Polyadic Algebras Research Articles

A formal model for verification of dynamic consistency of KBSs

This paper proposes a translation of the main concepts involved in Knowledge Based Systems Verification into a theoretical metalanguage based on Halmos and Leblanc's “Monadic and Polyadic Algebras.” These algebras are expressed in terms of a few basic concepts of preorder-category theory. Any Knowledge Base (KB) may be considered as a set of arrows that gives rise to a preorder category C, called “the N-category associated to the KB.” Two subsets are defined in C: E q = { x : x → q is an arrow in C}, and E p = { x : p → x is an arrow in C}, where q and p, respectively, are a goal and a conjunction of facts. A logic is a pair ( C, E p ); it is consistent if it is not the case that both a proposition and its negation are simultaneously in E p , which is equivalent to the inequality E p ≠ C. In this context, the two main ideas in the paper are the following. First, to characterize forward reasoning consistency of a KB with respect to a set of facts, in terms of consistency in ( C, E p ), where C is the N-category associated to the KB, and p is the conjunction of all the facts in the given set. Second, to characterize the absence of conflict in backward reasoning in terms of some relations among sets of the form E p and E q , for appropriate p and q. The aim of the paper is to present ideas for further discussion leading to the construction of formal logico-algebraic models for verification, rather than to propose a completed model. It presents just one possible approach which may suggest others.

Lambda abstraction algebras: representation theorems

Lambda abstraction algebras (LAAs) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic. Like combinatory algebras they can be defined by true identities and thus form a variety in the sense of universal algebra, but they differ from combinatory algebras in several important respects. The most natural LAAs are obtained by coordinatizing environment models of the lambda calculus. This gives rise to two classes of LAAs of functions of finite arity: functional LAAs (FLA) and point-relativized functional LAAs (RFA). It is shown that RFA is a variety and is the smallest variety including FLA. Dimension-complemented LAAs constitute the widest class of LAAs that can be represented as an algebra of functions and are known to have a natural intrinsic characterization. We prove that every dimension-complemented LAA is isomorphic to RFA. This is the crucial step in showing that RFA is a variety.

Lectures on cylindric set algebras

These notes are a corrected and revised version of notes which accompanied lectures given at the Banach Center in the fall of 1991. The intent is to give a self-contained introduction to cylindric algebras from the concrete point of view. I hope that after these lectures the reader will be able to digest the basic works on this subject (Henkin, Monk, Tarski [4], [5] and Henkin, Monk, Tarski, Andreka, Nemeti [6]) more easily, and that even research articles in this area will be readable by one who studies these notes carefully. As the title of the lectures indicates, we are mainly concerned with the topics in [6], which appear in a condensed form in [5]. One of the frightening things about both of these books is that they begin with a mass of definitions and proceed with very detailed discussion of the interrelationships of the defined notions. We are going to introduce just a few of these definitions, little by little, giving important (but not highly technical) results about them as we go along. And we will try to motivate the notions from logic. Cylindric algebras form the most developed form of algebraic logic. In general, algebraic logic is concerned with algebraic structures which correspond to logics of various sorts. Cylindric algebras correspond to ordinary first-order logics and to certain straightforward modifications of these logics. Other algebraic structures have a similar relationship to first-order logic; the most developed of these are relation algebras (in Tarski’s sense) and polyadic algebras. We will not be concerned with these, but the reader should be able to study them more easily after reading these notes. We will describe only the concrete aspect of cylindric algebras. The axiomatic version, fully developed in [4], will play only a minor role. Also, we will not deal with applications. Such applications exist in several other fields, such as combinatorics and theoretical computer science.

Polyadic algebras over nonclassical logics

The polyadic algebras that arise from the algebraization of the first-order extensions of a SIC are characterized and a representation theorem is proved. Standard implicational calculi (SIC)'s were considered by H. Rasiowa [19] and include classical and i

Weak products of universal algebras

Weak direct products of arbitrary universal algebras are introduced. The usual notion for groups and rings is a special case. Some universal algebraic properties are proved and applications to cylindric and polyadic algebras are considered.

What the finitization problem is not

0. Introduction. The finitization problem is one of the problems considered important in algebraic logic. Since it is often misunderstood, it seems desirable to try to develop a better understanding of what it is and what it is not about. The finitization problem is (at least partly) motivated by the following observations (1)–(3): (1) The natural algebraic counterpart of propositional logic is the variety of Boolean algebras which is axiomatizable by finitely many equations. (2) The natural algebraic counterpart of first-order logic Lωω is the variety RCAω of representable cylindric algebras which is very far from being axiomatizable by a finite schema of equations, cf. [HMT], Theorem 4.1.3, and Andreka [A91]. (3) The algebraic counterpart of Lωω without equality is the variety RQPAω of representable quasi-polyadic algebras which is also not axiomatizable by a finite schema, see Sain–Thompson [ST]. (Similar negative results apply to the finitevariable fragments of Lωω. The algebraic counterparts here are finite-dimensional representable cylindric or polyadic algebras and relation algebras.) The negative results (2) and (3) do have purely logical consequences motivating the (nonalgebraic) logician to look into the question, cf. e.g. §4 of Sain [S87], Simon [S90], [S91], Venema [V90], [V92], Nemeti [N91], but see also the present Appendix A. Observations (2) and (3) above are in contrast with (1). So there seems to be a sharp contrast between the behaviour of propositional logic and quantifier logics. It is natural to ask whether the nonfinitizability in (2) and (3) is an unavoidable

Completeness of the infinitary polyadic axiomatization

AbstractThe present note is a reworking and streamlining of Daigneault and Monk's Representation Theory for Polyadic Algebras. MSC: 03G15.

The polyadic completion of a transformation algebra

The (faithful) polyadic completion was constructed in a special case using { 0 , 1 } \{ 0,1\} -valued homomorphisms by Leblanc. Here a different method— adjoining extrema freely to the underlying Boolean algebra—succeeds in full generality. This is based on an apparently new axiomatization of locally finite polyadic algebras of infinite degree within the class of transformation algebras as those having certain extrema that the transformations preserve.

Dominical categories: recursion theory without elements

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 Gödel'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 T0 and R0 (the sets of Gödel numbers of the theorems and refutables of T) are recursively inseparable [39]; or(4) if all re sets are representable in T, then T0 is creative [28], [39]; or(5) if T is a Rosser theory (i.e., all disjoint re sets are strongly separable in T), then T0 and R0 are effectively inseparable [39].

Dominical categories: recursion theory without elements

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 developedwithout referencetoelements.By Gödel's generalized incompleteness theorem for consistent arithmetical systemTwe mean any statement of the following sort:(1) if every recursive set is definable inT, thenTis essentially undecidable [41]; or(2) if all recursive functions are definable inT, thenTis essentially undecidable [41]; or(3) if every recursive set is definable inT, thenT0andR0(the sets of Gödel numbers of the theorems and refutables ofT) are recursively inseparable [39]; or(4) if all re sets are representable inT, thenT0is creative [28], [39]; or(5) ifTis a Rosser theory (i.e., all disjoint re sets are strongly separable inT), thenT0andR0are effectively inseparable [39].

Algebraic analysis of the logic with the quantifier ?there exist uncountably many?

In this paper we define a class of polyadic algebras (Q-algebras) as adequate algebraic structures for the logic with the quantifier “there exist uncountably many”. The main result is a representation theorem for the countableQ-algebras of countable degree.

A categorical approach to polyadic algebras

It is shown that a locally finite polyadic algebra on an infinite set V of variables is a Boolean-algebra object, endowed with some internal supremum morphism, in the category of locally finite transformation sets on V. Then, this new categorical definition of polyadic algebras is used to simplify the theory of these algebras. Two examples are given: the construction of dilatations and the definition of terms and constants.

A simpler set of axioms for polyadic algebras

A simpler set of axioms for polyadic algebras

Copeland algebras

It is well known that the laws of logic governing the sentence connectives—“and”, “or”, “not”, etc.—can be expressed by means of equations in the theory of Boolean algebras. The task of providing a similar algebraic setting for the full first-order predicate logic is the primary concern of algebraic logicians. The best-known efforts in this direction are the polyadic algebras of Halmos (cf. [2]) and the cylindric algebras of Tarski (cf. [3]), both of which may be described as Boolean algebras with infinitely many additional operations. In particular, there is a primitive operator, cκ, corresponding to each quantification, ∃υκ. In this paper we explore a version of algebraic logic conceived by A. H. Copeland, Sr., and described in [1], which has this advantage: All operators are generated from a finite set of primitive operations.Following the theory of cylindric algebras, we introduce, in the natural way, the classes of Copeland set algebras (SCpA), representable Copeland algebras (RCpA), and Copeland algebras of formulas. Playing a central role in the discussion is the set, Γ, of all equations holding in every set algebra. The reason for this is that the operations in a set algebra reflect the notion of satisfaction of a formula in a model, and hence an equation expresses the fact that two formulas are satisfied by the same sequences of objects in the model. Thus to say that an equation holds in every set algebra is to assert that a certain pair of formulas are logically equivalent.

Operations in Polyadic Algebras

A new treatment of P. R. Halmos’ theory of terms and operations in (locally finite) polyadic algebras (of infinite degree) is given that is considerably simpler than the original one.

Operations in polyadic algebras

A new treatment of P. R. Halmos’ theory of terms and operations in (locally finite) polyadic algebras (of infinite degree) is given that is considerably simpler than the original one.

Representation of Locally Finite Polyadic Algebras and Ultrapowers

Mathematical Logic QuarterlyVolume 17, Issue 1 p. 91-96 Article Representation of Locally Finite Polyadic Algebras and Ultrapowers Klaus Potthoff, Klaus Potthoff KielSearch for more papers by this author Klaus Potthoff, Klaus Potthoff KielSearch for more papers by this author First published: 1971 https://doi.org/10.1002/malq.19710170115Citations: 2AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume17, Issue11971Pages 91-96 RelatedInformation

Amalgamation of Polyadic Algebras

Amalgamation of Polyadic Algebras

Amalgamation of polyadic algebras

The main result of the paper is that for I an infinite set, the class of polyadic I-algebras (with equality) has the strong amalgamation property; i.e., if two polyadic I-algebras have a given common subalgebra they can be embedded in another algebra in such a way that the intersection of the images of the two algebras is the given common subalgebra. Polyadic algebras were introduced by Halmos to provide an algebraic reflection of the study of first order logic without equality; later the algebras were enriched to allow the discussion of equality. That the notion is an adequate reflection of first order logic was demonstrated by Halmos' representation theorem for locally finite polyadic algebras of infinite degree (with or without equality). Daigneault and Monk have proved a strong extension of Halmos' theorem; namely, every polyadic algebra of infinite degree (without equality) is representable. Thus the notion of polyadic algebra is an adequate reflection of Keisler's predicate logic having infinitary predictates. It is an interesting question to ask for algebraic versions of various model theoretic results. Daigneault has been successful in stating and proving algebraic versions of Beth's and Craig's theorems. This was done by proving the algebraic analogue of Robinson's consistency theorem: Localy finite polyadic I-algebras (with equality) of infinite degree have the amalgamation property. The major result of the present work is to remove the locally finite condition from Daigneault's result. With the stronger result, Robinson's, Beth's, and Craig's theorems follow for Keisler's logic though we shall defer this to a later paper. We shall preface our work with an outline of the basic theory of polyadic algebras including theorems of Halmos [10] and important dilation and compression results of Daigneault and Monk [5]. Our set theoretic notation is standard, but it is perhaps worthwhile to outline some of our conventions. If X and Y are two sets, we write YX for the set of all functions from Y into X. We shall often identify 2X with Xx X-the cartesian Presented to the Society, April 8, 1967; received by the editors February 27, 1969. AMS Subject Classifications. Primary 0240, 0248; Secondary 0235, 0250.

Nonfinitizability of classes of representable polyadic algebras

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.