Finiteness in Infinite-Valued Łukasiewicz Logic

In this paper we deepen Mundici's analysis on reducibility of the decision problem from infinite-valued ukasiewicz logic to a suitable m-valued ukasiewicz logic _m , where m only depends on the length of the formulas to be proved. Using geometrical arguments we find a better upper bound for the least integer m such that a formula is valid in if and only if it is also valid in _m. We also reduce the notion of logical consequence in to the same notion in a suitable finite set of finite-valued ukasiewicz logics. Finally, we define an analytic and internal sequent calculus for infinite-valued ukasiewicz logic.     

More Proofs of an Axiom of Łukasiewicz

This paper reports results and some new problems in one of the domains to which automatic first-order theorem provers have been most successfully applied: axiomatics of nonclassical propositional logics. It is well known that one of the standard axioms of the denumerable-valued pure implication logic of ukasiewicz becomes derivable from the remainder in the presence of negation. Here it is shown that the same axiom is similarly derivable using conjunction and disjunction instead of negation. This closes a problem left open by Harris and Fitelson (Journal of Automated Reasoning 27, 2001). Related problems are discussed, and five such open problems are proposed as challenges to the automated reasoning community.     

Theory of truth degrees of formulas in ■ukasiewicz n-valued propositional logic and a limit theorem

~~Theory of truth degrees of formulas in ■ukasiewicz n-valued propositional logic and a limit theorem1. Rosser, J. B., Turquette, A. R., Many-Valued Logics, Amsterdam: North-Holland, 1952. 2. Pavelka, J., On fuzzy logic Ⅰ,Ⅱ,Ⅲ, Z. Math. Logik Grundlagen Math., 1979, 25: 45-52; 119-134; 447-464. 3. Ying Mingsheng, Deduction theorem for many-valued inference, Z. Math. Logik Grundlagen Math., 1991, 37: 533-537. 4. Ying Mingsheng, The fundamental theorem of ultraproduct in …     

Linear optimization with bipolar fuzzy relational equation constraints using the Łukasiewicz triangular norm

This paper discusses the linear optimization problem constrained by a system of bipolar fuzzy relational equations with max- \(T\) composition, where the involved triangular norm is the ?ukasiewicz t-norm. Although it is in general NP-hard, such an optimization problem can be reformulated in polynomial time into a 0-1 integer linear optimization problem and then solved taking advantage of well developed techniques in integer optimization.     

FC-normal and extended stratified logic program

This paper investigates the consistency property of FC-normal logic program and presents an equivalent deciding condition whether a logic program P is an FC-normal program. The deciding condition describes the characterizations of FC-normal program. By the Petri-net presentation of a logic program, the characterizations of stratification of FC-normal program are investigated. The stratification of FC-normal program motivates us to introduce a new kind of stratification, extended stratification, over logic program. It is shown that an extended (locally) stratified logic program is an FC-normal program. Thus, an extended (locally) stratified logic program has at least one stable model. Finally, we have presented algorithms about computation of consistency property and a few equivalent deciding methods of the finite FC-normal program.     

Normal forms in BL and ŁΠ algebras of functions

Three types of normal forms are introduced for fuzzy logic functions: disjunctive, conjunctive and additive. Disjunctive and conjunctive normal forms are considered in two variants: infinite and finite. It is proved that infinite normal forms are universal representation formulas whereas finite normal forms are universal approximation formulas for any L-valued function where L is a support set of any complete BL-algebra. The additive normal form lies in the middle of the two others. For all of them the estimation of the quality of approximation is suggested.     

Revising Z: Part I – logic and semantics

This is the first of two related papers. We introduce a simple specification logic Z _C comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within Z _C . As a result we obtain a sound logic for Z, including a basic schema calculus. Received March 1998 / Accepted in revised form April 1999     

A Class of Implicative Expansions of Kleene’s Strong Logic,a Subclass of Which Is Shown Functionally Complete Via the Precompleteness of Łukasiewicz’s 3-Valued Logic Ł3

Journal of Logic, Language and Information - The present paper is a sequel to Robles et al. (J Logic Lang Inf 29(3):349–374, 2020. https://doi.org/10.1007/s10849-019-09306-2). A class of...     

Choquet integral with respect to ?ukasiewicz filters, and its modifications

In many decision problems a set of actions is evaluated with respect to a set of points of view, called criteria. This paper follows two aims - first to compare the so-called level-dependent Choquet integral introduced recently by Greco et al. with another transformation of Choquet integral, proposed by Havranová and Kalina. The other aim of this paper is to look for an appropriate utility function in a given setting. We illustrate our approach on a practical example, utilizing the level-dependent Choquet integral.     

Theory of truth degrees of formulas in Łukasiewicz<Emphasis Type="Italic">n</Emphasis>-valued propositional logic and a limit theorem

The concept of truth degrees of formulas in Łukasiewiczn-valued propositional logicL _n is proposed. A limit theorem is obtained, which says that the truth functionτ _n induced by truth degrees converges to the integrated truth functionτ whenn converges to infinite. Hence this limit theorem builds a bridge between the discrete valued Łukasiewicz logic and the continuous valued Łukasiewicz logic. Moreover, the results obtained in the present paper is a natural generalization of the corresponding results obtained in two-valued propositional logic.     

Boosting learning and inference in Markov logic through metaheuristics

Markov Logic (ML) combines Markov networks (MNs) and first-order logic by attaching weights to first-order formulas and using these as templates for features of MNs. State-of-the-art structure learning algorithms in ML maximize the likelihood of a database by performing a greedy search in the space of structures. This can lead to suboptimal results because of the incapability of these approaches to escape local optima. Moreover, due to the combinatorially explosive space of potential candidates these methods are computationally prohibitive. We propose a novel algorithm for structure learning in ML, based on the Iterated Local Search (ILS) metaheuristic that explores the space of structures through a biased sampling of the set of local optima. We show through real-world experiments that the algorithm improves accuracy and learning time over the state-of-the-art algorithms. On the other side MAP and conditional inference for ML are hard computational tasks. This paper presents two algorithms for these tasks based on the Iterated Robust Tabu Search (IRoTS) metaheuristic. The first algorithm performs MAP inference and we show through extensive experiments that it improves over the state-of-the-art algorithm in terms of solution quality and inference time. The second algorithm combines IRoTS steps with simulated annealing steps for conditional inference and we show through experiments that it is faster than the current state-of-the-art algorithm maintaining the same inference quality.     

A modal contrastive logic: The logic of ‘but’

In this paper we present a modal approach to contrastive logic, the logic of contrasts as these appear in natural language conjunctions such as but. We use a simple modal logic, which is an extension of the well-knownS5 logic, and base the contrastive operators proposed by Francez in [2] on the basic modalities that appear in this logic. We thus obtain a logic for contrastive operators that is more in accord with the tradition of intensional logic, and that, moreover — we argue — has some more natural properties. Particularly, attention is paid to nesting contrastive operators. We show that nestings of but give quite natural results, and indicate how nestings of other contrastive operators can be done adequately. Finally, we discuss the example of the Hangman's Paradox and some similarities (and differences) with default reasoning. But but us no buts, as they say.Also partially supported by Nijmegen University, Toernooiveld, 6525 ED Nijmegen, The Netherlands.     

Why nonmonotonic logic?

The motivation for nonmonotonic logic is to produce a machine representation for default reasoning, broadly construed. In this paper we argue that all nonmonotonic logics have (by definition) inference rules that fail to preserve truth, and this fact leads to several undesirable features. In response to these problems, but recognizing the importance of the original motivation for nonmonotonic logic, we propose an alternative to nonmonotonic logic, which achieves nonmonotonicity of reasoning without abandoning in any way truth preserving inferences. This approach is based on a possible worlds framework, which we illustrate with a small Prolog program. Motivating this approach is an important distinction, which we believe the advocates of nonmonotonic logic to be ignoring: that between inferencing and making decisions, or equivalently that between inferencing and theory construction.     

Virtual spaces and virtual layers – governing the ungovernable?

Online multi-user platforms like World of Warcraft and Twitter have one common regulatory mechanism; the End-User License Agreement (EULA). This document forms the cornerstone of the regulatory system within each of these spaces. Yet it is regularly contravened by users and providers alike. These agreements are very often the only forms of control or regulation that are present in online environments and therefore control more than user behaviour. Yet these platforms also share another feature: virtual disputes, but these are no longer confined online. Threats of violence and other criminal offences arise too, with examples including the abuse issued to Criado-Perez, and more recently, Flipovic. Criado-Perez suffered Twitter abuse and Flipovic was victimised on online message boards. Cyberspace was once deemed to be free from governmental control but the increasing disputes suggest there is now a need to consider how users of spaces such as online games, virtual worlds and social media are protected. Is it fair and practical to leave regulation to EULAs? How do users achieve redress for wrongs – through online and in-site governance mechanisms or wider controls? This work will consider some of these issues, and will suggest that there is now a need for additional layers of regulation to fill the ‘responsibility gap’ left between EULAs and the offline legal mechanisms.     

Tool support for learning Büchi automata and linear temporal logic

We introduce a graphical interactive tool, named GOAL, that can assist the user in understanding Büchi automata, linear temporal logic, and their relation. Büchi automata and linear temporal logic are closely related and have long served as fundamental building blocks of linear-time model checking. Understanding their relation is instrumental in discovering algorithmic solutions to model checking problems or simply in using those solutions, e.g., specifying a temporal property directly by an automaton rather than a temporal formula so that the property can be verified by an algorithm that operates on automata. One main function of the GOAL tool is translation of a temporal formula into an equivalent Büchi automaton that can be further manipulated visually. The user may edit the resulting automaton, attempting to optimize it, or simply run the automaton on some inputs to get a basic understanding of how it operates. GOAL includes a large number of translation algorithms, most of which support past temporal operators. With the option of viewing the intermediate steps of a translation, the user can quickly grasp how a translation algorithm works. The tool also provides various standard operations and tests on Büchi automata, in particular the equivalence test which is essential for checking if a hand-drawn automaton is correct in the sense that it is equivalent to some intended temporal formula or reference automaton. Several use cases are elaborated to show how these GOAL functions may be combined to facilitate the learning and teaching of Büchi automata and linear temporal logic. This work was partially supported by the National Science Council, Taiwan (R.O.C.) under grants NSC94-2213-E-002-089, NSC95-2221-E-002-127, NSC95-3114-P-001-001-Y02 (iCAST 2006), NSC96-3114-P-001-002-Y (iCAST 2007), and NSC97-2221-E-002-074-MY3.     

(L)ukasiewicz逻辑系统中的随机化研究

利用赋值集的随机化方法,在(L)ukasiewicz逻辑中提出了公式的随机真度,证明了所有公式的随机真度之集在[0,1]中没有孤立点;给出了两公式间的DL-相似度与伪距离的概念,并建立了DL-逻辑度量空间,证明了此空间没有孤立点.     

Computability of probability measures and Martin-Löf randomness over metric spaces

In this paper, we investigate algorithmic randomness on more general spaces than the Cantor space, namely computable metric spaces. To do this, we first develop a unified framework allowing computations with probability measures. We show that any computable metric space with a computable probability measure is isomorphic to the Cantor space in a computable and measure-theoretic sense. We show that any computable metric space admits a universal uniform randomness test (without further assumption).     

From linear temporal logic and limit-deterministic Büchi automata to deterministic parity automata

Controller synthesis for general linear temporal logic (LTL) objectives is a challenging task. The standard approach involves translating the LTL objective into a deterministic parity automaton (DPA) by means of the Safra-Piterman construction. One of the challenges is the size of the DPA, which often grows very fast in practice, and can reach double exponential size in the length of the LTL formula. In this paper, we describe a single exponential translation from limit-deterministic Büchi automata (LDBA) to DPA and show that it can be concatenated with a recent efficient translations from LTL to LDBA to yield a double exponential, ‘Safraless’ LTL-to-DPA construction. We also report on an implementation and a comparison with other LTL-to-DPA translations on several sets of formulas from the literature.