多值Łukasiewicz逻辑公式的范式表示和计数问题

将符号化计算树逻辑中的Shannon展开式做了推广,在n值Łukasiewicz逻辑系统Łn中,研究了由逻辑公式导出的n值McNaughton函数的展开式,给出了m元n值McNaughton函数的准析取范式和准合取范式.在此基础上,给出了m元n值McNaughton函数的计数问题,并在n值Łukasiewicz逻辑系统Łn中,给出了m元逻辑公式的构造方法及其逻辑等价类的计数问题.     

Theory of truth degrees of formulas in Łukasiewicz<Emphasis Type="Italic">n</Emphasis>-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Łukasiewiczn-valued propositional logicL _n is proposed. A limit theorem is obtained, which says that the truth functionτ _n induced by truth degrees converges to the integrated truth functionτ whenn converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Łukasiewicz logic and the continuous valued Łukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.     

Interval-valued fuzzy n-ary subhypergroups of n-ary hypergroups

This paper provides a continuation of ideas presented by Davvaz and Corsini (J Intell Fuzzy Syst 18(4):377–382, 2007). Our aim in this paper is to introduce the concept of quasicoincidence of a fuzzy interval value with an interval-valued fuzzy set. This concept is a generalized concept of quasicoincidence of a fuzzy point within a fuzzy set. By using this new idea, we consider the interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup of a n-ary hypergroup. This newly defined interval-valued (∈, ∈ ∨q)-fuzzy n-ary subhypergroup is a generalization of the usual fuzzy n-ary subhypergroup. Finally, we consider the concept of implication-based interval-valued fuzzy n-ary subhypergroup in an n-ary hypergroup; in particular, the implication operators in ￡ukasiewicz system of continuous-valued logic are discussed.     

■ukasiewicz命题逻辑系统中有限命题集的约简理论

在n值■ukasiewicz命题逻辑中提出了命题集Γ的约简理论,引入由命题集Γ所诱导的形式背景的概念,从Γ及其子集的关系出发给出了n值命题逻辑中有限命题集Γ约简的判定定理以及求Γ约简的方法。说明了无穷值■ukasiewicz命题逻辑中命题集Γ的约简可转化为n值情形。     

Strengthening Brady’s Paraconsistent 4-Valued Logic BN4 with Truth-Functional Modal Operators

?ukasiewicz presented two different analyses of modal notions by means of many-valued logics: (1) the linearly ordered systems ?3,..., Open image in new window ,..., \(\hbox {L}_{\omega }\); (2) the 4-valued logic ? he defined in the last years of his career. Unfortunately, all these systems contain “?ukasiewicz type (modal) paradoxes”. On the other hand, Brady’s 4-valued logic BN4 is the basic 4-valued bilattice logic. The aim of this paper is to show that BN4 can be strengthened with modal operators following ?ukasiewicz’s strategy for defining truth-functional modal logics. The systems we define lack “?ukasiewicz type paradoxes”. Following Brady, we endow them with Belnap–Dunn type bivalent semantics.     

Extremal states on bounded residuated \ell-monoids with general comparability

Bounded residuated lattice ordered monoids (RlR\ell-monoids) are a common generalization of pseudo-BLBL-algebras and Heyting algebras, i.e. algebras of the non-commutative basic fuzzy logic (and consequently of the basic fuzzy logic, the Łukasiewicz logic and the non-commutative Łukasiewicz logic) and the intuitionistic logic, respectively. We investigate bounded RlR\ell-monoids satisfying the general comparability condition in connection with their states (analogues of probability measures). It is shown that if an extremal state on Boolean elements fulfils a simple condition, then it can be uniquely extended to an extremal state on the RlR\ell-monoid, and that if every extremal state satisfies this condition, then the RlR\ell-monoid is a pseudo-BLBL-algebra.     

Partiality and Nondeterminacy in Program Proofs

Specifications and programs make much use of nondeterministic and/or partial expressions, i.e. expressions which may yield several or no outcomes for some values of their free variables. Traditional 2-valued logics do not comfortably accommodate reasoning about undefined expressions, and do not cater at all for nondeterministic expressions. We seek to rectify this with a 4-valued typed logic E4 which classifies formulae as either “true”, “false”, “neither true nor false”, or “possibly true, possibly false”. The logic is derived in part from the 2-valued logic E and the 3-valued LPF, and preserves most of the theorems of E. Indeed, the main result is that nondeterminacy can be added to a logic covering partiality at little cost. Received July 1996 / Accepted in revised form April 1998     

A note on the representation of McNaughton lines by basic literals

 We introduce a syntactically simple subclass of formulas of the infinite-valued logic of Łukasiewicz, the class of basic literals, whose associated McNaughton functions are truncated lines. We present some properties of these formulas and an application to states of MV-algebras.     

On copulas, quasicopulas and fuzzy logic

We investigate (quasi)copulas as possible truth functions of fuzzy conjunction which is not necessarily associative and present some axiom systems for such fuzzy logics. In particular, we study an expansion of Łukasiewicz (infinite valued propositional) logic by a new connective interpreted as an arbitrary quasicopula (and also by a new connective interpreted as the residuum of the copula). Main results concern standard completeness.     

Limitations of the Program Memory and the Expressive Power of Dynamic Logics

For a dynamic logic L we study dynamic logics L_n for which programs allowed in formulas cannot use more than n variables. We prove that there exists a structure A of a finite signature such that for a wide class of dynamic logics L and for every natural number n the logic L_n+1 is more expressive over A than L_n. This result is based on a construction of some canonical form for the formulas of L_n over a free one-generated groupoid.     

n-Fold implicative basic logic is G?del logic

We prove that Haveshki’s and Eslami’s n-fold implicative basic logic is G?del logic and n-fold positive implicative basic logic is a fragment of ukasiewicz logic.     

Eliminability of cut in hypersequent calculi for some modal logics of linear frames

Hypersequent calculi, introduced independently by Pottinger and Avron, provide a powerful generalization of ordinary sequent calculi. In the paper we present a proof of eliminability of cut in hypersequent calculi for three modal logics of linear frames: K4.3, KD4.3 and S4.3. Our cut-free calculus is based on Avron's HC formalization for Gödel–Dummett's logic. The presented proof of eliminability of cut is purely syntactical and based on Ciabattoni, Metcalfe, Montagna's proof of eliminability of cut for hypersequent calculi for some fuzzy logics with modalities.     

A Legacy Recalled and a Tradition Continued

Since the first presentation of classical sentential logic as an axiomatic system by Frege in 1879, the study of a variety of sentential calculi has flourished. One major area of investigation, initiated by ukasiewicz and his colleagues in the first half of the twentieth century and carried into the second half by Meredith, Thomas, Prior, et al., focuses on alternative axiom sets for such logics, and on formal proofs within them. This paper recalls a sampling of the results obtained heretofore, noting along the way how the papers in this special issue of the Journal of Automated Reasoning fit into that larger tradition of which they now form a part. It also suggests a number of further questions, open problems, and projects to which the methods developed in these papers seem ideally suited and might well be fruitfully applied.     

Resolution with Order and Selection for Hybrid Logics

We investigate labeled resolution calculi for hybrid logics with inference rules restricted via selection functions and orders. We start by providing a sound and refutationally complete calculus for the hybrid logic H(@,ˉ,A)\mathcal{H}(@,{\downarrow},\mathsf{A}), even under restrictions by selection functions and orders. Then, by imposing further restrictions in the original calculus, we develop a sound, complete and terminating calculus for the H(@)\mathcal{H}(@) sublanguage. The proof scheme we use to show refutational completeness of these calculi is an adaptation of a standard completeness proof for saturation-based calculi for first-order logic that guarantees completeness even under redundancy elimination. In fact, one of the contributions of this article is to show that the general framework of saturation-based proving for first-order logic with equality can be naturally adapted to saturation-based calculi for other languages, in particular modal and hybrid logics.     

Cut-free sequent systems for temporal logic

Currently known sequent systems for temporal logics such as linear time temporal logic and computation tree logic either rely on a cut rule, an invariant rule, or an infinitary rule. The first and second violate the subformula property and the third has infinitely many premises. We present finitary cut-free invariant-free weakening-free and contraction-free sequent systems for both logics mentioned. In the case of linear time all rules are invertible. The systems are based on annotating fixpoint formulas with a history, an approach which has also been used in game-theoretic characterisations of these logics.     

Characterizations of ordered semigroups in terms of (∈, ∈ ∨ q)-fuzzy interior ideals

In this paper, we give characterizations of ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals. We characterize different classes regular (resp. intra-regular, simple and semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy interior ideals (resp. (∈, ∈ ∨q)-fuzzy ideals). In this regard, we prove that in regular (resp. intra-regular and semisimple) ordered semigroups the concept of (∈, ∈ ∨q)-fuzzy ideals and (∈, ∈ ∨q)-fuzzy interior ideals coincide. We prove that an ordered semigroup S is simple if and only if it is (∈, ∈ ∨q)-fuzzy simple. We characterize intra-regular (resp. semisimple) ordered semigroups in terms of (∈, ∈ ∨q)-fuzzy ideals (resp. (∈, ∈ ∨q)-fuzzy interior ideals). Finally, we consider the concept of implication-based fuzzy interior ideals in an ordered semigroup, in particular, the implication operators in Lukasiewicz system of continuous-valued logic are discussed.     

Dynamic Hotlinks

Consider a directed rooted tree T=(V,E) representing a collection V of n web pages connected via a set E of links all reachable from a source home page, represented by the root of T. Each web page i carries a weight w _i representative of the frequency with which it is visited. By adding hotlinks, shortcuts from a node to one of its descendants, we are interested in minimizing the expected number of steps needed to visit pages from the home page. We give the first linear time algorithm for assigning hotlinks so that the number of steps to access a page i from the root of the tree reaches the entropy bound, i.e. is at most O(log (W/w _i )) where W=∑ _i∈T w _i . The best previously known algorithm for this task runs in time O(n ²). We also give the first efficient data structure for maintaining hotlinks when nodes are added, deleted or their weights modified, in amortized time O(log (W/w _i )) per update. The data structure can be made adaptive, i.e. reaches the entropy bound in the amortized sense without knowing the weights w _i in advance.     

A spatial equational logic for the applied π-calculus

Spatial logics have been proposed to reason locally and modularly on algebraic models of distributed systems. In this paper we define the spatial equational logic A π L whose models are processes of the applied π-calculus. This extension of the π-calculus allows term manipulation and records communications as aliases in a frame, thus augmenting the predefined underlying equational theory. Our logic allows one to reason locally either on frames or on processes, thanks to static and dynamic spatial operators. We study the logical equivalences induced by various relevant fragments of A π L, and show in particular that the whole logic induces a coarser equivalence than structural congruence. We give characteristic formulae for some of these equivalences and for static equivalence. Going further into the exploration of A π L’s expressivity, we also show that it can eliminate standard term quantification.     

Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

Conquering the Meredith Single Axiom

For more than three and one-half decades, beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems studied range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for two-valued sentential (or propositional) calculus and whose conclusion is the ukasiewicz three-axiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps contains a term of the form n(n(t)) for any term t. As evidence of the power of the new methodology, the article also discusses OTTER's success in obtaining the first known proof of a theorem concerning a single axiom of ukasiewicz.