Basic Rules of Arithmetic

The authors study the decidability of redundancy detection in logic programs which are composed of inference rules with functions in their arguments. They formally define redundancies based on the solution sets in logic programs, and represent solution sets in well-defined domains. They prove that redundancy detection in inference rules with functions is undecidable. The theoretical results obtained here complete the previously unknown properties of redundancy detection of logic programs. Except for nonrecursive inference rules, redundancy detection is undecidable in general. Although general results reveal the inherent difficulty of efficient execution of logic programs through redundancy detection, special redundant cases can still be detected and pruned. Tradeoffs between compile-time analysis of decidable cases and run-time checking of undecidable cases can still be made. >

The Semantic Web is a vision for the future of the Web in which information is given explicit meaning, making it easier for machines to automatically process and integrate information available on the Web[1]. In semantic web, Knowledge can be encoded using a formal language termed as OWL (Ontology Web Language). To keep this language alive Progression of OWL2 from OWL is one such major step. OWL2 is a strong Modeling language and extends the W3C OWL with a small but useful set of features. The objective of this study is an attempt to highlight those features and to provide and indepth analysis on: a. Introduction to three profiles of OWL2 with a particular focus on how these profiles differ and why the differences are important. b. Comparison of OWL Profiles on the basis of Usage, Reasoning Time, Complexity and Algorithms. c. Reasoning in OWL RL Profile and comparison of Reasoners in OWL RL Profiles. Keywords: OWL – Ontology Web Language, OWL axioms – Inference rules

Does Frege have a metatheory for his logic? There is an obvious and uncontroversial sense in which he does. Frege introduces and discusses his new logic in natural language; he argues, in response to criticisms of Begriffsschrift, that his logic is superior to Boole's by discussing formal features of both systems. In so far as the enterprise of using natural language to introduce, discuss, and argue about features of a formal system is metatheoretic, there can be no doubt: Frege has a metatheory. There is also an obvious and uncontroversial sense in which Frege does not have a metatheory for his logic. The model theoretic semantics with which we are familiar today are a post-Fregean development. The question I address in this paper is, does Frege have a metatheory in the following sense: do his justifications of his basic laws and rules of inference employ, or even require, ineliminable use of a truth predicate and metalinguistic variables? My answer is ‘no’ on both counts. I argue that Frege neither uses, nor has any need to use, a truth predicate or metalinguistic variables in his justifications of his basic laws and rules of inference. Quine's famous explanation of the need for semantic ascent simply does not apply to Frege's logic. The purpose of the discussions that are typically understood as constituting Frege's metatheory is, rather, elucidatory. And once we see what the aim of these particular elucidations is, we can explain Frege's otherwise puzzling eschewal of the truth predicate in his discussions of the justification of the laws and rules of inference.

Logic programming, as exemplified by datalog, defines the meaning of a program as its unique smallest model: the deductive closure of its inference rules. However, many problems call for an enumeration of models that vary along some set of choices while maintaining structural and logical constraints—there is no single canonical model. The notion of stable models for logic programs with negation has successfully captured programmer intuition about the set of valid solutions for such problems, giving rise to a family of programming languages and associated solvers known as answer set programming. Unfortunately, the definition of a stable model is frustratingly indirect, especially in the presence of rules containing free variables. We propose a new formalism, finite-choice logic programming, that uses choice, not negation, to admit multiple solutions. Finite-choice logic programming contains all the expressive power of the stable model semantics, gives meaning to a new and useful class of programs, and enjoys a least-fixed-point interpretation over a novel domain. We present an algorithm for exploring the solution space and prove it correct with respect to our semantics. Our implementation, the Dusa logic programming language, has performance that compares favorably with state-of-the-art answer set solvers and exhibits more predictable scaling with problem size.

Multiset rewriting has proved to be a useful presentation of process synchronization [1, 2, 3, 6]. Since sequent calculus presentations of logics that do not use the structural rules of contractions and weakening are based on using multisets of formulas as left and right contexts, it is natural to identify processes with formulas, multisets with sequent contexts, and multiset rewriting as an inference rule. Given earlier work on using sequent calculus to describe logic programming as goal-directed search for proofs [8], it is most natural to use right-hand contexts of sequents to represent multisets of processes. This choice requires the identification of the multiset constructor and the empty multiset with the multiplicative disjunction and false (the and ⊥ of linear logic [4]), and backchaining with a single step of multiset rewriting. While the logic programming language λProlog [10] and its linear logic refinement Lolli [5] contain rich sources of abstraction (such as modular programming, abstract data types, and higher-order programming), they contain no primitives for specifying concurrency, communications, or synchronization. If multiset rewriting is added to Lolli via the logical connectives and ⊥, the result is a language that contains primitives for both abstraction and concurrency. Surprisingly, the resulting logic, called Forum [7], is a presentation of all of linear logic in the sense that all of the connectives of linear logic can be defined via logical equivalences using only the connectives of Forum. Thus the rich meta-theory of linear logic, for example, the de Morgan dualities and cut-elimination, can be applied to the analysis of Forum programs. Several examples to illustrate the expressiveness of this presentation of linear logic will be given. These examples will involve a specification of sequent calculi for object-level logics, a specification of the π-calculus [9], and a specification of a functional programming language that contains side-effects and concurrency operators. In each of these examples, we shall argue that the specification is perspicuous and modular and that the meta-theory of linear logic can be used to derive properties of the specification.

This paper considers the possibility of designing AI that can learn logical or non-logical inference rules from data. We first provide an abstract framework for learning logics. In this framework, an agent $${{{\mathcal {A}}}}$$ provides training examples that consist of formulas S and their logical consequences T. Then a machine $${{{\mathcal {M}}}}$$ builds an axiomatic system that makes T a consequence of S. Alternatively, in the absence of an agent $$\mathcal{A}$$ , a machine $${{{\mathcal {M}}}}$$ seeks an unknown logic underlying given data. We next consider the problem of learning logical inference rules by induction. Given a set S of propositional formulas and their logical consequences T, the goal is to find deductive inference rules that produce T from S. We show that an induction algorithm LF1T, which learns logic programs from interpretation transitions, successfully produces deductive inference rules from input data. Finally, we consider the problem of learning non-logical inference rules. We address three case studies for learning abductive inference, frame axioms, and conversational implicature. Each case study uses machine learning techniques together with metalogic programming.

The performance of any fuzzy logic controller (FLC) is greatly dependent on its inference rules. In most cases, the closed-loop control performance and stability are enhanced if more rules are added to the rule base of the FLC. However, a large set of rules requires more on-line computational time and more parameters need to be adjusted. In this paper, a robust PD-type FLC is driven for a class of MIMO second order nonlin- ear systems with application to robotic manipulators. The rule base consists of only four rules per each de- gree of freedom (DOF). The approach implements fuzzy partition to the state variables based on Lyapunov synthesis. The resulting control law is stable and able to exploit the dynamic variables of the system in a lin- guistic manner. The presented methodology enables the designer to systematically derive the rule base of the control. Furthermore, the controller is decoupled and the procedure is simplified leading to a computationally efficient FLC. The methodology is model free approach and does not require any information about the sys- tem nonlinearities, uncertainties, time varying parameters, etc. Here, we present experimental results for the following controllers: the conventional PD controller, computed torque controller (CTC), sliding mode con- troller (SMC) and the proposed FLC. The four controllers are tested and compared with respect to ease of design, implementation, and performance of the closed-loop system. Results show that the proposed FLC has outperformed the other controllers.

PREFACE, ABBREVIATIONS, 1. THE NEED FOR A LOGICAL CALCULUS FOR THE THEORY OF SINN AND BEDEUTUNG, Introduction, Frege's Project: Logicism and the Notion of Begriffsschrift, The Theory of Sinn and Bedeutung, The Limitations of the Begriffsschrift, Filling the Gap, 2. THE LOGIC OF THE GRUNDGESETZE, Logical Language and the Content of Logic, Functionality and Predication, Quantifiers and Gothic Letters, Roman Letters: An Alternative Notation for Generality, Value-Ranges and Extensions of Concepts, The Syntactic Rules of the Begriffsschrift, The Axiomatization of Frege's System, Responses to the Paradox, 3. SINNE AND GEDANKEN, The Nature of Sinne Generally, The Sinne of Incomplete Expressions and the Composition of Gedanken, Context Principle, Priority Thesis, and Multiple Analyses: Challenges to Compositionality, The Identity Conditions of Sinne and Gedanken, 4. CHURCH'S LOGIC OF SENSE AND DENOTATION, Overview, The Method of Transparent Intensional Logic, Alternatives (0), (1) and (2) and Synonymous Isomorphism, The Formulation of the Systems, Problems in Church's Logic of Sense and Denotation, The UnFregean Elements of Church's Logic of Sense and Denotation, 5. A LOGICAL CALCULUS FOR THE THEORY OF SINN AND BEDEUTUNG, Basic Features, Names and Variables for Sinne, The New Constants of the Expanded Language, Syntactic Rules of the Expanded Language, Axioms and Inference Rules of the Expanded Calculi, Inferences Involving Propositional Attitudes and Quantifying In, Difficulties with the Expanded Calculi, 6. COMPARISON WITH RUSSELL AND OTHER THINKERS, Solatium miseris, socios habuisse malorum., Russell's Ontology of Propositions, Russell and the Principles of Mathematics Appendix B Paradox, Frege and the Principles of Mathematics Appendix B Paradox, Lessons Learned from Russell, Other Systems of Intensional Logic, 7. POSSIBLE REVISIONS TO FREGE'S PHILOSOPHY, Adequate versus Ad Hoc Solutions, A Cantorian Analysis of the Difficulties, Dropping Classes and/or Concepts, Reducing the Number of Sinne and Gedanken, Ramification as a Solution to the Paradoxes, The Justification of Ramification, Conclusion, APPENDIX: SUMMARY OF DEFINITIONS, AXIOMS AND INFERENCE RULES, BIBLIOGRAPHY, INDEX

One of the central questions in 20th-century discussions within logic and philosophy of language is where logical constants, specifically propositional connectives, get their meaning from. What kind of determination/justification bond is found between the inference rules peculiar to a given connective and the meaning of that connective? Are these rules justified by it, or rather do they contribute to its construction? An observation made by G. Gentzen, who founded in the 1930s the proof-theoretical approach at large, triggered a view (called by some logical inferentialism) that gives a remarkable answer to the above question: the meaning of a logical constant in a logical language is provided, not by some sort of representational content, but by the inferential norms that govern its overall use. In 1960 A. N. Prior fictionalized as a counter-instance the connective tonk solely using a couple of inference rules, a connective capable of overthrowing the system of deduction; N. Belnap’s 1962 reply in the form of an analysis of the tonk problem opens the way to discussions in logic-cum-philosophy of language with important outcomes. The present little study can be read as some further deflation of the tonk problem with a relatively unconstrained inferentialistic view of the matter. The two main theses of the study are (i) that the problem posed by tonk-like connectives can be captured, more simply than in Belnap’s (otherwise correct) analysis, through inferential relations of a certain type which will be dubbed alethic relations; and (ii) that Prior’s challenge, brought to completion in whichever way, cannot give any result against the inferentialist conception.

Formal verification involves the use of logical and computational methods to establish claims that are expressed in precise mathematical terms. These can include ordinary mathematical theorems, as well as claims that pieces of hardware or software, network protocols, and mechanical and hybrid systems meet their specifications. In practice, there is not a sharp distinction between verifying a piece of mathematics and verifying the correctness of a system: formal verification requires describing hardware and software systems in mathematical terms, at which point establishing claims as to their correctness becomes a form of theorem proving. Conversely, the proof of a mathematical theorem may require a lengthy computation, in which case verifying the truth of the theorem requires verifying that the computation does what it is supposed to do. The gold standard for supporting a mathematical claim is to provide a proof, and twentieth-century developments in logic show most if not all conventional proof methods can be reduced to a small set of axioms and rules in any of a number of foundational systems. With this reduction, there are two ways that a computer can help establish a claim: it can help find a proof in the first place, and it can help verify that a purported proof is correct. Automated theorem proving focuses on the “finding” aspect. Resolution theorem provers, tableau theorem provers, fast satisfiability solvers, and so on provide means of establishing the validity of formulas in propositional and first-order logic. Other systems provide search procedures and decision procedures for specific languages and domains, such as linear or nonlinear expressions over the integers or the real numbers. Architectures like SMT (“satisfiability modulo theories”) combine domain-general search methods with domain-specific procedures. Computer algebra systems and specialized mathematical software packages provide means of carrying out mathematical computations, establishing mathematical bounds, or finding mathematical objects. A calculation can be viewed as a proof as well, and these systems, too, help establish mathematical claims. Automated reasoning systems strive for power and efficiency, often at the expense of guaranteed soundness. Such systems can have bugs, and it can be difficult to ensure that the results they deliver are correct. In contrast, interactive theorem proving focuses on the “verification” aspect of theorem proving, requiring that every claim is supported by a proof in a suitable axiomatic foundation. This sets a very high standard: every rule of inference and every step of a calculation has to be justified by appealing to prior definitions and theorems, all the way down to basic axioms and rules. In fact, most such systems provide fully elaborated “proof objects” that can be communicated to other systems and checked independently. Constructing such proofs typically requires much more input and interaction from users, but it allows us to obtain deeper and more complex proofs. The Lean Theorem Prover aims to bridge the gap between interactive and automated theorem proving, by situating automated tools and methods in a framework that supports user interaction and the construction of fully specified axiomatic proofs. The goal is to support both mathematical reasoning and reasoning about complex systems, and to verify claims in both domains.

In this paper we propose an approach to manage access rights in a content management systems which relies on semantic web models and technologies. We present the AMO ontology which consists (1) in a set of classes and properties dedicated to the annotation of resources whose access should be controlled and (2) in a base of inference rules modeling the access management strategy to carry out. When applied to the annotations of the resources whose access should be controlled, these rules enable to manage access according to a given strategy. This modelisation is flexible, extendable and ensures the adaptability of the AMO ontology to any access management strategy. We illustrate the use of AMO on the documents of a collaborative website managed by the semantic wiki SweetWiki in the ANR ISICIL project. We show how to annotate documents with AMO, we explain which AMO inference rules can be applied and which semantic queries finally enable to control access to SweetWiki documents.

We present a general storage model that reflects features of object oriented (OO) languages with pure reference semantics. Based on this model, we develop an OO Separation Logic (OOSL) to specify and verify OO programs. Many inference rules in the Separation Logic still hold in OOSL. Additionally, OOSL has certain properties important to OO reasoning. We introduce Hoare-Triple for a small OO language, and use the Schorr-Waite Marking Algorithm as a verification example.

In general, landscape architecture includes analysis, planning, design, administration and management of natural and artificial. An important aspect is the formation of so-called sustainable landscapes that allow maximum use of the environment, natural resources and promote sustainable restoration of ecosystems. For such purposes, a designer needs a complete database with existing and suitable plants, but no designing tool has one. Therefore it is presented the structure and the development of on ontology suitable for storing and managing all information and knowledge about plants. The advantage is that the format of the ontology allows the storage of any plant species (e.g. live or fossil) and automated reasoning. Ontology is a formal conceptualization of a particular knowledge about the world, through the explicit representation of basic concepts, relations, and inference rules about themselves. Therefore the ontology may be used by a design tool for helping the designer and choosing the best options for a sustainable landscape.

Preface: Volume 59, Issue 4

The “fuzzy dot” (or fuzzy relation) representation of fuzzy rules in fuzzy rule based systems, in case of classical fuzzy reasoning methods (e.g. the Zadeh-Mamdani- Larsen Compositional Rule of Inference (CRI) (Zadeh, 1973) (Mamdani, 1975) (Larsen, 1980) or the Takagi - Sugeno fuzzy inference (Sugeno, 1985) (Takagi & Sugeno, 1985)), are assuming the completeness of the fuzzy rule base. If there are some rules missing i.e. the rule base is “sparse”, observations may exist which hit no rule in the rule base and therefore no conclusion can be obtained. One way of handling the “fuzzy dot” knowledge representation in case of sparse fuzzy rule bases is the application of the Fuzzy Rule Interpolation (FRI) methods, where the derivable rules are deliberately missing. Since FRI methods can provide reasonable (interpolated) conclusions even if none of the existing rules fires under the current observation. From the beginning of 1990s numerous FRI methods have been proposed. The main goal of this article is to give a brief but comprehensive introduction to the existing FRI methods.