scholarly journals A STUDY OF TRUTH PREDICATES IN MATRIX SEMANTICS

2018 ◽  
Vol 11 (4) ◽  
pp. 780-804 ◽  
Author(s):  
TOMMASO MORASCHINI
Keyword(s):  
Systematic Study ◽  
Open Problem ◽  
Algebraic Logic ◽  
Abstract Algebraic Logic ◽  
Theoretic Property ◽  
Intermediate Condition ◽  
Leibniz Operator ◽  
Matrix Semantics ◽  
Image Position ◽  
Order Reflection

AbstractAbstract algebraic logic is a theory that provides general tools for the algebraic study of arbitrary propositional logics. According to this theory, every logic ${\cal L}$ is associated with a matrix semantics $Mo{d^{\rm{*}}}{\cal L}$. This article is a contribution to the systematic study of the so-called truth sets of the matrices in $Mo{d^{\rm{*}}}{\cal L}$. In particular, we show that the fact that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ can be defined by means of equations with universally quantified parameters is captured by an order-theoretic property of the Leibniz operator restricted to deductive filters of ${\cal L}$. This result was previously known for equational definability without parameters. Similarly, it was known that the truth sets of $Mo{d^{\rm{*}}}{\cal L}$ are implicitly definable if and only if the Leibniz operator is injective on deductive filters of ${\cal L}$ over every algebra. However, it was an open problem whether the injectivity of the Leibniz operator transfers from the theories of ${\cal L}$ to its deductive filters over arbitrary algebras. We show that this is the case for logics expressed in a countable language, and that it need not be true in general. Finally we consider an intermediate condition on the truth sets in $Mo{d^{\rm{*}}}{\cal L}$ that corresponds to the order-reflection of the Leibniz operator.

COMPATIBILITY OPERATORS IN ABSTRACT ALGEBRAIC LOGIC

2016 ◽  
Vol 81 (2) ◽  
pp. 417-462 ◽  
Author(s):  
HUGO ALBUQUERQUE ◽  
JOSEP MARIA FONT ◽  
RAMON JANSANA
Keyword(s):  
Algebraic Logic ◽  
Special Kind ◽  
Integrated Approach ◽  
Abstract Algebraic Logic ◽  
Unified Framework ◽  
Recent Developments ◽  
Leibniz Operator ◽  
General Correspondence ◽  
Correspondence Theorem ◽  
Leibniz Hierarchy

AbstractThis paper presents a unified framework that explains and extends the already successful applications of the Leibniz operator, the Suszko operator, and the Tarski operator in recent developments in abstract algebraic logic. To this end, we refine Czelakowski’s notion of an S-compatibility operator, and introduce the notion of coherent family of S-compatibility operators, for a sentential logic S. The notion of coherence is a restricted property of commutativity with inverse images by surjective homomorphisms, which is satisfied by both the Leibniz and the Suszko operators. We generalize several constructions and results already existing for the mentioned operators; in particular, the well-known classes of algebras associated with a logic through each of them, and the notions of full generalized model of a logic and a special kind of S-filters (which generalizes the less-known notion of Leibniz filter). We obtain a General Correspondence Theorem, extending the well-known one from the theory of protoalgebraic logics to arbitrary logics and to more general operators, and strengthening its formulation. We apply the general results to the Leibniz and the Suszko operators, and obtain several characterizations of the main classes of logics in the Leibniz hierarchy by the form of their full generalized models, by old and new properties of the Leibniz operator, and by the behaviour of the Suszko operator. Some of these characterizations complete or extend known ones, for some classes in the hierarchy, thus offering an integrated approach to the Leibniz hierarchy that uncovers some new, nice symmetries.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
Author(s):  
SAMUEL R. BUSS ◽  
LESZEK ALEKSANDER KOŁODZIEJCZYK ◽  
NEIL THAPEN
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

An abstract algebraic logic approach to tetravalent modal logics

2000 ◽  
Vol 65 (2) ◽  
pp. 481-518 ◽  
Author(s):  
Josep Maria Font ◽  
Miquel Rius
Keyword(s):  
Algebraic Logic ◽  
Modal Logics ◽  
The Other ◽  
Abstract Algebraic Logic ◽  
Algebraic Semantics ◽  
Modal Algebras ◽  
Logic Approach ◽  
Abstract Logics ◽  
Joint Study

AbstractThis paper contains a joint study of two sentential logics that combine a many-valued character, namely tetravalence, with a modal character; one of them is normal and the other one quasinormal. The method is to study their algebraic counterparts and their abstract models with the tools of Abstract Algebraic Logic, and particularly with those of Brown and Suszko's theory of abstract logics as recently developed by Font and Jansana in their “A General Algebraic Semantics for Sentential Logics”. The logics studied here arise from the algebraic and lattice-theoretical properties we review of Tetravalent Modal Algebras, a class of algebras studied mainly by Loureiro, and also by Figallo. Landini and Ziliani, at the suggestion of the late Antonio Monteiro.

scholarly journals Categorical Abstract Algebraic Logic: Equivalential π-Institutions

2008 ◽  
Vol 6 ◽  
Author(s):  
George Voutsadakis
Keyword(s):  
Matrix Theory ◽  
Algebraic Logic ◽  
Characterization Theorem ◽  
Abstract Algebraic Logic ◽  
Transfer Theorem ◽  
Deductive Systems

The theory of equivalential deductive systems, as introduced by Prucnal and Wrónski and further developed by Czelakowski, is abstracted to cover the case of logical systems formalized as π-institutions. More precisely, the notion of an N-equivalence system for a given π-institution is introduced. A characterization theorem for N-equivalence systems, previously proven for N-parameterized equivalence systems, is revisited and a “transfer theorem” for N-equivalence systems is proven. For a π-institution I having an N-equivalence system, the maximum such system is singled out and, then, an analog of Herrmann’s Test, characterizing those N-protoalgebraic π-institutions having an N-equivalence system, is formulated. Finally, some of the rudiments of matrix theory are revisited in the context of π-institutions, as they relate to the existence of N-equivalence systems.

scholarly journals A COMPUTABLE FUNCTOR FROM GRAPHS TO FIELDS

2018 ◽  
Vol 83 (1) ◽  
pp. 326-348 ◽  
Author(s):  
RUSSELL MILLER ◽  
BJORN POONEN ◽  
HANS SCHOUTENS ◽  
ALEXANDRA SHLAPENTOKH
Keyword(s):  
Model Theory ◽  
Open Problem ◽  
Category Theory ◽  
Computable Model ◽  
Computable Model Theory ◽  
Arbitrary Characteristic ◽  
Faithful Functor ◽  
New Construction ◽  
Image Position ◽  
Countable Structure

AbstractFried and Kollár constructed a fully faithful functor from the category of graphs to the category of fields. We give a new construction of such a functor and use it to resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure${\cal S}$, there exists a countable field${\cal F}$of arbitrary characteristic with the same essential computable-model-theoretic properties as${\cal S}$. Along the way, we develop a new “computable category theory”, and prove that our functor and its partially defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.

The Hilbert transform of Schwartz distributions. II

1987 ◽  
Vol 102 (3) ◽  
pp. 553-559 ◽  
Author(s):  
M. Aslam Chaudhry ◽  
J. N. Pandey
Keyword(s):  
Hilbert Transform ◽  
Open Problem ◽  
Compact Support ◽  
Schwartz Space ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Schwartz Distributions ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

AbstractLet D(R) be the Schwartz space of C∞ functions with compact support on R and let H(D) be the space of all C∞ functions defined on R for which every element is the Hilbert transform of an element in D(R), i.e.where the integral is defined in the Cauchy principal-value sense. Introducing an appropriate topology in H(D), Pandey [3] defined the Hilbert transform Hf of f ∈ (D(R))′ as an element of (H(D))′ by the relationand then with an appropriate interpretation he proved that.In this paper we give an intrinsic description of the space H(D) and its topology, thereby providing a solution to an open problem posed by Pandey ([4], p. 90).

scholarly journals A Survey on Bandwidth Management in MANET

2018 ◽  
Vol 7 (3.4) ◽  
pp. 38 ◽  
Author(s):  
Nibedita Jagadev ◽  
Binod Kumar Pattanayak ◽  
Debabrata Singh ◽  
Sipra Sahoo
Keyword(s):  
Systematic Study ◽  
Open Problem ◽  
Ad Hoc Network ◽  
Ad Hoc ◽  
Mobile Ad Hoc Network ◽  
Vital Role ◽  
Network Throughput ◽  
Bandwidth Management ◽  
Overall Performance ◽  
Mobile Ad Hoc

Mobile ad-hoc network (MANET) are self-configuring without any fixed infrastructure. In MANET the challenge of providing QoS support during multimedia application is an open problem.  To enhance the network throughput adequate amount of bandwidth is required. Bandwidth management in MANET plays a vital role in improving the networks overall performance. In this paper a systematic study has been carried out about challenging issues associated with bandwidth management during routing in MANET and analysis has been done about different models of bandwidth management. 

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