ALBERT

Beschreibung: The main ‘philosophical’ outcome of this article is to demonstrate that the structural description of residuated lattices requires the use of the co-residuated setting. A construction, called skew symmetrization, which generalizes the well-known representation of an ordered Abelian group obtained from the positive (or negative) cone of the algebra is introduced here. Its definition requires leaving the accustomed residuated setting and entering the co-residuated setting. It is shown that every uninorm on [0, 1] with an involution defined by the residual complement with respect to the unit and having the unit as the fixed point of the involution can be described as the skew symmetrization of its underlying t-norm or underlying t-conorm.

Beschreibung: This article deals with some probabilistic model theory for intuitionistic predicate logic. We introduce the notions of intuitionistic probability, probabilistic structure for intuitionistic predicate logic and model of an intuitionistic probability. We prove a Gaifman-style completeness theorem: any intuitionistic probability has a weak probabilistic model.

Beschreibung: We introduce a new approach to conditional probability over many-valued events, which is based on bets. Then we show that this approach fits with Kroupa's approach, and we give two characterizations of coherence for books on conditional many-valued events, the first one based on states, and the second one based on logical coherence of a suitable theory on many-valued logic.

Beschreibung: We solve the minimization problem for finitely axiomatizable theories in Gödel infinite-valued propositional logic. That is, we obtain an algorithm that when input a formula α( X 1 ,..., X n ) outputs a formula β( X 1 ,..., X m ) such that (i) the theories singly axiomatized by {α} and {β} have isomorphic algebraic semantics, and (ii) if β'( X 1 ,..., X m ' ) is any formula satisfying (i), then m '≥ m .

Beschreibung: Lax modalities occur in intuitionistic logics concerned with hardware verification, the computational lambda calculus and access control in secure systems. They also encapsulate the logic of Lawvere–Tierney–Grothendieck topologies on topoi. This article provides a complete semantics for quantified lax logic by combining the Beth–Kripke–Joyal cover semantics for first-order intuitionistic logic with the classical relational semantics for a ‘diamond’ modality. The main technique used is the lifting of a multiplicative closure operator (nucleus) from a Heyting algebra to its MacNeille completion, and the representation of an arbitrary locale as the lattice of ‘propositions’ of a suitable cover system. In addition, the theory is worked out for certain constructive versions of the classical logics K and S4. An alternative completeness proof is given for (non-modal) first-order intuitionistic logic itself with respect to the cover semantics, using a simple and explicit Henkin-style construction of a characteristic model whose points are principal theories rather than prime-saturated ones. The article provides further evidence that there is more to intuitionistic modal logic than the generalization of properties of boxes and diamonds from Boolean modal logic.

Beschreibung: In Baaz and Iemhoff (2006, Annals of Pure and Applied Logic, 142, 269–295), an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore, eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in Baaz and Iemhoff (2006, Annals of Pure and Applied Logic, 142, 269–295) were semantical. In this article, a proof-theoretic proof of the completeness of eskolemization with respect to existential quantifiers is presented.

Beschreibung: The intuitionistic sequent calculus (at most one formula on the right-hand side of sequents) comes with a natural dual system: the dual-intuitionistic sequent calculus (at most one formula on the left-hand side). We explain how the duality between these two systems exactly corresponds to the intensively studied duality between call-by-value systems and call-by-name systems for classical logic. Relying on the uniqueness of the computational behaviour underlying these four logics (intuitionistic, dual-intuitionistic, call-by-value classical and call-by-name classical), we define a generic syntax of nets which can be used for any of these logics.

Beschreibung: The study of propositional realizability logic was initiated in the 50th of the last century. Unfortunately, no description of the class of realizable propositional formulas is found up to now. Nevertheless, some attempts of such a description were made. In 1974, the author proved that every known realizable propositional formula has the property that every one of its closed arithmetical instances is deducible in the system obtained by adding Extended Church's Thesis and Markov Principle as axiom schemes to Intuitionistic Arithmetic. Visser calls this system Markov's Arithmetic. In 1990, another attempt of describing the class of realizable propositional formulas was made by Varpakhovskii who proposed a calculus in an extended propositional language and proved that all known realizable propositional formulas are deducible in this calculus. In this article we prove that every propositional formula deducible in Varpakhovskii's calculus has the property that each of its closed arithmetical instances is deducible in Markov's Arithmetic.

Beschreibung: Is it possible to give a coordinate-free formulation of the Second Incompleteness Theorem? We pursue one possible approach to this question. We show that (i) cutfree consistency for finitely axiomatized theories can be uniquely characterized modulo EA -provable equivalence and (ii) consistency for finitely axiomatized sequential theories can be uniquely characterized modulo EA -provable equivalence. The case of infinitely axiomatized RE theories is more delicate. We carefully discuss this in the article.

Beschreibung: In this article, we study the canonical model for the closed fragment of GLP and establish its precise relationship with a universal model constructed by Ignatiev. In particular, we effectively characterize the canonical model in terms of a coordinate system based on sequences of ordinals up to 0 .We then define a simple topological model of this logic by defining a natural polytopology on the ordinal 0 itself.

Beschreibung: We give a sound and complete labelled natural deduction system for a bundled branching temporal logic, namely the until-free version of BCTL *. The logic BCTL * is obtained by referring to a more general semantics than that of CTL *, where we only require that the set of paths in a model is closed under taking suffixes (i.e. is suffix-closed) and is closed under putting together a finite prefix of one path with the suffix of any other path beginning at the same state where the prefix ends (i.e. is fusion-closed). In other words, this logic does not enjoy the so-called limit-closure property of the standard CTL * validity semantics. We give both a classical and an intuitionistic version of our labelled natural deduction system for the until-free version of BCTL *, and carry out a proof-theoretical analysis of the intuitionistic system: we prove that derivations reduce to a normal form, which allows us to give a purely syntactical proof of consistency (for both the intuitionistic and classical versions) of the deduction system.

Beschreibung: The existence of states and probabilities on effect algebras as logical structures when events may be non-compatible, unsharp, fuzzy or imprecise is still an open question. Only a few families of effect algebras possessing states are known. We are going to show some families of effect algebras, the existence of a pseudocomplementation on which implies the existence of states. Namely, those are Archimedean atomic lattice effect algebras, which are sharply dominating or s-compactly generated or extendable to complete lattice effect algebras.

Beschreibung: One approach to moderating the expected behaviour of agents in open societies is the use of explicit languages for defining norms, conditional commitments and/or social expectations, together with infrastructure supporting conformance checking. This article presents a logical account of the fulfilment and violation of social expectations modelled as conditional rules over a hybrid linear propositional temporal logic. Our semantics captures the intuition that the fulfilment or violation of an expectation must be determined without recourse to information from later states.We define a means of updating expectations from one state to the next based on formula progression, and show how conformance checking was implemented by combining the MCFULLmodel checking algorithm of Franceschet and de Rijke and the semantics for LTL over truncated paths proposed by Eisner et al. We present algorithms for both traditional offline model checking, where the complete model is available at once, and online model checking, where states are added to the model sequentially at runtime.

Beschreibung: Among the class of finite integral commutative residuated chains (ICRCs), we identify those algebras which can be obtained as a nuclear retraction of a conuclear contraction of a totally ordered Abelian -group. We call the ICRCs satisfying this condition regular. Then we discuss the structure of finite regular ICRCs. Finally, we prove that the class of regular members generate a strictly smaller variety than the variety generated by ICRCs.

Beschreibung: We provide a constructive, direct and simple proof of the completeness of the cut-free part of the hypersequential calculus HG for Gödel logic (thereby proving both completeness of the calculus for its standard semantics, and the admissibility of the cut rule in the full calculus). We then extend the results and proofs to derivations from assumptions, showing that such derivations can be confined to those in which cuts are made only on formulas which occur in the assumptions. The article is self-contained, and no previous knowledge concerning HG (or even Gödel logic) is needed for understanding it.

Beschreibung: A particular notion of an interpretation of a theory over a fuzzy predicate logic in another such theory is discussed. For interpretability with the domain defined by a provably crisp formula, which is of course a syntactical notion, a semantic characterization is established. In the last section, we discuss the question of whether the extension of a decidable theory by a single new axiom is decidable and present an erratum to the paper (Hájek, 2007, Fundamenta informaticae , 81 , 155–163).

Beschreibung: This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

Beschreibung: The article deals with Codd's relational model of data and its fuzzy logic extensions. Our main purpose is to examine, from the point of view of fuzzy logic in the narrow sense, some of the extensions proposed in the literature and the relationships between them. We argue that fuzzy logic in the narrow sense is important for the fuzzy logic extensions because it provides conceptual and methodological foundations, clarity and simplicity. We present several comparative observations as well as new technical results.

Beschreibung: This article shows how derivations in the deep inference system SKS for classical propositional logic can be translated into proof nets. Since an SKS derivation contains more information about a proof than the corresponding proof net, we observe a loss of information which can be understood as ‘eliminating bureaucracy’. Technically, this is achieved by cut reduction on proof nets. As an intermediate step between the two extremes, SKS derivations and proof nets, we will see proof graphs representing derivations in ‘Formalism A’.

Beschreibung: The superintuitionistic predicate logics (without or with equality) of all predicate Kripke frames with nested domains over a fixed poset W (a set of possible worlds) are embeddable in the logic (without equality) of all Kripke frames with constant domains over W . Therefore, Takano's result [13] on finite axiomatizability of the logic of Kripke frames with constant domains over the set of real numbers implies the recursive axiomatizability of the corresponding logics with nested domains. Other consequences are mentioned as well.

Beschreibung: This article formalizes and compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system.The first system, LKID, supports traditional proof by induction, with induction rules formulated as rules for introducing inductively defined predicates on the left of sequents.We show LKID to be cut-free complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system, LKID , uses infinite (non-well-founded) proofs to represent arguments by infinite descent. In this system, the left-introduction rules for inductively defined predicates are simple case-split rules, and an infinitary, global condition on proof trees is required in order to ensure soundness.We show LKID to be cut-free complete with respect to standard models, and again infer the eliminability of cut. The infinitary system LKID is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs, which is so suited. We demonstrate that this restricted ‘cyclic’ proof system, CLKID , subsumes LKID, and conjecture that CLKID and LKID are in fact equivalent, i.e. that proof by induction is equivalent to regular proof by infinite descent.

Beschreibung: Measuring inconsistency in knowledge bases has been recognized as an important problem in several research areas. Many methods have been proposed to solve this problem and a main class of them is based on some kind of paraconsistent semantics. However, existing methods suffer from two limitations: (i) they are mostly restricted to propositional knowledge bases; (ii) very few of them discuss computational aspects of computing inconsistency measures. In this article, we try to solve these two limitations by exploring algorithms for computing an inconsistency measure of first-order knowledge bases. After introducing a four-valued semantics for first-order logic, we define an inconsistency measure of a first-order knowledge base, which is a sequence of inconsistency degrees. We then propose a precise algorithm to compute our inconsistency measure. We show that this algorithm reduces the computation of the inconsistency measure to classical satisfiability checking. This is done by introducing a new semantics, named S [ n ] -4 semantics, which can be calculated by invoking a classical SAT solver. Moreover, we show that this auxiliary semantics also gives a direct way to compute upper and lower bounds of inconsistency degrees. That is, it can be easily revised to compute approximating inconsistency measures. The approximating inconsistency measures converge to the precise values if enough resources are available. Finally, by some nice properties of the S [ n ] -4 semantics, we show that some upper and lower bounds can be computed in P-time, which says that the problem of computing these approximating inconsistency measures is tractable.

Beschreibung: The abstract nature of Dung’s theory of argumentation accounts for its wide-spread application as a general framework for various species of non-monotonic reasoning, and, more generally, reasoning in the presence of conflict. In this article, we formalize reasoning about argumentation within the Dung argumentation paradigm itself. A metalevel Dung argumentation framework is itself instantiated by arguments that make statements about arguments, their interactions, and their evaluation in an object-level argumentation framework.We show how Dung’s theory, and object-level extensions of Dung’s theory, such as those intended to accommodate preferences, can then be uniformly characterised by metalevel argumentation in a Dung framework. We then discuss how this provides for application of the full range of theoretical and practical developments of Dung’s theory, to extensions of Dung’s theory, and provides for integration and further augmentation of these extensions.

Beschreibung: Classical logic of proofs LP naturally extends propositional calculus to the language enriched with formulas meaning t is a proof of formula F . Intuitionistic logic of proofs iLP introduced by Artemov and Iemhoff was conjectured to be complete with respect to intuitionistic arithmetic HA. The article presents a proof of this conjecture.

Beschreibung: Using Pauly’s majority logic, a modal-like language that describes decisions of a collective of agents, this article develops an explicit relationship between logic and social choice theory, particularly judgement aggregation. Majority logic is, within certain limits, able to express properties of decision procedures that are used to reach collective judgements. Those properties are often facts about the validity of logical inference rules at the level of collective decision making. From the perspective of social choice theory, formulae of majority logic can be regarded as axioms about the possibility of correct collective inference. Exploring the link with modal logic, and using so-called simple games as the mathematical structures representing decision procedures, we argue that the axiomatizations of social choice theorists mimic definability results from the perspective of modal logicians. Sets of formulae of majority logic define classes of decision procedures in this sense. Based on simple games, we give a game-theoretic characterization of properties of decision procedures that can be expressed using majority logic. From this, we deduce the closure conditions on classes of decision procedures that are definable using majority logic. This establishes a concrete link between a language in which axioms may be formulated, and the properties of decision procedures that it is able to characterize. Closure conditions also allow us to determine limits to the expressive and axiomatic power of majority logic. Since much of social choice theory is occupied with finding decision procedures that admit valid collective inference, and this is the domain of majority logic, quite a few familiar classes of decision procedures are definable. However, we show that majority logic is too weak to express non-dictatorship—in effect a language-relative impossibility result. We relate these definability results to recent results obtained in the judgement aggregation literature, in particular to some of the impossibility results.

Beschreibung: Recent work on Alternating-Time Temporal Logic and Coalition Logic has allowed the expression of many interesting properties of coalitions and strategies. However, there is no natural way of expressing resource requirements in these logics. In this article, we present a Resource-Bounded Coalition Logic ( RBCL ) that has explicit representation of resource bounds in the language. We give a complete and sound axiomatization of RBCL , a procedure for deciding satisfiability of RBCL formulas, and a model-checking algorithm.

Beschreibung: We show that propositional linear temporal logic with knowledge modalities but without common knowledge has an undecidable satisfiability problem when interpreted in a ‘concrete’ semantics with perfect recall or with perfect recall and synchrony.We then conclude that this concrete semantics is not axiomatizable in the semantics, based on local states.

Beschreibung: Justification Logic is a framework for reasoning about evidence and justification in multi-agent systems. Most accounts of Justification Logic are essentially static, in that the (justified) beliefs of agents are immutable. In this article, we add public communication, a dynamic operation of belief change studied in the area of Dynamic Epistemic Logic , to the language of Justification Logic. Introducing notions of bisimulation for the languages of Justification Logic with and without public communication, we catalogue the expressive relationships that exist between almost all of the well-known static fragments of Justification Logic and then determine whether the addition of public communication affects the various expressive relationships existing between these fragments.

Beschreibung: The study of constraint satisfaction problems (CSPs) definable in various fragments of Datalog has recently gained considerable importance.We consider CSPs that are definable in the smallest natural recursive fragment of Datalog—monadic linear Datalog with at most one EDB (extensional database predicates) per rule, and also in the smallest nonlinear extension of this fragment. We give combinatorial and algebraic characterizations of such problems, in terms of homomorphism dualities and lattice operations, respectively.We then apply our results to study graph H -colouring problems.