unction\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} ${\cal D}:T\to \omega$\end{document} so that for (closed) terms \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a,b$\end{document} of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} we have: If \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $a$\end{document} reduces to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $b$\end{document} then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\omega>{\cal D}(a)>{\cal D}(b).$\end{document} By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence new proof of strongly uniform termination of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}, hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. A new proof of Troelstra's result on strong normalization for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document}. Additionally, a slow growing analysis of Gödel's \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $T$\end{document} is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\lambda$\end{document}-calculus. (shrink)

Assume [Formula: see text]. If [Formula: see text] is an ordinal and X is a set of ordinals, then [Formula: see text] is the collection of order-preserving functions [Formula: see text] which have uniform cofinality [Formula: see text] and discontinuous everywhere. The weak partition properties on [Formula: see text] and [Formula: see text] yield partition measures on [Formula: see text] when [Formula: see text] and [Formula: see text] when [Formula: see text]. The following almost everywhere continuity properties for functions (...) on partition spaces with respect to these partition measures will be shown. For every [Formula: see text] and function [Formula: see text], there is a club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. For every [Formula: see text] and function [Formula: see text], there is an [Formula: see text]-club [Formula: see text] and a [Formula: see text] so that for all [Formula: see text], if [Formula: see text] and [Formula: see text], then [Formula: see text]. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of [Formula: see text]. [Formula: see text]. [Formula: see text]. [Formula: see text]. It will also be shown that [Formula: see text] has the Jónsson property: For every [Formula: see text], there is an [Formula: see text] with [Formula: see text] so that [Formula: see text]. (shrink)

Two real-valued deduction schemes are introduced, which agree on $\vdash \triangle$ but not on $\Gamma \vdash \triangle$ , where Δ and ▵ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and (...) the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other. (shrink)

We consider games over a finite alphabet with Gurevich-Harrington's winning conditions and restraints as in Yakhnis-Yakhnis . The game tree, the Gurevich-Harrington's kernels of the winning condition and the restraints are defined by finite automata. We give an effective criterion to determine the winning player and an effective presentation of a class of finite automata defined winning strategies.Our approach yields an alternative solution to the games considered by Büchi and Landweber . The BL algorithm is an important tool for solving (...) effectively presented games. The differences between the BL and our algorithms are as follows. Our algorithm is uniformly applicable to winning sets that correspond to boolean combinations of BL winning conditions, while the BL algorithm requires a transformation of such boolean combinations into a single BL condition. Our algorithm directly applies to a larger class of game trees for a given game alphabet than does the BL one. We describe Büchi-Landweber's class of winning conditions in a more transparent way by using finite automata and Gurevich-Harrington's form of winning conditions. Our algorithms that determine the winner of the game and find a winning strategy, are based on a translation of the events in the game in terms of the transition table of the automaton representing the game. This automaton differs from the ones that represent BL winning conditions. Both algorithms need further investigation.The algorithm determining the winner in the game, can be used as a verifier of concurrent programs. This is because programs can be viewed as strategies in Gurevich-Harrington's games and program specifications could be regarded as winning conditions for such games. The algorithm which finds a winning strategy could be used as a basis for automated concurrent program design.The present paper is a sequel to Annals of Pure and Applied Logic 48 277–297. (shrink)

If S ⊆{0,1}; * and S ′ = {0,1} * \sb S are both recognized within a certain nondeterministic time bound T then, in not much more time, one can write down tautologies A n → A′ n with unique interpolants I n that define S ∩{0,1} n ; hence, if one can rapidly find unique interpolants, then one can recognize S within deterministic time T p for some fixed p \s>0. In general, complexity measures for the problem of finding (...) unique interpolants in sentential logic yield new relations between circuit depth and nondeterministic Turing time, as well as between proof length and the complexity of decision procedures of logical theories. (shrink)