A temporal semantics for Nilpotent Minimum logic

In [Ban97] a connection among rough sets (in particular, pre-rough algebras) and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95, EG01]), in which the logic of every instant is given by {\L}3: a completeness theorem will be shown. This is the prosecution of the work initiated in [AGM08] and [ABM09], in which the authors construct a temporal semantics for the many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table

A Theory of Continuum Economies with Independent Shocks and Matchings

Numerous economic models employ a continuum of negligible agents with a sequence of idiosyncratic shocks and random matchings. Several attempts have been made to build a rigorous mathematical justification for such models, but these attempts have left many questions unanswered. In this paper, we develop a discrete time framework in which the major, desirable properties of idiosyncratic shocks and random matchings hold. The agents live on a probability space, and the probability distribution for each agent is naturally replaced by the population distribution. The novelty of this approach is in the assumption of unknown identity. Each agent believes that initially he was randomly and uniformly placed on the agent space, i.e., the agent's identity (the exact location on the agent space) is unknown to the agent.random matching, idiosyncratic shocks, the Law of Large Numbers, aggregate uncertainty, mixing

A Theory of Continuum Economies with Idiosyncratic Shocks and Random Matchings

Many economic models use a continuum of negligible agents to avoid considering one person's effect on aggregate characteristics of the economy. Along with a continuum of agents, these models often incorporate a sequence of independent shocks and random matchings. Despite frequent use of such models, there are still unsolved questions about their mathematical justification. In this paper we construct a discrete time framework, in which major desirable properties of idiosyncratic shocks and random matchings hold. In this framework the agent space constitutes a probability space, and the probability distribution for each agent is replaced by the population distribution. Unlike previous authors, we question the assumption of known identity - the location on the agent space. We assume that the agents only know their previous history - what had happened to them before, - but not their identity. The construction justifies the use of numerous dynamic models of idiosyncratic shocks and random matchings.random matching; idiosyncratic shocks; the Law of Large Numbers; aggregate uncertainty; mixing

Hyperstates of Involutive MTL-Algebras that Satisfy (2x)2=2(x2)(2x)^2 = 2(x^2)

States of MV-algebras have been the object of intensive study and attempts of generalizations. The aim of this contribution is to provide a preliminary investigation for states of prelinear semihoops and hyperstates of algebras in the variety generated by perfect and involutive MTL-algebras (IBP0-algebras for short). Grounding on a recent result showing that IBP0-algebras can be constructed from a Boolean algebra, a prelinear semihoop and a suitably defined operator between them, our first investigation on states of prelinear semihoops will support and justify the notion of hyperstate for IBP0- algebras and will actually show that each such map can be represented by a probability measure on its Boolean skeleton, and a state on a suitably defined abelian l-group.Comment: 12 page

Homomorphism Preservation Theorems for Many-Valued Structures

A canonical result in model theory is the homomorphism preservation theorem which states that a first-order formula is preserved under homomorphisms on all structures if and only if it is equivalent to an existential-positive formula, standardly proved compactness. Rossman (2008) established that the h.p.t. remains valid when restricted to finite structures. This is a significant result in the field of finite model theory, standing in contrast to other preservation theorems and as an theorem which remains true in the finite but whose proof uses entirely different methods. It also has importance to the field of constraint satisfaction due to the equivalence of existential-positive formulas and unions of conjunctive queries. Adjacently, Dellunde and Vidal (2019) established that a version of the h.p.t. holds for a collection of first-order many-valued logics, those whose (possibly infinite) structures are defined over a fixed finite MTL-chain. In this paper we unite these two strands, showing how one can extend Rossman's proof of a finite h.p.t. to a very wide collection of many-valued predicate logics and simultaneously establishing a finite variant to Dellunde and Vidal's result, one which not only applies to structures defined over algebras more general than MTL-chains but also where we allow for those algebra to vary between models. This investigation provides a starting point in a wider development of finite model theory for many-valued logics and, just as the classical finite h.p.t. has implications for constraint satisfaction, the many-valued finite h.p.t. has implications for valued constraint satisfaction problems.Comment: 22 page

On Termination for Faulty Channel Machines

A channel machine consists of a finite controller together with several fifo channels; the controller can read messages from the head of a channel and write messages to the tail of a channel. In this paper, we focus on channel machines with insertion errors, i.e., machines in whose channels messages can spontaneously appear. Such devices have been previously introduced in the study of Metric Temporal Logic. We consider the termination problem: are all the computations of a given insertion channel machine finite? We show that this problem has non-elementary, yet primitive recursive complexity

Rational and Delta expansions of the Nilpotent Minimum Logic

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona. Curs: 202x-202x. Tutor: xxx[eng] The aim of this thesis is to study some expansions of the Nilpotent minimum logic (denoted by NML), focusing on their lattices of axiomatic and finitary extensions and, additionally, exploring the structural completeness of these logics, alongside their variants (active structural completeness, passive structural completeness, ... ). The project includes research about the rational Nilpotent minimum logic (designated by RNML), which is obtained by adding rational constants to the language of NML. Moreover, we also study the Δ-core fuzzy logic obtained by expanding the language of NML with the Baaz Delta connective and examine the impact of the incorporation of rational constants to the language of this logic (which is equivalent to the addition of the Baaz Delta connective to RNML). The thesis culminates with the corresponding analysis of an extension of the later logic which is obtained by introducing bookkeeping axioms for the Δ operator, motivated by the aim for the algebra of constants to form a subalgebra. In the project, through comparative analysis, the differences and similarities between the lattices of axiomatic and finitary extensions among the previously mentioned expansions are evaluated, as well as how the structural completeness results obtained may vary from one logic to another.[spa] El objetivo de esta tesis es estudiar algunas expansiones de la lógica del Nilpotente mínimo (denotada por NML), centrándonos en sus retículos de extensiones axiomáticas y finitas y, además, explorando la completitud estructural de estas lógicas, junto con sus variantes (completitud estructural activa, completitud estructural pasiva, ...). El proyecto abarca la lógica racional del Nilpotente mínimo (designada por RNML), que se obtiene añadiendo constantes racionales al lenguaje de NML. También se estudia la lógica fuzzy Δ-core obtenida mediante la expansión del lenguaje de NML con el operador Delta de Baaz, y se examina el impacto de la incorporaci´on de constantes racionales al lenguaje de esta lógica (lo que equivale a añadir el operador Delta de Baaz a RNML). La tesis culmina con el correspondiente análisis de una extensión de la ´ultima lógica presentada, resultante de la introducción de bookkeeping axioms para el operador Δ, motivada por el objetivo de que el álgebra de constantes forme una subálgebra. En el proyecto, a través de un análisis comparativo, se evalúan las diferencias y similitudes entre los retículos de extensiones axiomáticas y finitas de las distintas expansiones mencionadas anteriormente, así como la forma en que varían los resultados de completitud estructural de una lógica a otra.[cat] L’objectiu d’aquesta tesi és estudiar algunes expansions de la lògica del Nilpotent mínim (denotada per NML), centrant-nos en els seus reticles d’extensions axiomàtiques i finites i, a més, explorant la completitud estructural d’aquestes lògiques, juntament amb les seves variants (completitud estructural activa, completitud estructural passiva, ...). El projecte abasta la lògica racional del Nilpotent mínim (designada per RNML), que s’obté afegint constants racionals al llenguatge de NML. També s’estudia la lògica fuzzy Δ-core obtinguda mitjançant l’expansió del llenguatge de NML amb l’operador Delta de Baaz, i s’examina l’impacte de la incorporació de constants racionals al llenguatge d’aquesta lògica (el que equival a afegir l’operador Delta de Baaz a RNML). La tesi culmina amb l’anàlisi corresponent d’una extensió de l'última lògica presentada, que resulta de la introducció de bookkeeping axioms per a l’operador Δ, motivada per l’objectiu que l'àlgebra de constants formi una subàlgebra. En el projecte, mitjançant una anàlisi comparativa, s’avaluen les diferències i similituds entre els reticles d’extensions axiomàtiques i finites de les diferents expansions esmentades anteriorment, així com la manera com varien els resultats de completitud estructural d’una lògica a una altra