A Shared Framework for Consequence Operations and Abstract Model Theory

In this paper we develop an abstract theory of adequacy. In the same way as the theory of consequence operations is a general theory of logic, this theory of adequacy is a general theory of the interactions and connections between consequence operations and its sound and complete semantics. Addition of axioms for the connectives of propositional logic to the basic axioms of consequence operations yields a unifying framework for different systems of classical propositional logic. We present an abstract model-theoretical semantics based on model mappings and theory mappings. Between the classes of models and theories, i.e., the set of sentences verified by a model, it obtains a connection that is well-known within algebra as Galois correspondence. Many basic semantical properties can be derived from this observation. A sentence A is a semantical consequence of T if every model of T is also a model of A. A model mapping is adequate for a consequence operation if its semantical inference operation is identical with the consequence operation. We study how properties of an adequate model mapping reflect the properties of the consequence operation and vice versa. In particular, we show how every concept of the theory of consequence operations can be formulated semantically.     

A denotational semantics ofLC2

The aim of this paper is to extend the classical sequent calculusLC to the second order. This task is realized by a semantical approach mixing the correlation spaces semantics ofLC on the one hand, and the analogy with the interpretation of systemF in coherent spaces on the other hand. This relies on the introduction of a new semantical object:noetherian correlation spaces.From the semantics we deduce the syntax of the second order classical sequent calculusLC2.     

Communication and understanding mathematical foundations and practical applications

Since Shannon??s and Weaver??s ??Mathematical theory of Communication?? it is well known that mathematical definitions of information or the degree of information respectively is possible. The great problem for a complete theory of communication is the exact definition of meaning in mathematical terms. I shall demonstrate how such a definition can be achieved in terms of complex systems theory. In particular it is possible to derive exact definitions of the degree of meaning, applied to semantical networks, and of the degree of information that is suited for the analysis of human communication. The degree of information as well as the degree of meaning is dependent on the geometry of the receiving systems, which are modeled as semantical networks. It can be shown that the knowledge about the geometrical structure allows predictions about the degrees of information and of meaning a message has for a certain receiving system, although of course only on the average. In a semantical application of these fundamental concepts it will be shown how a new self organized leaning neural network that we have developed is able to deal with linguistic ambiguities. Additional applications of the communication theory will be demonstrated, in particular an Internet meta search engine based on computing the degrees of information and meaning, and a computer based discourse analysis.     

On the synthesis of finite sequential machines

Starting from the studies of Kleene and Mealy on sequential machines, in this paper is presented a formalism which, in a sense, unifies their treatments. From the specification of the required machine behaviour in terms of events and associated output states, a uniform procedure is given for obtaining a transition table and from that a minimal machine, whenever such a complete reduction is possible. The various steps of the synthesis procedure are so stated that they can be easily programmed on a computer.     

Su alcuni aspetti assiomatici del criterio di Nagata

Nagata's criterion for the openness of the regular loci can be stated for any propertyP of locally noetherian schemes and was proved to be true for many of them. We give reasonable sets of axioms forP, which are satisfied by the usual properties (Reg, Cohen-Macaulay, Gorenstein etc.…), from which a formal proof of Nagata's criterion is possible.     

Relating and extending semantical approaches to possibilistic reasoning

Two main semantical approaches to possibilistic reasoning with classical propositions have been proposed in the literature. Namely, Dubois-Prade's approach known as possibilistic logic, whose semantics is based on a preference ordering in the set of possible worlds, and Ruspini's approach that we redefine and call similarity logic, which relies on the notion of similarity or resemblance between worlds. In this article we put into relation both approaches, and it is shown that the monotonic fragment of possibilistic logic can be semantically embedded into similarity logic. Furthermore, to extend possibilistic reasoning to deal with fuzzy propositions, a semantical reasoning framework, called fuzzy truth-valued logic, is also introduced and proved to capture the semantics of both possibilistic and similarity logics.     

Mathematical developments arising from ‘semantical implication’ and the evaluation of membership characteristic functions

In this paper, the actuation of a semantical ordering in a property set is exposed. Such an ordering follows from the concept of semantical implication: the relation that takes place between two properties, when the values of the details forming one of these are the same as of the other excepting those which, in the implied property, are irrelevant. Mainly, the argumentation deals with the analysis of the operations which, due to the relations in subject, are allowed in the property set considered. This, with the direct purpose of attaining to the structure of the set itself.     

A Routley–Meyer Semantics for Gödel 3-Valued Logic and Its Paraconsistent Counterpart

Routley–Meyer semantics (RM-semantics) is defined for Gödel 3-valued logic G3 and some logics related to it among which a paraconsistent one differing only from G3 in the interpretation of negation is to be remarked. The logics are defined in the Hilbert-style way and also by means of proof-theoretical and semantical consequence relations. The RM-semantics is defined upon the models for Routley and Meyer’s basic positive logic B₊, the weakest positive RM-semantics. In this way, it is to be expected that the models defined can be adapted to other related many-valued logics.     

Sull'estensione del principio di Lagrange-d'Alembert nella dinamica relativistica di un fluido ideale

The aim of this paper is to extend to the ideal relativistic fluid dynamics the classical principle of Lagrange-d'Alembert when the external body forces are absent. There are considered separately the adiabatic fluxes and the fluxes involving an entropy variation along the stream line of generic particle of the medium.     

Implementazione di tests probabilistici di primalità

Sunto In questo lavoro si considerano due tests probabilistici di primalità per interi disparim di forma qualsiasi. Gli algoritmi sono tali che sem è dichiarato composto allora lo è certamente, mentre sem è dichiarato primo il risultato ha un certo margine di errore, che può essere reso arbitrariamente piccolo. Un programma scritto e compilato in linguaggio FORTRAN, applicabile ad interi fino a 10²⁰⁰⁰ ed adatto anche a personal computer, permette un confronto dei due tests sulla base del tipo dei risultati e del tempo di elaborazione, fornendo diverse opzioni ed una stima del limite superiore per l’eventuale errore in una dichiarazione di probabile primalità.

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Summary In this paper we consider two probabilistic tests for determining the primality of any odd integerm. Both algorithms have the property that ifm is declared to be composite than it is guaranteed to be so. On the other hand, ifm is declared to be prime than it is sure to be prime only within a certain margin of error which can be made arbitrarily small. A program, written and compiled in the computer language FORTRAN enable this tests to be applied to integers up to 10²⁰⁰⁰, allowing a comparison between the two tests with respect to their running time and with an esplicit bound for the maximum possible error whenm is declared prime. The program is sufficiently compact to be implemented on a personal computer such as an IBM PS/2 with an arithmetic co-processor

     

Shaking table experimental study on the effectiveness of polymer bearings for seismic isolation of structures

Seismic isolation has been recognised to be a very effective way of protecting structures from damage during earthquakes. It allows us to extend the natural period of the structure and therefore avoid resonance with the ground motion. Moreover, by increasing damping in the isolation devices, more energy can be dissipated and thus the structural response can be further reduced. The aim of this paper is to show the results of the study focused on verification of the applicability of the innovative method of seismic isolation by installing bearings made of a polymer mass, which is an especially designed flexible elastoplastic two-component grout based on polyurethane resin. In the study, a model structure was tested experimentally on a shaking table under earthquake excitation. First, the structure was fixed directly to the table and its response was recorded. Then, the response of the structure with polymer bearings installed at its base was observed and both responses were compared. The results of the study show that equipping the structure with the polymer bearings can considerably reduce the structural response under earthquake excitation. The innovative method considered in the study has been verified to be an effective seismic isolation technique. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Categories with families and first-order logic with dependent sorts

First-order logic with dependent sorts, such as Makkai's first-order logic with dependent sorts (FOLDS), or Aczel's and Belo's dependently typed (intuitionistic) first-order logic (DFOL), may be regarded as logic enriched dependent type theories. Categories with families (cwfs) is an established semantical structure for dependent type theories, such as Martin-Löf type theory. We introduce in this article a notion of hyperdoctrine over a cwf, and show how FOLDS and DFOL fit in this semantical framework. A soundness and completeness theorem is proved for DFOL. The semantics is functorial in the sense of Lawvere, and uses a dependent version of the Lindenbaum-Tarski algebra for a DFOL theory. Agreement with standard first-order semantics is established. Applications of DFOL to constructive mathematics and categorical foundations are given. A key feature is a local propositions-as-types principle.     

Densification via polynomials,languages, and frames

It is known that every countable totally ordered set can be embedded into a countable dense one. We extend this result to totally ordered commutative monoids and to totally ordered commutative residuated lattices (the latter result fails in the absence of commutativity). The latter has applications to density elimination of semilinear substructural logics. In particular we obtain as a corollary a purely algebraic proof of the standard completeness of uninorm logic; the advantage over the known proof-theoretic proof and the semantical proof is that it is extremely short and transparent and all details can be verified easily using standard algebraic constructions.     

Some properties of the basis finite automaton

The present paper discusses non-deterministic finite Rabin-Scott’s automata. The majority of works recently dealing with this subject were, in fact, concerned only with properties of a canonical form automata or of some objects equivalent to it. This article continues the series of works in which the authors state a different point of view, describing the finite automata as just another invariant of the given regular language called basis finite automaton. In this article the authors argue on some new properties for the basis finite automaton. One of them is included into basis automaton’s table of binary relations. It is stated that this table can not contain either identical strings or identical columns. Another property depicts a possibility to obtain any finite automaton for a given regular language by the process of duplicating or combining some of its states.     

On the Toeplitz pencil conjecture

The validity of a conjecture on a singular Toeplitz pencil over ?[x] stated by Schmale and Sharma in connection with an old problem in control theory is presented in this article. The aim of this work is to give a proof of this conjecture.     

A semantical proof of the strong normalization theorem for full propositional classical natural deduction

We give in this paper a short semantical proof of the strong normalization for full propositional classical natural deduction. This proof is an adaptation of reducibility candidates introduced by J.-Y. Girard and simplified to the classical case by M. Parigot.     

THE OPERATIONAL VIEWPOINT AND SOME OF ITS CONSEQUENCES: A SHORT REVIEW

This paper presents first results on fuzzy set theory (with the emphasis on probability theoretical aspects) from an operational viewpoint, a new approach that claims to be a way out of some of the difficulties and to present a consistent framework for the set-theoretical aspects of fuzzy set theory (e.g.the complementation paradoxes vanish, and fuzzy sets become a topos under this interpretation). As a preliminary conclusion of this approach we conjecture that the concept of fuzzy set has two essentially different semantical dimensions, which have not sufficiently been separated in the traditional theory, they may be considered either as an evaluation over a universe (i.e. as a connotation) or as a collection of evaluated object (i.e.as a denotation).     

Scalar restriction for formal groups

The aim of this work is to generalize the results on the scalar restriction functor for formal groups stated in the paper “Explicit classification of formal groups over local fields” by the first two authors. These results are extended to formal groups over an arbitrary commutative ring, which we assume to be an integral domain or a ring of characteristic zero if necessary. Bibliography: 6 titles. Deceased. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 319, 2004, pp. 59–70.     

Teoremi di normalizzazione per l'omotopia regolare dei grafi

Given a metric compact spaceS and a finite graphG we show that:

On several results about convex set functions

In 1979, in an interesting paper, R.J. Morris introduced the notion of convex set function defined on an atomless finite measure space. After a short period this notion, as well as generalizations of it, began to be studied in several papers. The aim was to obtain results similar to those known for usual convex (or generalized convex) functions. Unfortunately several notions are ambiguous and the arguments used in the proofs of several results are not clear or not correct. In this way there were stated even false results. The aim of this paper is to point out that using some simple ideas it is possible, on one hand, to deduce the correct results by means of convex analysis and, on the other hand, to emphasize the reasons for which there are problems with other results.