Showing papers in "Journal of Symbolic Logic in 1987"

Polynomial Size Proofs of the Propositional Pigeonhole Principle

Abstract: Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic.

Systematization of Finite Many-Valued Logics Through the Method of Tableaux

Abstract: This paper presents a unified treatment of the propositional and first-order many-valued logics through the method of tableaux. It is shown that several important results on the proof theory and model theory of those logics can be obtained in a general way.We obtain, in this direction, abstract versions of the completeness theorem, model existence theorem (using a generalization of the classical analytic consistency properties), compactness theorem and Lowenheim-Skolem theorem.The paper is completely self-contained and includes examples of application to particular many-valued formal systems.

A theory of properties

Abstract: Frege's attempts to formulate a theory of properties to serve as a foundation for logic, mathematics and semantics all dissolved under the weight of the logicial paradoxes. The language of Frege's theory permitted the representation of the property which holds of everything which does not hold of itself. Minimal logic, plus Frege's principle of abstraction, leads immediately to a contradiction. The subsequent history of foundational studies was dominated by attempts to formulate theories of properties and sets which would not succumb to the Russell argument. Among such are Russell's simple theory of types and the development of various iterative conceptions of set. All of these theories ban, in one way or another, the self-reference responsible for the paradoxes; in this sense they are all “typed” theories. The semantical paradoxes, involving the concept of truth, induced similar nightmares among philosophers and logicians involved in semantic theory. The early work of Tarski demonstrated that no language that contained enough formal machinery to respresent the various versions of the Liar could contain a truth-predicate satisfying all the Tarski biconditionals. However, recent work in both disciplines has led to a re-evaluation of the limitations imposed by the paradoxes.In the foundations of set theory, the work of Gilmore [1974], Feferman [1975], [1979], [1984], and Aczel [1980] has clearly demonstrated that elegant and useful type-free theories of classes are feasible. Work on the semantic paradoxes was given new life by Kripke's contribution (Kripke [1975]). This inspired the recent work of Gupta [1982] and Herzberger [1982]. These papers demonstrate that much room is available for the development of theories of truth which meet almost all of Tarski's desiderata.

A constructive analysis of RM

Abstract: The system RM is the most well-understood (and to our opinion, also the most important) system among the logics developed by the Anderson and Belnap school. In this paper we investigate RM from a constructive point of view. For example, we give a new proof of the completeness of RM relative to the Sugihara matrix (first shown by Meyer), a proof in which a p.r. procedure is presented, applying which to a sentence A in RM language yields either a proof of it in RM or a refuting valuation for it in the Sugihara matrix S Z . Two topics dealt with in this work deserve a special attention. a) The admissibility of γ . This is a famous theorem of Meyer and Dunn. In [1] Anderson and Belnap emphasize that “the Meyer-Dunn argument … guarantees the existence of a proof of B , but there is no guarantee that the proof of B is related in any sort of plausible way to the proofs of A and Ā ∨ B .” In §2 we provide such a guarantee for the RM -case. In fact, we give there a direct method of obtaining a proof of B from given proofs of A and Ā ∨ B . b) The relationships between RM and its full negation-implication fragment . RM is known ([1, pp. 148–149], and [3]) to be a conservative extension of (Sobocinski 3-valued logic; see [4]). Anderson and Belnap admit [1, p. 149] that this fact came to them as a distinct surprise, since RM as a whole is far from being three-valued. In this paper, however, this “surprising” fact appears quite natural (see III.3). In fact, we show that , is the “hard core” of RM , since our proof of the completeness of RM is based in an essential way on the completeness of relative to the Sobocinski matrix, and since the Gentzen-type calculus we develop for RM is a direct extension of a similar (but much simpler) calculus for . Because of the importance has in this work, we devote the first section to a constructive investigation of it. We note, finally, that the Gentzen-type calculus mentioned above admits cut-elimination and normal-form techniques. (Such calculi were found till now only for RM without distribution.)

Categorical Semantics for Higher Order Polymorphic Lambda Calculus

Abstract: A categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete. In fact, there is an equivalence between the theories and the categories. Also presented is a definitional extension of PLC including “subtypes”, for example, equality subtypes, together with a construction providing models of the extended language, and a context for Girard's extension of the Dialectica interpretation.

Classifying the Computational Complexity of Problems

Abstract: One of the more significant achievements of twentieth century mathematics, especially from the viewpoints of logic and computer science, was the work of Church, Godel and Turing in the 1930's which provided a precise and robust definition of what it means for a problem to be computationally solvable, or decidable, and which showed that there are undecidable problems which arise naturally in logic and computer science. Indeed, when one is faced with a new computational problem, one of the first questions to be answered is whether the problem is decidable or undecidable. A problem is usually defined to be decidable if and only if it can be solved by some Turing machine, and the class of decidable problems defined in this way remains unchanged if “Turing machine” is replaced by any of a variety of other formal models of computation. The division of all problems into two classes, decidable or undecidable, is very coarse, and refinements have been made on both sides of the boundary. On the undecidable side, work in recursive function theory, using tools such as effective reducibility, has exposed much additional structure such as degrees of unsolvability. The main purpose of this survey article is to describe a branch of computational complexity theory which attempts to expose more structure within the decidable side of the boundary.Motivated in part by practical considerations, the additional structure is obtained by placing upper bounds on the amounts of computational resources which are needed to solve the problem. Two common measures of the computational resources used by an algorithm are time, the number of steps executed by the algorithm, and space, the amount of memory used by the algorithm.

Von Mises' definition of random sequences reconsidered

Abstract: We review briefly the attempts to define random sequences (§0). These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence (§§1–3 and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests (§4).

First Order Topological Structures and Theories

Abstract: In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

Semi-De Morgan Algebras

Abstract: The purpose of this paper is to define and investigate a new (equational) class of algebras, which we call semi-De Morgan algebras, as a common abstraction of De Morgan algebras and distributive pseudocomplemented lattices. We were first led to this class of algebras in 1979 (in Brazil) as a result of our attempt to extend both the well-known theorem of Glivenko (see [4, Theorem 26]) and Lakser's characterization of principal congruences to a setting more general than that of distributive pseudocomplemented lattices. In subsequent years, our work in [20] on a subvariety of Ockham algebras, first considered by Berman [3], renewed our interest in semi-De Morgan algebras by providing new examples. It seems worth mentioning that these new algebras may also turn out to be useful in resolving a conjecture made in [22] to unify certain strikingly similar results on Heyting algebras with a dual pseudocomplement (see [21]) and Heyting algebras with a De Morgan negation (see [22]). In §2 we introduce semi-De Morgan algebras and prove the main theorem, which, roughly speaking, states that certain elements of a semi-De Morgan algebra form a De Morgan algebra. Several applications then follow, including new axiomatizations of distributive pseudocomplemented lattices, Stone algebras and De Morgan algebras.

Dominical categories: Recursion theory without elements

Abstract: Dominical categories are categories in which the notions of partial morphisms and their domains become explicit, with the latter being endomorphisms rather than subobjects of their sources. These categories form the basis for a novel abstract formulation of recursion theory, to which the present paper is devoted. The abstractness has of course its usual concomitant advantage of generality: it is interesting to see that many of the fundamental results of recursion theory remain valid in contexts far removed from their classic manifestations. A principal reason for introducing this new formulation is to achieve an algebraization of the generalized incompleteness theorem, by providing a category-theoretic development of the concepts and tools of elementary recursion theory that are inherent in demonstrating the theorem. Dominical recursion theory avoids the commitment to sets and partial functions which is characteristic of other formulations, and thus allows for an intrinsic recursion theory within such structures as polyadic algebras. It is worthy of notice that much of elementary recursion theory can be developed without reference to elements . By Godel's generalized incompleteness theorem for consistent arithmetical system T we mean any statement of the following sort: (1) if every recursive set is definable in T , then T is essentially undecidable [41]; or (2) if all recursive functions are definable in T , then T is essentially undecidable [41]; or (3) if every recursive set is definable in T , then T 0 and R 0 (the sets of Godel numbers of the theorems and refutables of T ) are recursively inseparable [39]; or (4) if all re sets are representable in T , then T 0 is creative [28], [39]; or (5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T ), then T 0 and R 0 are effectively inseparable [39].

Monadic second order definable relations on the binary tree

Abstract: Let S2S [WS2S] respectively be the strong [weak] monadic second order theory of the binary tree T in the language of two successor functions. An S2S-formula whose free variables are just individual variables defines a relation on T (rather than on the power set of T). We show that S2S and WS2S define the same relations on T, and we give a simple characterization of these relations. ?1. The infinite binary tree T is given by the set {0, 1}* of all finite (0,1)-words, called the nodes of the tree. Every node x has two successor nodes, s0(x):= xO and sl(x):= xl. S2S is the monadic second order theory of (Tsosl) in the language of two successor functions: In addition to the first order theory there are set variables ranging over subsets of T, existential and universal quantifier over set variables and the membership relation. WS2S is the corresponding monadic weak second order theory: Set variables range only over finite subsets of T. An S2S-formula with free set variables defines a relation on P(T), the power set of T, while an S2S-formula with just free individual variables defines a relation on T. The following results are due to M. 0. Rabin (see [7] and [8]): (I) There are S2S-definable even one-place relations on P(T) which are not WS2Sdefinable. (II) A subset of T is S2S-definable iff it is regular; in particular, S2S and WS2S define the same one-place relations on T. A slightly simpler proof of (II), based on [4], is given in [10]. In this paper we give a simple characterization of the S2S-definable relations on T. In particular, we prove THEOREM 1. For n E ca) and R c Tn, R is S2S-definable if it is WS2S-definable. The corresponding result for SI S, the monadic second order theory of the natural numbers with successor function, is due to J. R. Bichi [2] and answers a question raised by R. M. Robinson in [9]. Monadic second-order definability and weak monadic second-order definability are known to be equivalent even for relations on P(w) (see W. Thomas [11]). Our proof is based on Rabin's characterization of S2S-definability in terms of finite tree automata. Received October 17, 1985; revised May 9, 1986. C) 1987, Association for Symbolic Logic 0022-4812/87/5201-0021 /$01 .80

Elimination of Quantifiers for Ordered Valuation Rings

Abstract: Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

Reduced Coproducts of Compact Hausdorff Spaces

Abstract: By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

Semiproper forcing axiom implies Martin maximum but not PFA

Abstract: We prove that MM (Martin maximum) is equivalent (in ZFC) to the older axiom SPFA (semiproper forcing axiom). We also prove that SPFA does not imply SPFA + or even PFA + (using the consistency of a large cardinal).

A model in which the base-matrix tree cannot have cofinal branches

Abstract: A model of ZFC is constructed in which the distributivity cardinal h is , and in which there are no ω 2-towers in [ω]ω. As an immediate corollary, it follows that any base-matrix tree in this model has no cofinal branches. The model is constructed via a form of iterated Mathias forcing, in which a mixture of finite and countable supports is used.

Natural deduction and sequent calculus for intuitionistic relevant logic

Abstract: Relevance logic began in an attempt to avoid the so-called fallacies of relevance. These fallacies can be in implicational form or in deductive form. For example, Lewis's first paradox can beset a system in implicational form, in that the system contains as a theorem the formula (A & ∼A) → B; or it can beset it in deductive form, in that the system allows one to deduce B from the premisses A, ∼A. Relevance logic in the tradition of Anderson and Belnap has been almost exclusively concerned with characterizing a relevant conditional. Thus it has attacked the problem of relevance in its implicational form. Accordingly for a relevant conditional → one would not have as a theorem the formula (A & ∼A) → B. Other theorems even of minimal logic would also be lacking. Perhaps most important among these is the formula (A → (B → A)). It is also a well-known feature of their system R that it lacks the intuitionistically valid formula ((A ∨ B) & ∼A) → B (disjunctive syllogism). But it is not the case that any relevance logic worth the title even has to concern itself with the conditional, and hence with the problem in its implicational form. The problem arises even for a system without the conditional primitive. It would still be an exercise in relevance logic, broadly construed, to formulate a deductive system free of the fallacies of relevance in deductive form even if this were done in a language whose only connectives were, say, &, ∨ and ∼. Solving the problem of relevance in this more basic deductive form is arguably a precondition for solving it for the conditional, if we suppose (as is reasonable) that the relevant conditional is to be governed by anything like the rule of conditional proof.

Combinatorics on Ideals and Forcing with Trees

Abstract: Classes of forcings which add a real by forcing with branching conditions are examined, and conditions are found which guarantee that the generic real is of minimal degree over the ground model. An application is made to almost-disjoint coding via a real of minimal degree.

Modal Sequents and Definability

Abstract: The language of propositional modal logic is extended by the introduction of sequents. Validity of a modal sequent on a frame is defined, and modal sequent-axiomatic classes of frames are introduced. Through the use of modal algebras and general frames, a study of the properties of such classes is begun.

The degree of the set of sentences of predicate provability logic that are true under every interpretation

Abstract: The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows. Pr( x ), alias Bew( x ), is the standard provability predicate of PA. For any formula F of PA, Pr[ F ] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F , i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[ F ], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[ F ] = Pr(‘ F ’), the sentence that expresses the PA-provability of F . Let υ 1 , υ 2 ,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P * of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P .

Provability in Finite Subtheories of Pa and Relative Interpretability: A Modal Investigation

Abstract: By Solovay's theorem [16], the modal logic of provability GL gives a complete description of the propositional schemata involving the provability predicate PrPA(x) for Peano arithmetic PA, which are provable in PA. However, many important aspects of provability cannot be fully expressed in terms of PrPA(x). For this reason, many authors have introduced extensions of GL which take account either of Rosser constructions or of other important metamathematical formulas (see, for example, [5], [6], [14], [16], and [19]). In this paper, we concentrate on the modal logic of the provability predicate for finitely axiomatizable subtheories of PA; the interest of this modal logic is based on the following facts. First of all, it provides a modal translation of a very important property of PA, namely the essential reflexiveness. Secondly, in view of Orey's theorem [10] it constitutes a possible approach to the study of interpretability of finite extensions of PA. Indeed, by Orey's theorem PA + θ is interpretable in PA + θ′ iff for every n, and, therefore, relative interpretability of finitely axiomatizable extensions of PA can be expressed by means of the provability predicate for finitely axiomatizable subtheories of PA.In §1, we introduce three modal logics extending GL and discuss their arithmetical interpretations; §2 deals with Kripke semantics for two of these logics. In §3, a theorem on arithmetical completeness is shown, which characterizes the logic of the provability predicate for finitely axiomatizable subtheories of PA; a uniform version of this theorem is proved in §4.

A Constructive Interpretation of the Full Set Theory

Abstract: The interpretation of the ZF set theory reported in this paper is, actually, part of a wider effort, namely, a new approach to the foundation of mathematics, which is referred to as The Cybernetic Foundation. A detailed exposition of the Cybernetic Foundation will be published elsewhere. Our approach leads to a full acceptance of the formalism of the classical set theory, but interprets it using only the idea of potential, but not actual (completed) infinity, and dealing only with finite objects that can actually be constructed. Thus we have a finitist proof of the consistency of ZF. This becomes possible because we set forth a metatheory of mathematics which goes beyond the classical logic and set theory and, of course, cannot be formalized in ZF, yet yields proofs which are as convincing—at least, from the author's viewpoint—as any mathematical proof can be.Our metatheory is based on the following two ideas. Firstly, we define the semantics of the mathematical language using the cybernetical concept of knowledge. According to this concept, to say that a cybernetic system (a human being, in particular) has some knowledge is to say that it has some models of reality. In the Cybernetic Foundation we consider mathematics as the art of constructing linguistic models of reality.

Omitting Types and the Real Line

Abstract: Etude de quelques relations entre l'omission des types d'une theorie denombrable et quelques notions definies en termes de la droite reelle, comme par exemple, l'ideal des sous-ensembles maigres de R

PAS d'imaginaires dans l'infini!

Abstract: Dans Poizat [1981], le second auteur a montre qu'un sous-groupe infiniment definissable d'un groupe stable etait intersection de sous-groupes definissables; il a pose la question de savoir si une relation d'equivalence E , infiniment definissable dans un modele M d'une theorie stable T , etait conjonction de relations d'equivalence definissables. Nous allons voir ici que c'est presque exact: c'est vrai si T est totalement transcendante, et, dans le cas general de stabilite E a toujours un raffinement E 1 (plus precisement, E 1 est la conjonction de E et de la relation “ x et y ont meme type”) qui a cette propriete; cela montre que cette relation E n'introduit pas d'imaginaires d'une nature vraiment differente de celle des imaginaires de Shelah: dans une theorie stable, un imaginaire infinitaire n'est rien d'autre qu'un ensemble d'imaginaires finis. La demonstration du theoreme principal de cette note s'appuie lourdement sur la construction M eq de Shelah, la machinerie de la deviation, les parametres imaginaires canoniques pour la definition d'un type stable, etc…. Pour tout cela, les references adequates sont Shelah [1978], Pillay [1983], et Poizat [1985, Chapitre 16]. Nouscommencons par preciser ce que nous entendons par “relation d'equivalence infiniment definissable”: une collection de formules e( , ȳ ), et ȳ etant de longueur n , telle que, pour tout modele M de T , les couples ( , ȳ ) qui les satisfont toutes forment une relation d'equivalence E .

Remarks on Weak Notions of Saturation in Models of Peano Arithmetic

Abstract: This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic . Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ -like for some regular κ . In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω -lofty but not continuously ω -lofty and others which are continuously ω -lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ 1 -saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ 1 ; every κ -saturated model of PA has a κ -saturated proper, simple extension which is not κ + -saturated. Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

On the Existence of Extensional Partial Combinatory Algebras

Abstract: The principal aim of this paper is to present a construction method for nontotal extensional combinatory algebras. This is done in §2. In §0 we give definitions of some basic notions for partial combinatory algebras from which the corresponding notions for (total) combinatory algebras are obtained as specializations. In §1 we discuss some properties of nontotal extensional combinatory algebras in general. §2 describes a “partial” variant of reflexive complete partial orders yielding nontotal extensional combinatory algebras. Finally, §3 deals with properties of the models constructed in §2, such as incompletability, having no total submodel and the pathological behaviour with respect to the interpretation of unsolvable λ -terms.

Limit ultrapowers and abstract logics

Abstract: We associate with any abstract logic L a family F(L) consisting, intuitively, of the limit ultrapowers which are complete extensions in the sense of L.For every countably generated [ω, ω]-compact logic L, our main applications are:(i) Elementary classes of L can be characterized in terms of ≡L only.(ii) If and are countable models of a countable superstable theory without the finite cover property, then .(iii) There exists the “largest” logic M such that complete extensions in the sense of M and L are the same; moreover M is still [ω, ω]-compact and satisfies an interpolation property stronger than unrelativized ⊿-closure.(iv) If L = Lωω(Qx), then cf(ωx) > ω and λω < ωx, for all λ < ωx.We also prove that no proper extension of Lωω generated by monadic quantifiers is compact. This strengthens a theorem of Makowsky and Shelah. We solve a problem of Makowsky concerning Lκλ-compact cardinals. We partially solve a problem of Makowsky and Shelah concerning the union of compact logics.