11 research outputs found
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
T-Norms Driven Loss Functions for Machine Learning
Neural-symbolic approaches have recently gained popularity to inject prior
knowledge into a learner without requiring it to induce this knowledge from
data. These approaches can potentially learn competitive solutions with a
significant reduction of the amount of supervised data. A large class of
neural-symbolic approaches is based on First-Order Logic to represent prior
knowledge, relaxed to a differentiable form using fuzzy logic. This paper shows
that the loss function expressing these neural-symbolic learning tasks can be
unambiguously determined given the selection of a t-norm generator. When
restricted to supervised learning, the presented theoretical apparatus provides
a clean justification to the popular cross-entropy loss, which has been shown
to provide faster convergence and to reduce the vanishing gradient problem in
very deep structures. However, the proposed learning formulation extends the
advantages of the cross-entropy loss to the general knowledge that can be
represented by a neural-symbolic method. Therefore, the methodology allows the
development of a novel class of loss functions, which are shown in the
experimental results to lead to faster convergence rates than the approaches
previously proposed in the literature
On triangular norms and uninorms definable in ĆÎ 12
AbstractIn this paper, we investigate the definability of classes of t-norms and uninorms in the logic ĆÎ 12. In particular we provide a complete characterization of definable continuous t-norms, weak nilpotent minimum t-norms, conjunctive uninorms continuous on [0,1), and idempotent conjunctive uninorms, and give both positive and negative results concerning definability of left-continuous t-norms (and uninorms). We show that the class of definable uninorms is closed under construction methods as annihilation, rotation and rotationâannihilation. Moreover, we prove that every logic based on a definable uninorm is in PSPACE, and that any finitely axiomatizable logic based on a class of definable uninorms is decidable. Finally we show that the Uninorm Mingle Logic (UML) and the Basic Uninorm Logic (BUL) are finitely strongly standard complete w.r.t. the related class of definable left-continuous conjunctive uninorms
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
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
Fitting aggregation operators to data
Theoretical advances in modelling aggregation of information produced a wide range of aggregation operators, applicable to almost every practical problem. The most important classes of aggregation operators include triangular norms, uninorms, generalised means and OWA operators.With such a variety, an important practical problem has emerged: how to fit the parameters/ weights of these families of aggregation operators to observed data? How to estimate quantitatively whether a given class of operators is suitable as a model in a given practical setting? Aggregation operators are rather special classes of functions, and thus they require specialised regression techniques, which would enforce important theoretical properties, like commutativity or associativity. My presentation will address this issue in detail, and will discuss various regression methods applicable specifically to t-norms, uninorms and generalised means. I will also demonstrate software implementing these regression techniques, which would allow practitioners to paste their data and obtain optimal parameters of the chosen family of operators.<br /
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