Automated prover for attribute dependencies in data with grades

We present a new axiomatization of logic for dependencies in data with grades, which includes ordinal data and data over domains with similarity relations, and an efficient reasoning method that is based on the axiomatization. The logic has its ordinary-style completeness characterizing the ordinary, bivalent entailment as well as the graded-style completeness characterizing the general, possibly intermediate degrees of entailment. A core of the method is a new inference rule, called the rule of simplification, from which we derive convenient equivalences that allow us to simplify sets of dependencies while retaining semantic closure. The method makes it possible to compute a closure of a given collection of attributes with respect to a collection of dependencies, decide whether a given dependency is entailed by a given collection of dependencies, and more generally, compute the degree to which the dependency is entailed by a collection of dependencies. We also present an experimental evaluation of the presented method.     

A framework for reasoning under uncertainty based on non-deterministic distance semantics

In this paper, we introduce a general and modular framework for formalizing reasoning with incomplete and inconsistent information. Our framework is composed of non-deterministic semantic structures and distance-based considerations. This combination leads to a variety of entailment relations that can be used for reasoning about non-deterministic phenomena and are inconsistency-tolerant. We investigate the basic properties of these entailments, as well as some of their computational aspects, and demonstrate their usefulness in the context of model-based diagnostic systems.     

Fuzzy Horn logic I

The paper presents generalizations of results on so-called Horn logic, well-known in universal algebra, to the setting of fuzzy logic. The theories we consider consist of formulas which are implications between identities (equations) with premises weighted by truth degrees. We adopt Pavelka style: theories are fuzzy sets of formulas and we consider degrees of provability of formulas from theories. Our basic structure of truth degrees is a complete residuated lattice. We derive a Pavelka-style completeness theorem (degree of provability equals degree of truth) from which we get some particular cases by imposing restrictions on the formulas under consideration. As a particular case, we obtain completeness of fuzzy equational logic.     

An anytime deduction algorithm for the probabilistic logic and entailment problems

We study two basic problems of probabilistic reasoning: the probabilistic logic and the probabilistic entailment problems. The first one can be defined as follows. Given a set of logical sentences and probabilities that these sentences are true, the aim is to determine whether these probabilities are consistent or not. Given a consistent set of logical sentences and probabilities, the probabilistic entailment problem consists in determining the range of the possible values of the probability associated with additional sentences while maintaining a consistent set of sentences and probabilities.This paper proposes a general approach based on an anytime deduction method that allows the follow-up of the reasoning when checking consistency for the probabilistic logic problem or when determining the probability intervals for the probabilistic entailment problem. Considering a series of subsets of sentences and probabilities, the approach proceeds by computing increasingly narrow probability intervals that either show a contradiction or that contain the tightest entailed probability interval. Computational experience have been conducted to compare the proposed anytime deduction method, called ad-psat with an exact one, psatcol, using column generation techniques, both with respect to the range of the probability intervals and the computing times.     

All finitely axiomatizable subframe logics containing the provability logic CSM_{0} are decidable

In this paper we investigate those extensions of the bimodal provability logic (alias or which are subframe logics, i.e. whose general frames are closed under a certain type of substructures. Most bimodal provability logics are in this class. The main result states that all finitely axiomatizable subframe logics containing are decidable. We note that, as a rule, interesting systems in this class do not have the finite model property and are not even complete with respect to Kripke semantics. Received July 15, 1997     

A sequent calculus for logic of knowledge and past time: Completeness and decidability

We consider logic of knowledge and past time. This logic involves the discrete-time linear temporal operators next, until, weak yesterday, and since. In addition, it contains an indexed set of unary modal operators agent i knows.We consider the semantic constraint of the unique initial states for this logic. For the logic, we present a sequent calculus with a restricted cut rule. We prove the soundness and completeness of the sequent calculus presented. We prove the decidability of provability in the considered calculus as well. So, this calculus can be used as a basis for automated theorem proving. The proof method for the completeness can be used to construct complete sequent calculi with a restricted cut rule for this logic with other semantical constraints as well. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 427–437, July–September, 2006.     

A theory,implementation and applications of human-like understanding

Operation logic is a formal logic with well-defined formulas as semantic language clauses and with modus ponens rules as a method of reasoning. Operation logic can be implemented on any database management system (as the so-called OLS) having a universal general knowledge database and enabling understanding of data stored in the database. Semantic language clauses have necessary and sufficient properties for being able to describe any process in the world. Semantic language is the deepest level of any natural language, the level of data storing, understanding and reasoning. OLS can be a tool for studying implementation possibilities of human-like consciousness, for building artificial experts and artificial encyclopedias and for constructing semantic mathematical theories of anthropoecosystems (which is such an exact theory that qualitative information can be used with meaning completely defined by the user). In the paper the theory (and complete information enabling implementation) is presented for human-like understanding, topic-focus division of clauses, for human-like problem solving (program synthesis and verification) and for semantic mathematical analyses. Many examples are presented.     

数理逻辑中的数值化方法

数理逻辑的本质是形式推理而不是数值计算,非此即彼式的严谨性是其特征,因而在一定意义下它是"两极化"的.比如,(1)一个逻辑理论或者是相容的,或者是不相容的,不存在"半相容的"理论.(2)"逻辑公式A是假设集T的推论"或者成立,或者不成立,说它近似成立是无意义的.(3)逻辑公式中有重言式和矛盾式,但没有0.8重言式.本文的目的在于为上述基本概念提供程度化的版本,并从而建立一种近似推理理论.     

A logic of soft constraints based on partially ordered preferences

Representing and reasoning with an agent’s preferences is important in many applications of constraints formalisms. Such preferences are often only partially ordered. One class of soft constraints formalisms, semiring-based CSPs, allows a partially ordered set of preference degrees, but this set must form a distributive lattice; whilst this is convenient computationally, it considerably restricts the representational power. This paper constructs a logic of soft constraints where it is only assumed that the set of preference degrees is a partially ordered set, with a maximum element 1 and a minimum element 0. When the partially ordered set is a distributive lattice, this reduces to the idempotent semiring-based CSP approach, and the lattice operations can be used to define a sound and complete proof theory. A generalised possibilistic logic, based on partially ordered values of possibility, is also constructed, and shown to be formally very strongly related to the logic of soft constraints. It is shown how the machinery that exists for the distributive lattice case can be used to perform sound and complete deduction, using a compact embedding of the partially ordered set in a distributive lattice.     

Preservativity logic: An analogue of interpretability logic for constructive theories

In this paper we study the modal behavior of Σ‐preservativity, an extension of provability which is equivalent to interpretability for classical superarithmetical theories. We explain the connection between the principles of this logic and some well‐known properties of HA, like the disjunction property and its admissible rules. We show that the intuitionistic modal logic given by the preservativity principles of HA known so far, is complete with respect to a certain class of frames.     

On a New Idiom in the Study of Entailment

This paper is an experiment in Leibnizian analysis. The reader will recall that Leibniz considered all true sentences to be analytically so. The difference, on his account, between necessary and contingent truths is that sentences reporting the former are finitely analytic; those reporting the latter require infinite analysis of which God alone is capable. On such a view at least two competing conceptions of entailment emerge. According to one, a sentence entails another when the set of atomic requirements for the first is included in the corresponding set for the other; according to the other conception, every atomic requirement of the entailed sentence is underwritten by an atomic constituent of the entailing one. The former conception is classical on the twentieth century understanding of the term; the latter is the one we explore here. Now if we restrict ourselves to the formal language of the propositional calculus, every sentence has a finite analysis into its conjunctive normal form. Semantically, then, every sentence of that language can be represented as a simple hypergraph, H, on the powerset of a universe of states. Entailment of the sort we wish to study can be represented as a known relation, subsumption between hypergraphs. Since the lattice of hypergraphs thus ordered is a DeMorgan lattice, the logic of entailment thus understood is the familiar system, FDE of first-degree entailment. We observe that, extensionalized, the relation of subsumption is itself a DeMorgan Lattice ordered by higher-order subsumption. Thus the semantic idiom that hypergraph-theory affords reveals a hierarchy of lattices capable of representing entailments of every finite degree.     

The omega-rule interpretation of transfinite provability logic

Given a computable ordinal Λ, the transfinite provability logic ${\mathsf{GLP}}_{\mathrm{\Lambda }}$ has for each $\xi <\mathrm{\Lambda }$ a modality $[\xi ]$ intended to represent a provability predicate within a chain of increasing strength. One possibility is to read $[\xi ]?$ as ? is provable in T using ω-rules of depth at most ξ, where T is a second-order theory extending ${\mathrm{ACA}}_0$.In this paper we will formalize such iterations of ω-rules in second-order arithmetic and show how it is a special case of what we call uniform provability predicates. Uniform provability predicates are similar to Ignatiev's strong provability predicates except that they can be iterated transfinitely. Finally, we show that ${\mathsf{GLP}}_{\mathrm{\Lambda }}$ is sound and complete for any uniform provability predicate.     

What is a Logic Translation?

We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both proof- theoretic and model-theoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions. In memoriam Joseph Goguen     

Proofs and Countermodels in Non-Classical Logics

Proofs and countermodels are the two sides of completeness proofs, but, in general, failure to find one does not automatically give the other. The limitation is encountered also for decidable non-classical logics in traditional completeness proofs based on Henkin’s method of maximal consistent sets of formulas. A method is presented that makes it possible to establish completeness in a direct way: For any given sequent either a proof in the given logical system or a countermodel in the corresponding frame class is found. The method is a synthesis of a generation of calculi with internalized relational semantics, a Tait–Schütte–Takeuti style completeness proof, and procedures to finitize the countermodel construction. Finitizations for intuitionistic propositional logic are obtained through the search for a minimal derivation, through pruning of infinite branches in search trees by means of a suitable syntactic counterpart of semantic filtration, or through a proof-theoretic embedding into an appropriate provability logic. A number of examples illustrates the method, its subtleties, challenges, and present scope.     

A focused approach to combining logics

We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicative-additive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut-elimination holds in such fragments. From cut-elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classical-linear hybrid logics.     

Bisimulations between generalized Veltman models and Veltman models

Interpretability logic is an extension of provability logic. Veltman models and generalized Veltman models are two semantics for interpretability logic. We consider a connection between Veltman semantics and generalized Veltman semantics. We prove that for a complete image‐finite generalized Veltman modelW there is a Veltman model W ′ that is bisimular to W. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Constrained Consequence

There are various contexts in which it is not pertinent to generate and attend to all the classical consequences of a given premiss—or to trace all the premisses which classically entail a given consequence. Such contexts may involve limited resources of an agent or inferential engine, contextual relevance or irrelevance of certain consequences or premisses, modelling everyday human reasoning, the search for plausible abduced hypotheses or potential causes, etc. In this paper we propose and explicate one formal framework for a whole spectrum of consequence relations, flexible enough to be tailored for choices from a variety of contexts. We do so by investigating semantic constraints on classical entailment which give rise to a family of infra-classical logics with appealing properties. More specifically, our infra-classical reasoning demands (beyond a\modelsb{\alpha\models\beta}) that Mod(β) does not run wild, but lies within the scope (whatever that may mean in some specific context) of Mod(α), and which can be described by a sentence ·a{\bullet\alpha} with b\models·a{\beta\models\bullet\alpha}. Besides being infra-classical, the resulting logic is also non-monotonic and allows for non-trivial reasoning in the presence of inconsistencies.     

Eliminating disjunctions by disjunction elimination

Completeness and other forms of Zorn’s Lemma are sometimes invoked for semantic proofs of conservation in relatively elementary mathematical contexts in which the corresponding syntactical conservation would suffice. We now show how a fairly general syntactical conservation theorem that covers plenty of the semantic approaches follows from an utmost versatile criterion for conservation due to Scott.To this end we work with multi-conclusion entailment relations as extending single-conclusion entailment relations. In a nutshell, the additional axioms with disjunctions in positive position can be eliminated by reducing them to the corresponding disjunction elimination rules, which turn out provable in a wealth of mathematical instances. In deduction terms this means to fold up branchings of proof trees by way of properties of the relevant mathematical structures.Applications include syntactical counterparts of the theorems or lemmas known under the names of Artin–Schreier, Krull–Lindenbaum and Szpilrajn, as well as of the spatiality of coherent locales. Related work has been done before on individual instances, e.g. in locale theory, dynamical algebra, formal topology and proof analysis.     

Comparative and restrictive inequalities

Inequalities are an important topic in school mathematics, yet the body of research exploring students’ meanings for inequalities largely points to difficulties they experience. Thus, there is a need to further explore students’ meanings for inequalities. Addressing this need, we conducted an exploratory teaching experiment with two seventh-grade students to investigate their developing meanings for inequalities. We distinguish between two types of inequalities in student thinking: comparative and restrictive inequalities. Whereas a student reasoning about a comparative inequality compares two quantities’ values or magnitudes, reasoning about a restrictive inequality entails reasoning about a range of one quantity’s magnitudes or values. We realized a complexity arose in our interactions with students due to our conceiving the use of inequality symbols across the two types of inequalities as polysemous, whereas the students did not. Attending to these two types of inequalities has important implications for the teaching and learning of inequality.     

Distance-based paraconsistent logics

We introduce a general framework that is based on distance semantics and investigate the main properties of the entailment relations that it induces. It is shown that such entailments are particularly useful for non-monotonic reasoning and for drawing rational conclusions from incomplete and inconsistent information. Some applications are considered in the context of belief revision, information integration systems, and consistent query answering for possibly inconsistent databases.