Decidability and Completeness

It is well known that an important task of the artificial intelligence is to make a computer simulate a human being in dealing with certainty and uncertainty in information. Logic gives a technique for laying the foundations of this task. Information processing dealing with certain information is based on the classical logic. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of the classical logic to handle information with various facets of uncertainty, such as fuzziness and randomness. Therefore, nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. For investigation of several properties, such logics have been represented as algebras, that is, sets with one, two, or more algebraic operations satisfying some conditions inspired by these logics. This inspiration is illustrated by the similarities between the names. We have BCK-algebras and a BCK positive logic, BCI-algebras and a BCI positive logic, and BL-algebras and a basic logic. In many cases, the connection between such algebras and their corresponding logics is much stronger. In this case, one can give a translation procedure which translates all well-formed formulas and all theorems of a given logic L into terms and theorems of the corresponding algebras. In some cases, one can give also an inverse translation. In this case, we say that the given logic and class of such algebras are isomorphic. Nevertheless, the study of algebras motivated by logics is interesting and very useful also in the case when these structures are not isomorphic. An important class of algebras inspired by logic is BL-algebras introduced by Hajek in order to provide an algebraic proof of the completeness theorem of Basic Logic. A classical example of a BL-algebra is the interval [0,1] endowed with the structure induced by a continuous t-norm. Other important examples of BL-algebras are MV-algebras introduced earlier as a model of some infinitely valued Łukasiewicz logic. All these algebras are also strongly connected with residuated lattices. Hence, the filter theory plays an important role in the study of these algebras. From logic point of view, various filters correspond to various sets of provable formulae in the respective logic. Below we present several papers on filters in various algebras. In one of these papers, it is proved that there is a one-to-one correspondence between the set of all vague filters and all vague congruence of a lattice implication algebra. In another, the relations among fuzzy t-filters on residuated lattices are described. Another paper is devoted to the filter topology on lattice implication algebras. In further articles is characterized the relationship between the main types of fuzzy filters in BE-algebras and EQ-algebras. The role of ideals and soft sets in BL-algebras is described in another two papers. In papers on pseudo-weak-R0 algebras, the most simplified axioms system is presented and it is proved that pseudo-weak-R0 algebras are categorically isomorphic to pseudo-IMTL algebras. Pseudo-R0 algebras are categorically isomorphic to pseudo-NM algebras. Several presented papers are devoted to the possible applications of algebras inspired by logic and fuzzy sets. For example, a robust fuzzy logic controller is proposed for stabilization and disturbance rejection in nonlinear control systems of a particular type. The dynamic feedback controller is designed as a combination of a control law that compensates for nonlinear terms in a control system and a dynamic fuzzy logic controller that addresses unknown model uncertainties and an unmeasured disturbance. A mathematical derivation is used to prove that the controller is able to achieve asymptotic stability by processing state measurements. Properties of some hesitant triangular fuzzy aggregation operators based on Bonferroni means also are discussed. In one paper, the method of a construction of a ranking of the physical matches is proposed. This method is based on the fuzzy clustering analysis. Decisions are often made by many decision experts having preference of risks. In view of the grey situation group decision-making problems, a grey situation group decision-making method on the basis of prospect theory is shown. The method takes the positive and negative ideal situation distance as reference points, defines positive and negative prospect value function, and introduces decision experts' risk preference into grey situation decision-making to make the final decision more in line with decision experts' psychological behavior. One possibility of determination of the weight of each decision expert sets up comprehensive prospect value matrix for decision experts' evaluation, and finally determination of the optimal situation is presented. Jianming Zhan Bijan Davvaz Wieslaw A. Dudek Young Bae Jun Hee Sik Kim

Dung's argumentation framework is an abstract framework based on a set of arguments and a binary attack relation defined over the set. One instantiation, among many others, of Dung's framework consists in constructing the arguments from a set of propositional logic formulas. Thus an argument is seen as a reason for or against the truth of a particular statement. Despite its advantages, the argumentation approach for inconsistency handling also has important shortcomings. More precisely, in some applications what one is interested in are not so much only the conclusions supported by the arguments but also the precise explications of such conclusions. We show that argumentation framework applied to classical logic formulas is not suitable to deal with this problem. On the other hand, intuitionistic logic appears to be a natural alternative candidate logic (instead of classical logic) to instantiate Dung's framework. We develop constructive argumentation framework. We show that intuitionistic logic offers nice and desirable properties of the arguments. We also provide a characterization of the arguments in this setting in terms of minimal inconsistent subsets when intuitionistic logic is embedded in the modal logic S4.

The development of non-classical logics brought to light the question of their philosophical interpretation. Given a non-classical logic L, what is the informal/philosophical interpretation of its vocabulary? We find in the literature arguments defending that logical systems do not have a canonical philosophical interpretation, so the same logical system L has different philosophical interpretations. Although the thesis that a logic can be philosophically interpreted in different ways is well known, the thesis that a specific philosophical interpretation can be captured by different logics has not been widely explored. In this paper, we argue that if Kreisel’s informal rigor method is adequate to show that formal notions of logical consequence capture informal notions of logical consequence, then a specific philosophical interpretation only corresponds to a unique logic.

We start from the algebraic method of theorem-proving based on the translation of logic formulas into polynomials over finite fields, and adapt the case of first-order formulas by employing certain rings equipped with infinitary operations. This paper defines the notion of M-ring, a kind of polynomial ring that can be naturally associated to each first-order structure and each first-order theory, by means of generators and relations. The notion of M-ring allows us to operate with some kind of infinitary version of Boolean sums and products, in this way expressing algebraically first-order logic with a new gist. We then show how this polynomial representation of first-order sentences can be seen as a legitimate algebraic semantics for first-order logic, an alternative to cylindric and polyadic algebras and closer to the primordial forms of algebraization of logic. We suggest how the method and its generalization could be lifted successfully to n-valued logics and to other non-classical logics helping to reconcile some lost ties between algebra and logic.

The issue of what consequences to draw from the existence of non-classical logical systems has been the subject of an interesting debate across a diversity of fields. In this paper the matter of alternative logics is considered with reference to a specific belief system and its propositions :the Azande are said to maintain beliefs about witchcraft which, when expressed propositionally, appear to be inconsistent. When the Azande have been presented with such inconsistencies, they either fail to see them as such or else accept them as non-problematical. Is our knowledge of logical truths a relative and culturally determined phenomenon, or is there some (transcendent) criterion that allows us to adjudicate between alternative logical systems? The authors propose an approach for resolving disputes about the status of Azande reasoning which assumes a paraconsistent framework, thus providing a new perspective on this debate

Non-classical logic and ontological non-commitment, avoiding abstract objects through modal operators

The construction of many-valued logical systems is commonly believed to have been one of the major achievements of the Warsaw School, and specifically of Łukasiewicz.1 The first mention of many-valued (three-valued) logic is to be found in Łukasiewicz’s farewell speech at Warsaw University on March 7,1918. He said: In 1910 I published a book on the principle of contradiction in Aristotle’s work, in which I strove to demonstrate that that principle is not so evident as it is believed to be. Even then I strove to construct non-Aristotelian logic, but in vain. Now I believe I have succeeded in this. ... Examination of that issue was the subject-matter of my last lectures. I have proved that in addition to true and false propositions there are possible propositions, to which objective possibility corresponds as a third value in addition to being and non-being. This gave rise to a system of three-valued logic, which I worked out in detail last summer. That system is as coherent and self-consistent as Aristotle’s logic, and is much richer in laws and formulae. That new logic, by introducing the concept of objective possibility, destroyed the former concept of science, based on necessity. Possible phenomena have no cause, although they themselves can be the beginning of a causal sequence. An act of a creative individual can be free and at the same time affect the course of the world (1918, p. 86).

Substitution Frege and extended Frege proof systems in non-classical logics

Automatic service composition is an important problem in service computing. Existing works on service composition assume different representation models with various expressiveness for component services as well as for the composition logic. This has made understanding these composition approaches and their applications difficult. This paper presents a novel language-based formal framework that acts as a basis for understanding the expressiveness of service models and composition logics. It covers the three common service representation models, i.e., input/output, precondition/effects, and automaton models. The novelty of this framework is in using the execution language of service composition, characterized by logic formulae and formal languages, as a common ground to analyze service models and composition logics. We have studied the composition capability of each service model in terms of enabling common workflow patterns, and analyzed the computational complexity of composition algorithms of different service models. The framework simplifies the job of understanding service representation models and composition logics for a given service composition task.

The past few decades have seen a resurgence of reasoning techniques in artificial intelligence involving both classical and non-classical logics. In his paper, ``Multi-valued Logics: A Uniform Approach to Reasoning in Artificial Intelligence'', Ginsberg has shown that through the use of bilattices, several reasoning techniques can be unified under a single framework. A bilattice is a structure that can be viewed as a class of truth values that can accommodate incomplete and inconsistent information and in certain cases default information. In bilattice theory, knowledge is ordered along two dimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truth spaces, Ginsberg has shown that the same theorem prover can be used to simulate reasoning in first order logic, default logic, prioritized default logic and assumption truth maintenance system. Although this is a significant contribution, Ginsberg's paper was lengthy and involved. This paper summarizes some of the essential concepts and foundations of bilattice theory. Furthermore, it discusses the connections of bilattice theory and several other existing multi-valued logics such as the various three-valued logics and Belnap's four-valued logic. It is noted that the set of four truth values in Belnap's logic form a lattice structure that is isomorphic to the simplest bilattice. Subsequently, Fitting proposed a conflation operation that can be used to select sub-sets of truth values from this and other bilattices. This method of selecting sub-sets of truth values provides a means for identifying sub-logic in a bilattice.

An open quantified boolean formula (QBF) is a QBF that contains free (unquantified) variables. A solution to such a QBF is a quantifier-free formula that is logically equivalent to the given QBF. Although most recent QBF research has focused on closed QBF, there are a number of interesting applications that require one to consider formulas with free variables. This article shows how clause/cube learning for DPLL-based closed-QBF solvers can be extended to solve QBFs with free variables. We do this by introducing sequents that generalize clauses and cubes and allow learning facts of the form “under a certain class of assignments, the input formula is logically equivalent to a certain quantifier-free formula”.KeywordsInference RuleFree VariableInitial SequentPartial AssignmentCurrent AssignmentThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Relevant logics are non-classical logics, whose motivation is to remove logical fallacies caused by the classical “implication≓. In this paper, we propose a method to build an interactive theorem prover for relevant logics. This is done first by translating the possible world semantics for relevant logics to the higher-order representation of HOL, and then under the HOL theory obtained by this translation, relevant formulas are shown to be valid using the powerful HOL proof capabilities such as backward reasoning with tactics and tacticals. Relevant logics we have dealt with so far includes Routley and Meyer's R system (originally Hilbert-type axiomatization) and Read's R system (basically Gentzentype axiomatization). Our various proof experiences of relevant formulas by HOL and their analyses yielded a powerful proof heuristics for relevant logics. It actually allowed us to prove a formula which has been known to be difficult for traditional theorem provers and even relevant logicians.KeywordsAtomic FormulaNatural DeductionProof TheoryRelevant LogicWorld SemanticThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a summary of some things that can be said about negation understood as an impossibility operator. To model negation one may use possible-worlds models in the style of Kripke that have an accessibility relation R N peculiar to negation: not A holds at a world x if and only if A doesn’t hold at any world accessible from x by R N . This method for modelling negation is applicable to classical logic, intu­itionistic logic, intermediate logics and, in general, nonclassical logics with models in the style of Kripke, like relevant, linear and other substructural logics. We con­centrate here on negation-like operators added to intuitionistic negationless logic, taken as a test case. One could later try to proceed analogously with other logics.KeywordsModal LogicUnary OperatorIntuitionistic LogicKripke ModelAxiom SchemaThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The question of what is it that traders actually do seems endlessly present in the emerging sociology of finance. For Hardie (2004, p. 251), joining, clarifying and critiquing Shleifer, the question is important but misplaced. Misplaced because it distracts attention onto particular types involved in financial markets to the detriment of a more general analysis of 'what investors actually think and do' (Shleifer, 2000, p. 183; our emphasis). For others, such as Zaloom (2006) and Knorr Cetina and Bruegger (2000, 2002), the enacting agents of 'global microstructures' merit considerable focus in their own right. It is the very specificity of the trading position, so obviously lacking in much prior research, that they believe is most worthy of attention. Still others, perhaps the sources of Hardie's ire, have gone further – delineating different types of traders and trading strategies, as the investor slips further into the background. Key here would be Buenza and Stark (2005) and their efforts to reinforce distinction between 'value' and 'momentum' traders in order to hold open a classificatory space for 'radically deconstructionist' agents (p. 88) who with their 'mathematical formulae' (p. 92) and 'evaluative principles' (p. 90), engage in the precise slicing and dicing (p. 88) of 'modern arbitrage' (p. 87). Calculation, entanglement, and immersion in screened flow have all been invoked, along with accompanying emotional repertoires (Hassoun, 2005) and bodily disciplines (Zaloom, 2006), which enable and sustain them, in order to grasp the specificity of the trading place and those inhabiting it.KeywordsFinancial MarketFund ManagerTrading PositionTrading PlaceCollateralized Debt ObligationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Vacuity arises when a logical formula is trivially true in a given model due, for example, to antecedent failure. Beer et al. have recently introduced a logic-independent notion of vacuity and shown that certain logics, i.e., those with polarity, admit an efficient decision procedure for vacuity detection. We show that the modal mu-calculus, a very expressive temporal logic, is a logic with polarity and hence the results of Beer et al. are applicable. We also extend the definition of vacuity to achieve a new notion of redundancy in logical formulas. Redundancy captures several forms of antecedent failure that escape traditional vacuity analysis, including vacuous actions in temporal modalities and unnecessarily strong temporal operators. Furthermore, we have implemented an efficient redundancy checker for the modal mu-calculus in the context of the XMC model checker. Our checker generates diagnostic information in the form of all maximal subformulas that are redundant and exploits the fact that XMC can cache intermediate results in memo tables between model-checking runs. We have applied our redundancy checker to a number of previously published case studies, and found instances of redundancy that have gone unnoticed till now. These findings provide compelling evidence of the importance of redundancy detection in the design process.KeywordsModel CheckPositive PolaritySyntax TreeModal FormulaAction LabelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.