103 research outputs found
On fuzzy reasoning schemes
In this work we provide a short survey of the most frequently used fuzzy
reasoning schemes. The paper is organized as follows: in the first section
we introduce the basic notations and definitions needed for fuzzy inference
systems; in the second section we explain how the GMP works under Mamdani,
Larsen and Gšodel implications, furthermore we discuss the properties
of compositional rule of inference with several fuzzy implications; and in
the third section we describe Tsukamotoâs, Sugenoâs and the simplified fuzzy
inference mechanisms in multi-input-single-output fuzzy systems
Weighted logics for artificial intelligence : an introductory discussion
International audienceBefore presenting the contents of the special issue, we propose a structured introductory overview of a landscape of the weighted logics (in a general sense) that can be found in the Artificial Intelligence literature, highlighting their fundamental differences and their application areas
A map of dependencies among three-valued logics
International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value
One-variable fragments of first-order logics
The one-variable fragment of a first-order logic may be viewed as an
"S5-like" modal logic, where the universal and existential quantifiers are
replaced by box and diamond modalities, respectively. Axiomatizations of these
modal logics have been obtained for special cases -- notably, the modal
counterparts S5 and MIPC of the one-variable fragments of first-order classical
logic and intuitionistic logic -- but a general approach, extending beyond
first-order intermediate logics, has been lacking. To this end, a sufficient
criterion is given in this paper for the one-variable fragment of a
semantically-defined first-order logic -- spanning families of intermediate,
substructural, many-valued, and modal logics -- to admit a natural
axiomatization. More precisely, such an axiomatization is obtained for the
one-variable fragment of any first-order logic based on a variety of algebraic
structures with a lattice reduct that has the superamalgamation property,
building on a generalized version of a functional representation theorem for
monadic Heyting algebras due to Bezhanishvili and Harding. An alternative
proof-theoretic strategy for obtaining such axiomatization results is also
developed for first-order substructural logics that have a cut-free sequent
calculus and admit a certain interpolation property.Comment: arXiv admin note: text overlap with arXiv:2209.0856
A modal theorem-preserving translation of a class of three-valued logics of incomplete information
International audienceThere are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, áŽukasiewicz and Nelson logics. We show that Priestâs logic of paradox, closely connected to Kleeneâs, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management
Propositional Logics Complexity and the Sub-Formula Property
In 1979 Richard Statman proved, using proof-theory, that the purely
implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He
showed a polynomially bounded translation from full Intuitionistic
Propositional Logic into its implicational fragment. By the PSPACE-completeness
of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic
Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle
for a deductive system for a logic L states that whenever F1,...,Fk proves A,
there is a proof in which each formula occurrence is either a sub-formula of A
or of some of Fi. In this work we extend Statman result and show that any
propositional (possibly modal) structural logic satisfying a particular
formulation of the sub-formula principle is in PSPACE. If the logic includes
the minimal purely implicational logic then it is PSPACE-complete. As a
consequence, EXPTIME-complete propositional logics, such as PDL and the
common-knowledge epistemic logic with at least 2 agents satisfy this particular
sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our
technique can be used to prove that any finitely many-valued logic has the set
of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192
Interpolation Methods for Binary and Multivalued Logical Quantum Gate Synthesis
A method for synthesizing quantum gates is presented based on interpolation
methods applied to operators in Hilbert space. Starting from the diagonal forms
of specific generating seed operators with non-degenerate eigenvalue spectrum
one obtains for arity-one a complete family of logical operators corresponding
to all the one-argument logical connectives. Scaling-up to n-arity gates is
obtained by using the Kronecker product and unitary transformations. The
quantum version of the Fourier transform of Boolean functions is presented and
a Reed-Muller decomposition for quantum logical gates is derived. The common
control gates can be easily obtained by considering the logical correspondence
between the control logic operator and the binary propositional logic operator.
A new polynomial and exponential formulation of the Toffoli gate is presented.
The method has parallels to quantum gate-T optimization methods using powers of
multilinear operator polynomials. The method is then applied naturally to
alphabets greater than two for multi-valued logical gates used for quantum
Fourier transform, min-max decision circuits and multivalued adders
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