An Alternative Definition of Quantifiers on Four-Valued Łukasiewicz Algebras

An alternative notion of an existential quantifier on four-valued ?ukasiewicz algebras is introduced. The class of four-valued ?ukasiewicz algebras endowed with this existential quantifier determines a variety which is denoted by \(\mathbb {M}_{\frac{2}{3}}\mathbb {L}_4\). It is shown that the alternative existential quantifier is interdefinable with the standard existential quantifier on a four-valued ?ukasiewicz algebra. Some connections between the new existential quantifier and the existential quantifiers defined on bounded distributive lattices and Boolean algebras are given. Finally, a completeness theorem for the monadic four-valued ?ukasiewicz predicate calculus corresponding to the dual of the alternative existential quantifier is proven.     

Product Ł ukasiewicz Logic

u logic plays a fundamental role among many-valued logics. However, the expressive power of this logic is restricted to piecewise linear functions. In this paper we enrich the language of u logic by adding a new connective which expresses multiplication. The resulting logic, P, is defined, developed, and put into the context of other well-known many-valued logics. We also deal with several extensions of this propositional logic. A predicate version of P logic is introduced and developed too.The work of the first author was supported by the Grant Agency of the Czech Republic under project GACR 201/02/1540, by the Grant Agency of the Czech Technical University in Prague under project CTU 0208613, and by Net CEEPUS SK-042.The work of the second author was supported by grant IAA1030004 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

In Memoriam Jan Łukasiewicz

Ohne Zusammenfassung     

Universal Properties of Łukasiewicz Consequence

Boolean logic deals with {0, 1}-observables and yes–no events, as many-valued logic does for continuous ones. Since every measurement has an error, continuity ensures that small measurement errors on elementary observables have small effects on compound observables. Continuity is irrelevant for {0, 1}-observables. Functional completeness no longer holds when n-ary connectives are understood as [0, 1]-valued maps defined on [0, 1] ⁿ . So one must envisage suitable selection criteria for [0, 1]-connectives. ?ukasiewicz implication has a well known characterization as the only continuous connective ${\Rightarrow}$ satisfying the following conditions: (i) ${x\Rightarrow(y\Rightarrow z)= y\Rightarrow(x\Rightarrow z)}$ and (ii) ${x\Rightarrow y=1\, \, {\rm iff} x\leq y}$ . Then syntactic consequence can be defined purely algorithmically using the ?ukasiewicz axioms and Modus Ponens. As discussed in this paper, to recover a strongly complete semantics one may use differential valuations.     

Complexity of fuzzy answer set programming under Łukasiewicz semantics

Fuzzy answer set programming (FASP) is a generalization of answer set programming (ASP) in which propositions are allowed to be graded. Little is known about the computational complexity of FASP and almost no techniques are available to compute the answer sets of a FASP program. In this paper, we analyze the computational complexity of FASP under Łukasiewicz semantics. In particular we show that the complexity of the main reasoning tasks is located at the first level of the polynomial hierarchy, even for disjunctive FASP programs for which reasoning is classically located at the second level. Moreover, we show a reduction from reasoning with such FASP programs to bilevel linear programming, thus opening the door to practical applications. For definite FASP programs we can show P-membership. Surprisingly, when allowing disjunctions to occur in the body of rules – a syntactic generalization which does not affect the expressivity of ASP in the classical case – the picture changes drastically. In particular, reasoning tasks are then located at the second level of the polynomial hierarchy, while for simple FASP programs, we can only show that the unique answer set can be found in pseudo-polynomial time. Moreover, the connection to an existing open problem about integer equations suggests that the problem of fully characterizing the complexity of FASP in this more general setting is not likely to have an easy solution.     

Comprehension contradicts to the induction within Łukasiewicz predicate logic

We introduce the simpler and shorter proof of Hajek’s theorem that the mathematical induction on ω implies a contradiction in the set theory with the comprehension principle within ?ukasiewicz predicate logic ? ${\forall}$ (Hajek Arch Math Logic 44(6):763–782, 2005) by extending the proof in (Yatabe Arch Math Logic, accepted) so as to be effective in any linearly ordered MV-algebra.     

Reverse triple I reasoning methods based on the Łukasiewicz implication

The reverse and α-reverse triple I reasoning methods based on ?ukasiewicz implication I _L are established in new manners which correct the mistakes in the existing literature. Furthermore, the α-reverse triple I reasoning method is extended to a new form, called α(u, ν)-reverse triple I reasoning method, which can contain the reverse triple I reasoning method as its particular case. This is another improved point to the existing results.     

A characterization of tribes with respect to the Łukasiewicz t-norm

We give a complete characterization of tribes with respect to the ukasiewicz t-norm, i. e., of systems of fuzzy sets which are closed with respect to the complement of fuzzy sets and with respect to countably many applications of the ukasiewicz t-norm. We also characterize all operations with respect to which all such tribes are closed. This generalizes the characterizations obtained so far for other fundamental t-norms, e. g., for the product t-norm.     

States in Łukasiewicz logic correspond to probabilities of rational polyhedra

It will be shown that probabilities of infinite-valued events represented by formulas in ?ukasiewicz propositional logic are in one-to-one correspondence with tight probability measures over rational polyhedra in the unit hypercube. This result generalizes a recent work on rational measures of polyhedra and provides an elementary geometric approach to reasoning under uncertainty with states in ?ukasiewicz logic.     

Distinguishing non-standard natural numbers in a set theory within Łukasiewicz logic

In , a set theory with the comprehension principle within Łukasiewicz infinite-valued predicate logic, we prove that a statement which can be interpreted as “there is an infinite descending sequence of initial segments of ω” is truth value 1 in any model of , and we prove an analogy of Hájek’s theorem with a very simple procedure.      

Algebras of quasi-Plücker coordinates are Koszul

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Plücker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gröbner bases.     

Infinite-Valued First-Order Łukasiewicz Logic: Hypersequent Calculi Without Structural Rules and Proof Search for Sentences in the Prenex Form

The rational first-order Pavelka logic is an expansion of the infinite-valued first-order ?ukasiewicz logic ?? by truth constants. For this logic, we introduce a cumulative hypersequent calculus G¹?? and a noncumulative hypersequent calculus G²?? without structural inference rules. We compare these calculi with the Baaz–Metcalfe hypersequent calculus G?? with structural rules. In particular, we show that every G??-provable sentence is G¹??-provable and a ??-sentence in the prenex form is G??-provable if and only if it is G²??-provable. For a tableau version of the calculus G²??, we describe a family of proof search algorithms that allow us to construct a proof of each G²??-provable sentence in the prenex form.     

Commutative Finite-Dimensional Algebras Satisfying x(x(xy)) = 0 are Nilpotent

We investigate the structure of commutative non-associative algebras satisfying the identity x(x(xy)) = 0. Recently, Correa and Hentzel proved that every commutative algebra satisfying above identity over a field of characteristic ≠ 2 is solvable. We prove that every commutative finite-dimensional algebra 𝔄 over a field F of characteristic ≠ 2, 3 which satisfies the identity x(x(xy)) = 0 is nilpotent. Furthermore, we obtain new identities and properties for this class of algebras.