94 research outputs found
First-order Nilpotent Minimum Logics: first steps
Following the lines of the analysis done in [BPZ07, BCF07] for first-order
G\"odel logics, we present an analogous investigation for Nilpotent Minimum
logic NM. We study decidability and reciprocal inclusion of various sets of
first-order tautologies of some subalgebras of the standard Nilpotent Minimum
algebra. We establish a connection between the validity in an NM-chain of
certain first-order formulas and its order type. Furthermore, we analyze
axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The
introduction section has been rewritten, and many modifications have been
done to improve the readability; moreover, numerous references have been
added. Concerning the technical side, some proofs has been shortened or made
more clear, but the mathematical content is substantially the same of the
previous versio
Valuations in Nilpotent Minimum Logic
The Euler characteristic can be defined as a special kind of valuation on
finite distributive lattices. This work begins with some brief consideration on
the role of the Euler characteristic on NM algebras, the algebraic counterpart
of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified
version of the Euler characteristic we call idempotent Euler characteristic. We
show that the new valuation encodes information about the formul{\ae} in NM
propositional logic
A note on drastic product logic
The drastic product is known to be the smallest -norm, since whenever . This -norm is not left-continuous, and hence it
does not admit a residuum. So, there are no drastic product -norm based
many-valued logics, in the sense of [EG01]. However, if we renounce standard
completeness, we can study the logic whose semantics is provided by those MTL
chains whose monoidal operation is the drastic product. This logic is called
in [NOG06]. In this note we justify the study of this
logic, which we rechristen DP (for drastic product), by means of some
interesting properties relating DP and its algebraic semantics to a weakened
law of excluded middle, to the projection operator and to
discriminator varieties. We shall show that the category of finite DP-algebras
is dually equivalent to a category whose objects are multisets of finite
chains. This duality allows us to classify all axiomatic extensions of DP, and
to compute the free finitely generated DP-algebras.Comment: 11 pages, 3 figure
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
On some axiomatic extensions of the monoidal T-norm based logic MTL : an analysis in the propositional and in the first-order case
The scientific area this book belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case. The book is divided in two parts: in the first one the necessary background about these logics, with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic: in chapter 9 the sets of tautologies of some NM-chains (subalgebras of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have been also presented in papers
ON SOME AXIOMATIC EXTENSIONS OF THE MONOIDAL T-NORM BASED LOGIC MTL: AN ANALYSIS IN THE PROPOSITIONAL AND IN THE FIRST-ORDER CASE
The scientific area this thesis belongs to are many-valued logics: in particular, the logic MTL and some of its extensions, in the propositional and in the first-order case (see [8],[9],[6],[7]). The thesis is divided in two parts: in the first one the necessary background about these logics,
with some minor new results, are presented. The second part is devoted to more specific topics: there are five chapters, each one about a different problem. In chapter 6 a temporal semantics for Basic Logic BL is
presented. In chapter 7 we move to first-order logics, by studying the supersoundness property: we have improved some previous works about this theme, by expanding the analysis to many extensions of the first-order version of MTL. Chapter 8 is dedicated to four different families of n-contractive axiomatic extensions of BL, analyzed in the propositional and in the first-order case: completeness, computational and arithmetical complexity, amalgamation and interpolation properties are studied. Finally, chapters 9 and 10 are about Nilpotent Minimum logic (NM, see [8]): in chapter 9 the sets of tautologies of some NM-chains (subalgebras
of [0,1]_NM) are studied, compared and the problems of axiomatization and undecidability are tackled. Chapter 10, instead, concerns some logical and algebraic properties of (propositional) Nilpotent Minimum logic. The results (or an extended version of them) of these last chapters have
been also presented in papers [1, 4, 5, 2, 3]. ---------------------------------References---------------------------------------------
[1] S. Aguzzoli, M. Bianchi, and V. Marra. A temporal semantics for Basic
Logic. Studia Logica, 92(2), 147-162, 2009. doi:10.1007/s11225-009-9192-3.
[2] M. Bianchi. First-order Nilpotent Minimum Logics: first steps. Submitted
for publication,2010.
[3] M. Bianchi. On some logical and algebraic properties of Nilpotent Minimum
logic and its relation with G\uf6del logic. Submitted for publication, 2010. [4] M. Bianchi and F. Montagna. Supersound many-valued logics and
Dedekind-MacNeille completions. Arch. Math. Log., 48(8), 719-736, 2009.
doi:10.1007/s00153-009-0145-3.
[5] M. Bianchi and F. Montagna. n-contractive BL-logics. Arch. Math. Log.,
2010. doi:10.1007/s00153-010-0213-8.
[6] P. Cintula, F. Esteva, J. Gispert, L. Godo, F. Montagna, and C. Noguera.
Distinguished algebraic semantics for t-norm based fuzzy logics: methods
and algebraic equivalencies. Ann. Pure Appl. Log., 160(1), 53-81, 2009.
doi:10.1016/j.apal.2009.01.012.
[7] P. Cintula and P. H\ue1jek. Triangular norm predicate fuzzy logics. Fuzzy Sets
Syst., 161(3), 311-346, 2010. doi:10.1016/j.fss.2009.09.006.
[8] F. Esteva and L. Godo. Monoidal t-norm based logic: Towards a
logic for left-continuous t-norms. Fuzzy sets Syst., 124(3), 271-288, 2001.
doi:10.1016/S0165-0114(01)00098-7.
[9] P. H\ue1jek. Metamathematics of Fuzzy Logic, volume 4 of Trends
in Logic. Kluwer Academic Publishers, paperback edition, 1998.
ISBN:9781402003707
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
This paper focuses on the issue of how generalizations of continuous and left-continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms
Nilpotent Minimum Logic NM and Pretabularity
This paper deals with pretabularity of fuzzy logics. For this, we first introduce two systems NMnfp and NM½, which are expansions of the fuzzy system NM (Nilpotent minimum logic), and examine the relationships between NMnfp and the another known extended system NM—. Next, we show that NMnfp and NM½ are pretabular, whereas NM is not. We also discuss their algebraic completeness.
 
DUALITIES AND REPRESENTATIONS FOR MANY-VALUED LOGICS IN THE HIERARCHY OF WEAK NILPOTENT MINIMUM.
In this thesis we study particular subclasses of WNM algebras.
The variety of WNM algebras forms the algebraic semantics of the
WNM logic, a propositional many-valued logic that generalizes some
well-known case in the setting of triangular norms logics.
WNM logic lies in the hierarchy of schematic extensions of MTL, which is
proven to be the logic of all left-continuous triangular norms and their residua.
In this work, I have extensively studied two extensions
of WNM logic, namely RDP logic and NMG logic, from the point of view of
algebraic and categorical logic.
We develop spectral dualities between the varieties of algebras
corresponding to RDP logic and NMG logic, and suitable defined combinatorial categories.
Categorical dualities allow to give algorithmic construction of products in
the dual categories obtaining computable descriptions of coproducts
(which are notoriously hard to compute working only in the algebraic side)
for the corresponding finite algebras. As a byproduct, representation theorems
for finite algebras and free finitely generated algebras in the considered varieties
are obtained. This latter characterization is especially useful to provide explicit
construction of a number of objects relevant from the point of view of the logical
interpretation of the varieties of algebras: normal forms, strongest deductive
interpolants and most general unifiers
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