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A note on Wansing's expansion of Nelson's logic : a correction to "An axiomatization of Wansing's expansion of Nelson's logic"

Omori H.

Reports on Mathematical Logic

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2016

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Vol. 51

133--144

EN

The present note corrects an error made by the author in answering an open problem of axiomatizing an expansion of Nelson's logic introduced by Heinrich Wansing. It also gives a correct axiomatization that answers the problem by importing some results on subintuitionistic logics presented by Greg Restall.

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An axiomatization of Wansing's expansion of Nelson's logic

Omori H.

Reports on Mathematical Logic

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2015

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Vol. 50

41--51

EN

The present note offers an axiomatization for an expansion of Nelson’s logic motivated by Heinrich Wansing which serves as a base logic for the framework of nonmonotonic reasoning considered by Dov Gabbay and Raymond Turner. We also show that the expansion of Wansing is not conservative over intuitionistic logic, but at least as strong as Jankov’s logic.

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An Axiomatization of the Token Game Based on Petri Algebras

Badouel E., Chenou J., Guillou G.

Fundamenta Informaticae

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2007

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Vol. 77, nr 3

187-215

EN

The firing rule of Petri nets relies on a residuation operation for the commutative monoid of non negative integers. We identify a class of residuated commutative monoids, called Petri algebras, for which one can mimic the token game of Petri nets to define the behaviour of generalized Petri nets whose flow relations and place contents are valued in such algebraic structures. The sum and its associated residuation capture respectively how resources within places are produced and consumed through the firing of a transition. We show that Petri algebras coincide with the positive cones of lattice-ordered commutative groups and constitute the subvariety of the (duals of) residuated lattices generated by the commutative monoid of non negative integers. We however exhibit a Petri algebra whose corresponding class of nets is strictly more expressive than the class of Petri nets. More precisely, we introduce a class of nets, termed lexicographic Petri nets, that are associated with the positive cones of the lexicographic powers of the additive group of real numbers. This class of nets is universal in the sense that any net associated with some Petri algebra can be simulated by a lexicographic Petri net. All the classical decidable properties of Petri nets however (termination, covering, boundedness, structural boundedness, accessibility, deadlock, liveness ...) are undecidable on the class of lexicographic Petri nets. Finally we turn our attention to bounded nets associated with Petri algebras and show that their dynamics can be reformulated in term of MV-algebras.

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Counter-Free Keys and Functional Dependencies in Higher-Order Datamodels

Sali A., Schewe K-D.

Fundamenta Informaticae

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2006

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Vol. 70, nr 3

277-301

EN

We investigate functional dependencies (FDs) in the presence of several constructors for complex values. These constructors are the tuple constructor, list-, set- and multiset-constructors, an optionality constructor, and a disjoint union constructor. The disjoint union constructor implies restructuring rules, which complicate the theory. In particular, they do not permit a straightforward axiomatisation of the class of all FDs without a detour via weak functional dependencies (wFDs), i.e. disjunctions of functional dependencies, and even the axiomatisation of wFDs is not yet completely solved. Therefore, we look at the restricted class of counter-free functional dependencies (cfFDs). That is, we ignore subattributes that only refer to counting the number of elements in sets or multisets or distinguish only between empty or non-empty sets. We present a finite axiomatisation for the class of cfFDs. Furthermore, we study keys ignoring again the counting subattributes. We show that such keys are equivalent with certain ideals called HL-ideals. Based on that we introduce an ordering between key sets, and investigate systems of minimal keys. We give a sufficient condition for a Sperner family of HL-ideals being a system of minimal keys, and determine lower and upper bounds for the size of the smallest Armstrong-instance.

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Algebraic Structures Related to Many Valued Logical Systems. Part 2, Equivalence Among some Widespread Structures

Cattaneo G., Ciucci D., Giuntini R., Konig M.

Fundamenta Informaticae

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2004

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Vol. 63, nr 4

357--373

EN

Several algebraic structures (namely HW, BZMVdM, Stonean MV and MVΔ algebras) related to many valued logical systems are considered and their equivalence is proved. Four propositional calculi whose Lindenbaum-Tarski algebra corresponds to the four equivalent algebraic structures are axiomatized and their semantical completeness is given.

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Axiomatization of utility, outranking and decision rule preference models for multiple-criteria classification problems under partial inconsistency with the dominance principle

Słowiński R., Greco S., Matarazzo B.

Control and Cybernetics

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2002

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Vol. 31, no 4

1005-1035

EN

Multiple-criteria classification (sorting) problem concerns assignment of actions (objects) to some pre-defined and preference-ordered decision classes. The actions are described by a finite set of criteria, i.e. attributes, with preference-ordered scales. To perform the classification, criteria have to be aggregated into a preference model which can be: utility (discriminant) function, or outranking relation, or "if..., then..." decision rules. Decision rules involve partial profiles on subsets of criteria and dominance relation on these profiles. A challenging problem in multiple-criteria decision making is the aggregation of criteria with ordinal scales. We show that the decision rule model we propose has advantages over a general utility function, over the integral of Sugeno, conceived for ordinal criteria, and over an outranking relation. This is shown by basic axioms characterizing these models. Moreover, we consider a more general decision rule model based on the rough set theory. The advantage of the rough set approach compared to competitive methodologies is the possibility of handling partially inconsistent data that are often encountered in preferential information, due to hesitation of decision makers, unstable character of their preferences, imprecise or incomplete knowledge and the like. We show that these inconsistencies can be represented in a meaningful way by "if..., then..." decision rules induced from rough approximations. The theoretical results reported in this paper show that the decision rule model is the most general aggregation model among all the considered models.

PL

Wielokryterialne zadania klasyfikacji (sortowania) dotyczą przypisania działan (obiektów) pewnym z góry określonym i uporządkowanym wzgl(c)dem preferencji klasom. Działania są opisane przez skończony zbiór kryteriów, tj. atrybutów o skalach uporządkowanych według preferencji. Aby dokonać klasyfikacji, kryteria muszą zostać zagregowane do modelu preferencji, którym może być: funkcja użyteczności (dyskryminująca), bąd" relacja przewyższania, bąd" reguła decyzyjna typu "jeśli..., to...". Reguły decyzyjne oparte są na cz(c)ściowych profilach na podzbiorach kryteriów i relacji dominacji na tych profilach. Wyzwaniem w wielokryterialnym podejmowaniu decyzji jest agregacja kryteriów o skalach porządkowych. Pokazujemy w artykule, że model reguł decyzyjnych, zaproponowany przez nas, jest korzystniejszy niż ogólna postać funkcji użyteczności, niż całka Sugeno zaproponowana dla kryteriów porządkowych, i niż relacja przewyższania. Pokazano to przy pomocy podstawowych aksjomatów charakteryzujących rozważane modele. Ponadto, rozważamy ogólniejszy model reguł decyzyjnych, oparty na teorii zbiorów przybliżonych. Korzyścią z zastosowania zbiorów przybliżonych w porównaniu do innych podejść jest możliwość uwzgl(c)dniania cz(c)ściowo niespójnych danych, jakie cz(c)sto spotyka si(c) informacji o preferencjach, w związku z wahaniami decydentów, niestabilnością ich preferencji, niedokładną wiedzą, itp. Pokazujemy, że takie niespójności mogą być reprezentowane w sposób sensowny poprzez reguły decyzyjne typu "jeśli..., to...", wyprowadzone z ocen dokonanych przy pomocy zbiorów przybliżonych. Wyniki teoretyczne przedstawione w pracy pokazują, że model reguł decyzyjnych jest najogólniejszym modelem agregacji spośrod wszystkich rozważanych modeli.

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£-rozstrzygalność pewnego trójwartościowego systemu

Piróg-Rzepecka K.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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1999

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Vol. 6

55--69

PL

Artykuł ten dotyczy logiki zdaniowej S* przedstawionej w pracy [5]. Rozważana logika zawiera funktory: ∨,∧,⇒,⟺,¬,*, z których cztery pierwsze są dwuargumentowe, dwa ostatnie - jednoargumentowe. Logika S* posiada adekwatną matrycę trójelementową, której uniwersum jest zbiór {l, 0,-l}, wartościami wyróżnionymi są elementy 1 i -1, zaś działania odpowiadające funktorom określone są następującymi tabelkami: [wzór] Aksjomatami systemu S* jest dowolny układ aksjomatów klasycznego rachunku zdań oraz następujące wyrażenia: Al. *p ⟺ *¬ A2. *(p ∧ q) ⟺*p *∧ q, A3.p ⇒ *p. Regułami pierwotnymi systemu są: reguła podstawiania w zwykłym sformułowaniu oraz reguła odrywania dla implikacji o następującym schemacie:[wzór] Zbiór tez systemu aksjomatycznego S* pokrywa się ze zbiorem tautologii tej logiki. W pracy podany jest dowód £-rozstrzygalności systemu S*, tzn., że system ten spełnia następujące warunki: (I) Zbiory wszystkich jego formuł uznanych (tez systemu) i wszystkich jego formuł odrzuconych są rozłączne; (II) Kazda formuła danego systemu bądź jest formułą uznaną, bądź odrzuconą. Dowód -rozstrzygalności dotyczy również wersji inwariantnej.

EN

This paper concerns the propositional logic S* which has been introduced in the paper [5]. The logic S* is defined in the sentential language which involves the following symbols: ∨,∧,⇒,⟺,¬,*, from which the first four are two-argument and the last two - one-argument. The systems S* is defined axiomatically by any axiom system for the classical propositional logic augmented with the following list of axioms: Al. *p ⟺ *¬ A2. *(p ∧ q) ⟺*p *∧ q, A3.p ⇒ *p. Furthemore, as the primitive rules for this system we have: - rule for substitution (in the ordinary formulation), - rule Modus Ponens (detachement) for implication given by the following scheme:[formula] We prove that the system S* has an adequate (in the weak sense) three-element matrix. Let M be the matrix for which the set {l,0, -l} forms its universe and {l,-l} is the set of the designated elements. The operations of M which correspond to the above symbols are defined by the following tables:[formula] It is shown that the set of logical theses of S* coincides with the content of , M i.e. with the set of all formulas valid in M We also prove that the systems S* is Ł-decidable, i.e. S* satisfies the following conditions: (I) The set of all provable and a11rejected formulas of S* are disjoint; (II) Each formula of the language od S* ie either provable or rejected (in S*).

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Aksjomatyzacja rachunku zdaniowego dualnego względem klasycznego implikacyjno-negacyjnego rachunku zdań

Górnicka A.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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1999

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Vol. 7

30--36

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Coinductive axiomatization of recursive type equality and subtyping

Brandt M., Henglein F.

Fundamenta Informaticae

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1998

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Vol. 33, nr 4

309-338

EN

We present new sound and complete axiomatizations of type equality and subtype inequality for a first-order type language with regular recursive types. The rules are motivated by coinductive characterizations of type containment and type equality via simulation and bisimulation, respectively. The main novelty of the axiomatization is the fixpoint rule (or coinduction principle). It states that from A,Pý P one may deduce Aý P, where P is either a type equality t = t1 or type containment t