Hajnal Andréka

A complete logic for reasoning about programs via nonstandard model theory II

In Part I of the present paper W~Z defined the first order dynamic language DLd (of type d). In Definition 13 ~3 defined a decidable proof concept (I-~, Prn) for DLd, and in Theorem 2 we proved that (kN, Pm) is a strongly complete inferenr:e system for DLd. That is, for every theory Th and formula qj of first order dynamic language we have Th + q iff Th I--~ cc). By j!?ynamic Logic of type d we understand (DLd, (bN, Pm)). Here we investigate further properties of our Dynamic Logic, its expressive power, how it can be used for various purposes, how it can be adapted to various situations. Then we investigate Floyd’s method using the framework of DL. A complete characterization of the amount of information implicitly contained in Floyd’s method will be found but several questions remain open in this line. The proof method I--.~ is proved to be strictly stronger than Floyd’s method in Section 6. Different semantics ? programming are compared in Section 7 within the framework of DL. Comkrisons with several approaches related ill several ways are given in Sections 7-9.

Logic of Space-Time and Relativity Theory

2 Special relativity 4 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Axiomatization Specrel of special relativity in first-order logic 15 2.4 Characteristic differences between Newtonian and special relativistic kinematics . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 Explicit description of all models of Specrel, basic logical investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6 Observer-independent geometries in relativity theory; duality and definability theory of logic . . . . . . . . . . . . . . . . . 47 2.7 Conceptual analysis and “reverse relativity” . . . . . . . . . . 59

Representations of distributive lattice-ordered semigroups with binary relations

LetR(∩, ∪, ¦) denote the class of all algebras isomorphic to ones whose elements are binary relations and whose operations are union, intersection, and relation composition (or relative product) of relations. We prove thatR(∪, ∩, ¦) is not a variety and is not finitely axiomatizable. LetDLOS denote the class of all structures (A, ∨, ∧, ∘) where (A, ∨, ∧) is a distributive lattice, (A, ∘) is a semigroup and ∘ is additive w.r.t. ∨. We prove thatDLOS is the variety generated byR(∪, ∩, ¦), and moreover, if (A, ∨, ∧, ∘) ∈DLOS then it is representable whenever we disregard one of its operations.

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we “derive” an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist.

Notions of density that imply representability in algebraic logic

Abstract Henkin and Tarski proved that an atomic cylindric algebra in which every atom is a rectangle must be representable (as a cylindric set algebra). This theorem and its analogues for quasi-polyadic algebras with and without equality are formulated in Henkin, Monk and Tarski [13]. We introduce a natural and more general notion of rectangular density that can be applied to arbitrary cylindric and quasi-polyadic algebras, not just atomic ones. We then show that every rectangularly dense cylindric algebra is representable, and we extend this result to other classes of algebras of logic, for example quasi-polyadic algebras and substitution-cylindrification algebras with and without equality, relation algebras, and special Boolean monoids. The results of op. cit. mentioned above are special cases of our general theorems. We point out an error in the proof of the Henkin-Monk-Tarski representation theorem for atomic, equality-free, quasi-polyadic algebras with rectangular atoms. The error consists in the implicit assumption of a property that does not, in general, hold. We then give a correct proof of their theorem. Henkin and Tarski also introduced the notion of a rich cylindric algebra and proved in op. cit. that every rich cylindric algebra of finite dimension (or, more generally, of locally finite dimension) satisfying certain special identities is representable. We introduce a modification of the notion of a rich algebra that, in our opinion, renders it more natural. In particular, under this modification richness becomes a density notion. Moreover, our notion of richness applies not only to algebras with equality, such as cylindric algebras, but also to algebras without equality. We show that a finite dimensional algebra is rich iff it is rectangularly dense and quasi-atomic; moreover, each of these conditions is also equivalent to a very natural condition of point density . As a consequence, every finite dimensional (or locally finite dimensional) rich algebra of logic is representable. We do not have to assume the validity of any special identities to establish this representability. Not only does this give an improvement of the Henkin-Tarski representation theorem for rich cylindric algebras, it solves positively an open problem in op. cit. concerning the representability of finite dimensional rich quasi-polyadic algebras without equality.

Complexity of equations valid in algebras of relations part I: Strong non-finitizability

Abstract We study algebras whose elements are relations, and the operations are natural “manipulations” of relations. This area goes back to 140 years ago to works of De Morgan, Peirce and Schroder (who expanded the Boolean tradition with extra operators to handle algebras of binary relations). Well known examples of algebras of relations are the varieties RCA n of cylindric algebras of n -ary relations, RPEA n of polyadic equality algebras of n -ary relations, and RRA of binary relations with composition. We prove that any axiomatization, say E , of RCA n has to be very complex in the following sense: for every natural number k there is an equation in E containing more than k distinct variables and all the operation symbols, if 2 n ω . A completely analogous statement holds for the case n ⩾ ω . This improves Monks famous non-finitizability theorem for which we give here a simple proof. We prove analogous non-finitizability properties of the larger varieties SNr n CA n + k . We prove that the complementation-free (i.e. positive) subreducts of RCA n do not form a variety. We also investigate the reason for the above “non-finite axiomatizability” behaviour of RCA n . We look at all the possible reducts of RCA n and investigate which are finitely axiomatizable. We obtain several positive results in this direction. Finally, we summarize the results and remaining questions in a figure. We carry through the same programme for RPEA n and for RRA . By looking into the reducts we also investigate what other kinds of natural algebras of relations are possible with more positive behaviour than that of the well known ones. Our investigations have direct consequences for the logical properties of the n -variable fragment L n of first order logic. The reason for this is that RCA n and RPEA n are the natural algebraic counterparts of L n while the varieties SNr n CA n + k are in connection with the proof theory of L n . This paper appears in two parts. This is the first part, it contains the non-finite axiomatizability results. The second part contains finite axiomatizability results together with a figure summarizing the results in this area and the problems left open.

Free algebras in discriminator varieties

We investigate V-free algebras onn generators,Fn=Fr(V, n), where V is a discriminator variety and, more specifically, where V is a variety of relation algebras or of cylindricalgebras. Sample questions are: (a) IsFn+1 embeddable inFn? (b) DoesFn contain an n-element set that generates it non-freely? The answer to (a) is affirmative in some varieties of relation algebras, but it is negative in every congruence extensile variety in which some nontrivial finite member is an absolute retract. The answer to (b) is affirmative in every variety of relation algebras that contains the full algebra of relations on an infinite set.

Logical Axiomatizations of Space-Time. Samples from the Literature

We study relativity theory as a theory in the sense of mathematical logic. We use first-order logic (FOL) as a framework to do so. We aim at an “analysis of the logical structure of relativity theories”. First we build up (the kinematics of) special relativity in FOL, then analyze it, and then we experiment with generalizations in the direction of general relativity. The present paper gives samples from an ongoing broader research project which in turn is part of a research direction going back to Reichenbach and others in the 1920’s. We also try to give some perspective on the literature related in a broader sense. In the perspective of the present work, axiomatization is not a final goal. Axiomatization is only a first step, a tool. The goal is something like a conceptual analysis of relativity in the framework of logic.

Lambek Calculus and its relational semantics: Completeness and incompleteness

The problem of whether Lambek Calculus is complete with respect to (w.r.t.) relational semantics, has been raised several times, cf. van Benthem (1989a) and van Benthem (1991). In this paper, we show that the answer is in the affirmative. More precisely, we will prove that that version of the Lambek Calculus which does not use the empty sequence is strongly complete w.r.t. those relational Kripke-models where the set of possible worlds,W, is a transitive binary relation, while that version of the Lambek Calculus where we admit the empty sequence as the antecedent of a sequent is strongly complete w.r.t. those relational models whereW=U×U for some setU. We will also look into extendability of this completeness result to various fragments of Girards Linear Logic as suggested in van Benthem (1991), p. 235, and investigate the connection between the Lambek Calculus and language models.

Completeness problems in verification of programs and program schemes

Thm 1 states a negative result about the classical semantics of program schemes. Thm 2 investigates the reason for this. We conclude that Thm 2 justifies the Henkin-type semantics ⊨ for which the opposite of the present Thm 1 was proved in Andreka-Nemeti[1],