1,721,002 research outputs found
On countermodels in basic logic
No full textIn [3] the tautology problem for Hájek's Basic Logic BL is proved to be co-NP-complete by showing that if a formula φ is not a tautology of BL then there exists an integer m > 0, polynomially bounded by the length of φ, such that φ fails to be a tautology in the infinite-valued logic mŁ corresponding to the ordinal sum of m copies of the Łukasiewicz t-norm. In this paper we state that if φ is not a tautology of BL then it already fails to be a tautology of a finite set of finite-valued logics, defined by taking the ordinal sum of m copies of k-valued Łukasiewicz logics, for effectively determined integers m, k > 0 only depending on polynomial-time computable features of φ. This result allows the definition of a calculus for mŁ along the lines of [1], [2], while the analysis of the features of functions associated with formulas of mŁ constitutes a step toward the characterization of finitely generated free BL-algebras as algebras of [0, 1]-valued functions
A logical framework for fuzzy collaborative filtering
No full textSystems which predict the items a user would like, on the basis of the preferences given by other users, are attracting growing attention as automated recommender services on the Internet. Collaborative filtering techniques are widely used to implement such recommender systems. We show how fuzzy sets and many-valued logics can be fruitfully applied in the description and design of collaborative filtering methods
Finite-valued reductions of infinite-valued logics
No full textIn this paper we present a method to reduce the decision problem of several infinite-valued propositional logics to their finite-valued counterparts. We apply our method to Łukasiewicz, Gödel and Product logics and to some of their combinations. As a byproduct we define sequent calculi for all these infinite-valued logics and we give an alternative proof that their tautology problems are in co-NP
Poset representation for Godel and Nilpotent Minimum logics
No full textMTL is the logic of all left-continuous t-norms and their residua. Its algebraic semantics is constituted by the variety V(MTL) of MTL-algebras. Among schematic extensions of MTL there are infinite-valued logics L such that the finitely generated free algebras in the corresponding subvariety V(L) of V(MTL) are finite. In this paper we focus on Godel and Nilpotent Minimum logics. We give concrete representations of their associated free algebras in terms of finite algebras of sections over finite posets
Comparing the Expressive Power of Some Fuzzy Logics Based on Residuated t-norms
No full textIn this paper we deal with the expressive power of some logics based on residuated left-continuous t-norms. We investigate the class of truth functions for Nilpotent Minimum, Gödel and NMG logics counting the number of different elements and describing normal forms which generalize the classical Boolean sum of minterms and product of maxterms. It turns out that the logics considered in the paper have much greater expressive power than Boolean propositional logic, while the complexity of their normal forms remains almost as manageable as Boolean normal forms
Probability measures in the logic of Nilpotent Minimum
No full textWe axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n-generated NM-algebra {{\fancyscript{NM}_{n}}} exactly correspond to integrals of elements of {{\fancyscript{NM}_{n}}} with respect to Borel probability measures
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