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Genuine Paraconsistent logics $$\mathbf {L3A}$$ and $$\mathbf {L3B}$$ were defined in 2016 by Beziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernandez-Tello et al, provide implications for both logics and define the logics $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ . In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ satisfy a restricted version of the Substitution Theorem, and that both of them are maximal with respect to Classical Propositional Logic. To conclude we make some comparisons between $$\mathbf {L3A_G}$$ and $$\mathbf {L3B_G}$$ and among other logics, for instance $${\mathbf {Int}}$$ and some $${\mathbf {LFI}}$$ s.

We revisit a problem faced by all programmers. Can one write a program that determines whether any given program is the shortest program? How does one prove that a given program is the shortest? After answering these questions, we discuss very briefly the Kolmogorov complexity of a string of zero and one, which leads to a barrier on any axiomatic system, known as Chaitin’s barrier.

To specify and verify the concurrent and reactive systems with the theorem proving approach, a complete axiomatization is formalized for first order projection temporal logic (PTL) with both finite and infinite time. To this end, PTL is restricted to a finite domain, and the syntax, semantics as well as the axiomatization of PTL are presented. Further, the techniques of labeled normal form and labeled normal form graph of PTL formulas are introduced respectively, with which a decision procedure for quantifier free PTL (QFPTL) formulas is given. Moreover, a generalized labeled normal form graph is defined and employed to transform a quantified PTL formula into its equivalent QFPTL formula. Finally, a decision procedure for PTL is formalized and the completeness of the axiomatic system is proved based on the decidability of PTL formulas.

In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations of it in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut-elimination and some standard consequences of it: the interpolation and definability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call Godel sentences, using a purely proof-theoretical method.

Pairwise comparison matrices often exhibit inconsistency, therefore many indices have been suggested to measure their deviation from a consistent matrix. A set of axioms has been proposed recently that is required to be satisfied by any reasonable inconsistency index. This set seems to be not exhaustive as illustrated by an example, hence it is expanded by adding two new properties. All axioms are considered on the set of triads, pairwise comparison matrices with three alternatives, which is the simplest case of inconsistency. We choose the logically independent properties and prove that they characterize, that is, uniquely determine the inconsistency ranking induced by most inconsistency indices that coincide on this restricted domain. Since triads play a prominent role in a number of inconsistency indices, our results can also contribute to the measurement of inconsistency for pairwise comparison matrices with more than three alternatives., Comment: 12 pages

Chapter 5 explained the relevance between the initial idea of expected utility theory and psychology and introduced some studies of utility measurement based on expected utility theory. This chapter will first explain the axiomatic system of expected utility theory, then how to approach the axiomatic system, with introduction of some counterexamples.

Blockchain data structures maintained via the longest-chain rule have emerged as a powerful algorithmic tool for consensus algorithms. The technique—popularized by the Bitcoin protocol—has proven to be remarkably flexible and now supports consensus algorithms in a wide variety of settings. Despite such broad applicability and adoption, current analytic understanding of the technique is highly dependent on details of the protocol’s leader election scheme. A particular challenge appears in the proof-of-stake setting, where existing analyses suffer from quadratic dependence on suffix length. We describe an axiomatic theory of blockchain dynamics that permits rigorous reasoning about the longestchain rule in quite general circumstances and establish bounds—optimal to within a constant—on the probability of a consistency violation. This settles a critical open question in the proof-of-stake setting where we achieve linear consistency for the first time. Operationally, blockchain consensus protocols achieve consistency by instructing parties to remove a suffx of a certain length from their local blockchain. While the analysis of Bitcoin guarantees consistency with error 2-k by removing O(k) blocks, recent work on proof-of-stake (PoS) blockchains has suffered from quadratic dependence: (PoS) blockchain protocols, exemplified by Ouroboros (Crypto 2017), Ouroboros Praos (Eurocrypt 2018) and Sleepy Consensus (Asiacrypt 2017), can only establish that the length of this suffx should be Θ(k2). This consistency guarantee is a fundamental design parameter for these systems, as the length of the suffix is a lower bound for the time required to wait for transactions to settle. Whether this gap is an intrinsic limitation of PoS—due to issues such as the “nothing-at-stake” problem—has been an urgent open question, as deployed PoS blockchains further rely on consistency for protocol correctness: in particular, security of the protocol itself relies on this parameter. Our general theory directly improves the required suffix length from Θ(k2) to Θ(k). Thus we show, for the first time, how PoS protocols can match proof-of-work blockchain protocols for exponentially decreasing consistency error. Our analysis focuses on the articulation of a two-dimensional stochastic process that captures the features of interest, an exact recursive closed form for the critical functional of the process, and tail bounds established for associated generating functions that dominate the failure events. Finally, the analysis provides an explicit polynomial-time algorithm for exactly computing the exponentially-decaying error function which can directly inform practice.

A given order type in the plane can be represented by a corresponding point set. However, it might be difficult to recognize the orientations of some point triples. Recently, Aichholzer et al. [3] introduced exit graphs for visualizing order types in the plane. We present a new class of geometric graphs, called OT-graphs, using abstract order types and their axioms described in the well-known book by Knuth [15]. Each OT-graph corresponds to a unique order type. We develop efficient algorithms for recognizing OT-graphs and computing a minimal OT-graph for a given order type in the plane. We provide experimental results on all order types of up to nine points in the plane including a comparative analysis of exit graphs and OT-graphs.

This paper presents a general framework for the study of $$({\mathcal {I}}, {\mathcal {N}})$$ -single valued neutrosophic rough sets from constructive and axiomatic perspectives. In the constructive approach, a pair of single valued neutrosophic rough approximation operators based on single valued neutrosophic implicator $${\mathcal {I}}$$ and single valued neutrosophic norm $${\mathcal {N}}$$ is first proposed. Moreover, some basic properties of $$({\mathcal {I}},{\mathcal {N}})$$ -single valued neutrosophic rough approximation operators are explored. In addition, connections between single valued neutrosophic relations and $$(\mathcal {I},{\mathcal {N}})$$ -single valued neutrosophic rough approximation operators are systematically discussed. In the axiomatic approach, axiomatic characterization of $$({\mathcal {I}},{\mathcal {N}})$$ -single valued neutrosophic approximation operators is studied. Specifically, different axiom sets characterizing the intrinsic properties of $$(\mathcal {I},{\mathcal {N}})$$ -single valued neutrosophic rough approximation operators associated with diverse single valued neutrosophic relations are investigated in detail.

International audience; Necessity (resp. possibility) measures are very simple min-decomposable (resp. max-decomposable) representations of epistemic uncertainty due to incomplete knowledge. They can be used in both quantitative and qualitative settings. In the present work, we revisit Choquet and Sugeno integrals as criteria for decision under uncertainty and propose new axioms when uncertainty is representable in possibility theory. First, a characterization of Choquet integral with respect to a possibility or a necessity measure is proposed. We respectively add an optimism or a pessimism axiom to the axioms of the Choquet integral with respect to a general capacity. This new axiom enforces the maxitivity or the minitivity of the capacity without requiring the same property for the functional. It essentially assumes that the decision-maker preferences only reflect the plausibility ordering between states of nature. The obtained pessimistic (resp. optimistic) criterion is an average maximin (resp. maximax) criterion of Wald across cuts of a possibility distribution on the state space. The additional axiom can be also used in the axiomatic approach to Sugeno integral and generalized forms thereof to justify possibility and necessity measures. The axiomatization of these criteria for decision under uncertainty in the setting of preference relations among acts is also discussed. We show that the new axiom justifying possibilistic Choquet integrals can be expressed in this setting. In the case of Sugeno integral, we correct a characterization proof for an existing set of axioms on acts, and study an alternative set of axioms based on the idea of non-compensation.

Three-way decision theory is an extension of the commonly used binary-decision model with an added third option. It is originally introduced to explain the three regions of probabilistic rough sets. Every object in a three-way decision model can be assigned to one of the three regions according to its evaluation value under an evaluation function. This paper first introduces three-way decision models based on subset-evaluation which generalize the original models. By the axiomatic approach, we characterize a matroid in terms of evaluation function and then define three-way matroids based on this characterization. Furthermore, three-way matroids are generalized to three-way fuzzy matroids and an equivalent description of three-way fuzzy matroid in terms of fuzzy independent set system is presented. Finally, we give the second description of three-way fuzzy matroid: a three-way fuzzy matroid is exactly the greatest element of an equivalence class. Additionally, relations of notions introduced in this paper are also pointed out.

Under the assumption of rational economics, the consistency of judgments is one of the important issues in multiple criteria decision making (MCDM) methods. We propose three axiomatic properties of the consistent judgments in relative measurements being considered from the perspective of strict logic and rationality: (A1) the reciprocal property of pairwise comparisons, (A2) the invariance of consistency with respect to permutations of alternatives, and (A3) the robustness of the ranking of alternatives. These properties are further applied to analyze the consistency of the judgments expressed by positive real numbers and triangular fuzzy numbers, respectively. The inconsistency of comparison matrices encountered in the Analytic Hierarchy Process (AHP) can be considered to be the case of weakening the axiomatic property (A3). Fuzzifying the judgments is due to the weakening of axiomatic property (A1). A method of weakening the axiomatic properties for triangular fuzzy reciprocal matrices is proposed and a new concept of approximation-consistency is put forward to distinguish from the typical consistency. By considering the permutations of alternatives, an illustrative example is presented to show the use of the proposed procedures for solving the decision making problems with triangular fuzzy reciprocal preference relations.

We propose an axiomatic approach to characterize centrality measures for which the centrality of an agent is recursively related to the centralities of the agents she is connected to. This includes the Katz–Bonacich and the eigenvector centrality. The core of our argument hinges on the power of the consistency axiom, which relates the properties of the measure for a given network to its properties for a reduced problem. In our case, the reduced problem only keeps track of local and parsimonious information. Our axiomatic characterization highlights the conceptual similarities among those measures.

The axiomatic approach is more appropriate than constructive approach for studying the algebraic structure of rough sets. In this paper, the more simple axiomatic characterizations of (υ σ)-fuzzy rough approximation operators are explored where υ is a residuated implicator and σis its dual implicator. Firstly, we review the existing independent axiomatic sets to characterize various types of υ-lower and σ-upper fuzzy rough approximation operators. Secondly, we present one-axiom characterizations of (υ σ)-fuzzy rough approximation operators constructed by a serial fuzzy relation on two universes. Furthermore, we show that (υ σ)-fuzzy rough approximation operators, corresponding to reexive, symmetric and T-transitive fuzzy relations, can be presented by only two axioms respectively. We conclude the paper by introducing some potential applications and future works.

The notion of three-way decisions was originally introduced based on the need to explain the three regions of probabilistic rough sets. In a three-way decision model, every object can be evaluated by a function and according to the evaluation value, the object can be arranged in one of the three regions (i.e., positive, negative, and boundary regions). In this study, we generalize Yao's three-way decision models to a case where every subset in the universe can be evaluated by the evaluation function, and we then propose generalized three-way models. The properties and examples of these new models are presented, as well as extensions of these models. We also give some remarks regarding Hu's three-way decision spaces. Three-way matroids are introduced based on Hu's axiomatic approach and our generalized three-way models. Furthermore, three-way matroids are generalized to three fuzzy matroids as an application of our new model. Finally, we suggest future research related to our new models and three-way fuzzy matroids. This paper generalizes three-way decision models from element-evaluation to subset-evaluation.The importance of axiomatic approach in three-way decision spaces is pointed out.The notions of three-way matroids and three-way fuzzy matroids are introduced.

Betweenness as a relation between three individual points has been widely studied in geometry and axiomatized by several authors in different contexts. The article proposes a more general notion of betweenness as a relation between three sets of points. The main technical result is a sound and complete logical system describing universal properties of this relation between sets of vertices of a graph.

Background Axiomatic approach using set function Theorems of probability The addition theorem The multiplication theorem Bayes’ theorem

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.

Smarandache initiated neutrosophic sets (NSs) as a tool for handling undetermined information. Wang et al. proposed single valued neutrosophic sets (SVNSs) that is an especial NSs and can be used expediently to deal with real-world problems. In this paper, we propose single valued neutrosophic rough sets by combining single valued neutrosophic sets and rough sets. We study the hybrid model by constructive and axiomatic approaches. Firstly, by using the constructive approach, we propose the lower/upper single valued neutrosophic approximation operators and illustrate the connections between special single valued neutrosophic relations (SVNRs) and the lower/upper single valued neutrosophic approximation operators. Then, by using the axiomatic approach, we discuss the operator-oriented axiomatic characterizations of single valued neutrosophic rough sets. We obtain that different axiom sets of the lower/upper single valued neutrosophic set-theoretic operators guarantee the existence of different classes of SVNRs which produce the same operators. Finally, we introduce single valued neutrosophic rough sets on two-universes and an algorithm of decision making based on single valued neutrosophic rough sets on two-universes, and use an illustrative example to demonstrate the application of the proposed model.

Local sentences and the formal languages they define were introduced by Ressayre in [Res88]. We prove that locally finite ω-languages and effective analytic sets have the same topological complexity: the Borel and Wadge hierarchies of the class of locally finite ω-languages are equal to the Borel and Wadge hierarchies of the class of effective analytic sets. In particular, for each non-null recursive ordinal α < ω_1^CK there exist some Σ¨0_α-complete and some Π ^0_α-complete locally finite ω-languages, and the supremum of the set of Borel ranks of locally finite ω-languages is the ordinal γ^1_2 , which is strictly greater than the first non-recursive ordinal ω_1^ CK. This gives an answer to the question of the topological complexity of locally finite ω-languages, which was asked by Simonnet [Sim92] and also by Duparc, Finkel, and Ressayre in [DFR01]. Moreover we show that the topological complexity of a locally finite ω-language defined by a local sentence ϕ may depend on the models of the Zermelo-Fraenkel axiomatic system ZFC. Using similar constructions as in the proof of the above results we also show that the equivalence, the inclusion, and the universality problems for locally finite ω-languages are Π^1_2-complete, hence highly undecidable.

We proposed a pair of rough approximation operators on a complete completely distributive lattice (CCD lattice for short) in 2015. In this paper, we further discuss its properties and study the axiomatic approaches to the rough approximation operators. Through these axioms, fuzzy rough approximation operators can be seen as special cases of rough approximation operators on a CCD lattice. We also discuss the axiomatic approaches to generalized rough sets on CCD lattices.

Allen's Interval Algebra is among the leading formalisms in the area of qualitative temporal reasoning. However, its applications are restricted to linear flows of time. While there is some recent work studying relations between intervals on branching structures, there is no rigorous study of the first-order theory of branching time. In this paper, we approach this problem under a general definition of time structures, namely, tree-like lattices. Allen's work proved that meets is expressively complete in the linear case. We also prove that, surprisingly, it remains complete for all unbounded tree-like lattices. This does not generalize to the case of all tree-like lattices, for which we prove that the smallest complete set of relations has cardinality three. We provide in this paper a sound and complete axiomatic system for both the unbounded and general case, in Allen's style, and we classify minimally complete and maximally incomplete sets of relations.

Gleason's theorem is a statement that, given some reasonable assumptions, the Born rule used to calculate probabilities in quantum mechanics is essentially unique [A. M. Gleason, Indiana Univ. Math. J. 6, 885 (1957)]. We show that Gleason's theorem contains within it also the structure of sequential measurements, and along with this the state update rule. We give a small set of axioms, which are physically motivated and analogous to those in Busch's proof of Gleason's theorem [P. Busch, Phys. Rev. Lett. 91, 120403 (2003)], from which the familiar Kraus operator form follows. An axiomatic approach has practical relevance as well as fundamental interest, in making clear those assumptions which underlie the security of quantum communication protocols. Interestingly, the two-time formalism is seen to arise naturally in this approach.

It is not possible to characterize an affine space only via its sets of points and lines without having any further information. In the literature, it is, moreover, postulated that certain lines are parallel in terms of an equivalence relation. In this paper, we state an axiomatic system for affine spaces in terms of an incidence geometry that does not only involve its points and lines but also the planes. The equivalence to an axiomatic system stated by Tamaschke is proved, where Tamaschke’s axiom concerning similar triangles is weakened by an axiom by E.M. Schroder.

We investigate the connection between a general form of Concurrent Separation Logic (CSL), a logic for modular reasoning about concurrent programs, and Concurrent Kleene Algebra (CKA), which provides an axiomatic approach to models of concurrency. We show how the proof theory of a general form of CSL can be embedded in a variation on the notion of CKA. Our embedding, however, induces models of a particular form based on predicate transformers. We also investigate the relation between concrete models of CSL based on interleaving of traces and CKA. We find, curiously, that the validity of CSL's Concurrency proof rule in these models does not follow from or otherwise utilize CKA structure, but that a CKA structure exists nonetheless which can give a different model of the CSL proof rules. Our results can be read as providing a completeness theorem showing a sense in which nothing is missing as far as proof power goes in (a variant on) the notion of CKA, while at the same time showing that CKAs impose constraints that rule out some natural CSL models.

The current chapter lays the foundation for the original axiomatic approach. First, we introduce the last (fourth) axiom on the invariance of preference relation.

In traditional propositional logic(PL), the atomic part of formulas are proposition symbols. In first-order logic(FL) the atomic part of formulas are terms, predicates are relations among terms, and quantifiers ∀,∃ are introduced to express the binding of variables ranging over a domain of discourse. We propose a new kind of logics the quantified propositional logic(QL), QL's atomic formulas are in the forms of c, X(p), F(X), where c, p are a propositional symbol, X is a first-order predicate variable and F is a second-order predicate. The quantifiers ∀,∃ are applied on first-order predicate variables. An axiomatic system is given so that the system is sound and complete with the quantified propositional logic. The translations about the quantified propositional logic are given.

In this paper, inspired by the previous work of Franco Montagna on infinitary axiomatizations for standard BL-algebras, we focus on a uniform approach to the following problem: given a left-continuous t-norm ∗ , find an axiomatic system (possibly with infinitary rules) which is strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.] This system will be an expansion of Monoidal t-norm-based logic. First, we introduce an infinitary axiomatic system L∗∞, expanding the language with Δ and countably many truth constants, and with only one infinitary inference rule, that is inspired in Takeuti–Titani density rule. Then we show that L∗∞ is indeed strongly complete with respect to the standard algebra [InlineEquation not available: see fulltext.]. Moreover, the approach is generalized to axiomatize expansions of these logics with additional operators whose intended semantics over [0, 1] satisfy some regularity conditions. © 2016, Springer-Verlag Berlin Heidelberg., The authors are thankful to an anonymous reviewer for his/her comments that have helped to improve the final layout of this paper. Vidal has been supported by the joint project of Austrian Science Fund (FWF) I1897-N25 and Czech Science Foundation (GACR) 15-34650L and by the institutional support RVO:67985807. Esteva and Godo have been funded by the FEDER/MINECO Spanish Project TIN2015-71799-C2-1-P and by the Grant 2014SGR-118 from the Catalan Government. Bou thanks the Grant 2014SGR-788 from the Catalan Government.

We consider Hilbert’s contributions to the foundations of mathematics, including his conception of the axiomatic method, his notion of mathematical proof, and his approach to infinity through finitary tools. We describe his “Programme”, and we comment its collapse due to Godel’s Incompleteness Theorems. Despite that defeat, Hilbert’s project remains topical today.

A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for formalizing algorithms that manipulate mathematical expressions. A theory graph is a network of theories connected by meaning-preserving theory morphisms that map the formulas of one theory to the formulas of another theory. Theory graphs are in turn well suited for formalizing mathematical knowledge at the most convenient level of abstraction using the most convenient vocabulary. We are interested in the problem of whether a body of mathematical knowledge can be effectively formalized as a theory graph of biform theories. As a test case, we look at the graph of theories encoding natural number arithmetic. We used two different formalisms to do this, which we describe and compare. The first is realized in \({\textsc {ctt}}_\mathrm{uqe}\), a version of Church’s type theory with quotation and evaluation, and the second is realized in Agda, a dependently typed programming language.

In view of the defects of existing axiomatic definitions of interval-valued intuitionistic fuzzy entropy, an improved axiomatic definition of the interval-valued intuitionistic fuzzy entropy is presented and the corresponding formula is structured in the paper. Firstly, defects in three kinds of the axiomatic system of entropy of IVIFS are analyzed in detail respectively. New axiomatic entropy definition is proposed for the sake of more reasonable depiction of the fuzziness of IVIFS. Based on the new axiomatic entropy definition, a new computational formula is mentioned. Finally, an example is showed to illustrate the effectiveness of this new formula. Overall, the results indicate that the improved axiomatic definition and the new proposed formula are reasonable and effective.

The article investigates information flow properties of symmetric multi-party protocols. It gives a sound and complete axiomatic system for properties of the functional dependence predicate that are common to all protocols with the same group of symmetries.

The aim of this paper is modern establishment of geometric theory of continuity, which is based on two, fourth group axioms - Archimedes ' and Cantor's axiom. Various consequences of Archimedes' and Cantor's axioms are proven such as Cantor's and Dedekind's theorem. The paper gives a special view on Hilbert 's axiomatic establishment of geometry which uses axiom of linear completeness instead of Cantor's axiom. Finally, the paper illustrates the use of the axioms of continuity and their consequences in proving some theorems.

We introduce a novel approach towards Harnackʼs inequality in the context of spaces of homogeneous type. This approach, based on the so-called critical density property and doubling properties for weights, avoids the explicit use of covering lemmas and BMO.

The Smets-Magrez axiomatic is usually used to define the class of fuzzy continuous implications which are both S and R-implications (Lukasiewicz implications). Another approach is the construction of such class starting from a basic implication and applying automorphisms. Literature has shown that there is a harmony between those approaches, however in this paper we show that the extension of the Lukasiewicz implication defined on [0,1] for interval values cannot be applied in a direct way. We show that the harmony between the Smets-Magrez axiomatic approach and the one that comes from the generation by automorphisms is not preserved when such extension is done. One of the main consequences lies on the fact that the automorphism approach induces the loss of R-implications from the resulting class of implicators. More precisely, we show that the interval version of such approaches produce two disjunct classes of implicators, meaning that, unlike the usual case, the choice of the respective approach is an important step.

Axiomatic approaches to study approximation operators are one of the primary directions for the investigation of rough set theory. In this paper, we provide some axiomatic systems of lower and upper approximation operators in rough set theory. We also apply the axiomatic systems of generalized rough sets for definitions of generalized lower and upper approximations with respect to an ideal of a ring and discuss some of their significant properties.

We investigate a first-order conditional probability logic with equality, which is, up to our knowledge, the first treatise of such logic. The logic, denoted LFPOIC=, allows making statements such as: CP≥s(Φ, θ), and CP≤s(Φ, θ), with the intended meaning that the conditional probability of Φ given θ is at least (at most) s. The corresponding syntax, semantic, and axiomatic system are introduced, and Extended completeness theorem is proven. [Projekat Ministarstva nauke Republike Srbije, br. III44006 i br. ON174026]

We consider multi-modal logic (with countably infinite number of basic modalities) extended with additional modalities and corresponding to the union and intersection of all basic modalities. We present complete and sound axiomatic systems and polynomial-space terminating tableau-based decision procedures for the basic logic in this language, and its deterministic counterpart . We also show that admits filtration, which can be used independently of our tableaux to establish its decidability.

International audience; This paper investigates an anti-realist theory of meaning suitable for both logical and proper axioms. Unlike other anti-realist accounts such as Dummett–Prawitz verificationism, the standard framework of classical logic is not called into question. This account also admits semantic features beyond the inferential ones: computational aspects play an essential role in the determination of meaning. To deal with these computational aspects, a relaxation of syntax is necessary. This leads to a general kind of proof theory, where the objects of study are not typed objects like deductions, but rather untyped ones, in which formulas are replaced by geometrical configurations.

The paper explores the proof theory of non-wellfounded mereology with binary fusions and provides a cut-free sequent calculus equivalent to the standard axiomatic system.

We propose a way of reasoning about minimal and maximal values of the weights of transitions in a weighted transition system (WTS). This perspective induces a notion of bisimulation that is coarser than the classic bisimulation: it relates states that exhibit transitions to bisimulation classes with the weights within the same boundaries. We propose a customized modal logic that expresses these numeric boundaries for transition weights by means of particular modalities. We prove that our logic is invariant under the proposed notion of bisimulation. We show that the logic enjoys the finite model property which allows us to prove the decidability of satisfiability and provide an algorithm for satisfiability checking. Last but not least, we identify a complete axiomatization for this logic, thus solving a long-standing open problem in this field. All our results are proven for a class of WTSs without the image-finiteness restriction, a fact that makes this development general and robust.

This book covers the material of a one year course in real analysis. It includes an original axiomatic approach to Lebesgue integration which the authors have found to be effective in the classroom. Each chapter contains numerous examples and an extensive problem set which expands considerably the breadth of the material covered in the text. Hints are included for some of the more difficult problems

In this paper, we propose new methods to represent interdependence among alternative attributes and experts’ opinions by constructing Choquet integral using interval-valued intuitionistic fuzzy numbers. In the sequel, we apply these methods to solve the multiple attribute group decision-making (MAGDM) problems under interval-valued intuitionistic fuzzy environment. First, the concept of interval-valued intuitionistic fuzzy Choquet integral is deï¬ned, and some elementary properties are studied in detail. Next, an axiomatic system of interval-valued intuitionistic fuzzy measure is established by delivering a series of mathematical proofs. Then, with fuzzy entropy and Shapely-values in game theory, we propose the interval-valued intuitionistic fuzzy measure development methods in order to form the importance measure of attributes and correlation measure of the experts, respectively. Based on the results of theoretical analysis, a new method is proposed to handle the interval-valued intuitionistic fuzzy group decision making problems. A numerical example illustrates the procedure of the proposed methods and veriï¬es the validity and effectiveness of our new proposed methods.

In Sect. 1, an extension to semigroup couple metric spaces is given for the fixed point result in Matkowski (Diss Math 127:1–68, 1975). In Sect. 2, we show that the simulation-type contractive maps in quasi-metric spaces introduced by Alsulami et al. (Discrete Dyn Nat Soc 2014, Article ID 269286, 2014) are in fact Meir–Keeler maps. Finally, in Sect. 3, the Brezis–Browder ordering principle (Adv Math 21:355–364, 1976) is used to get a proof, in the reduced axiomatic system (ZF-AC+DC), of a fixed point result [in the complete axiomatic system (ZF)] over Cantor complete ultrametric spaces due to Petalas and Vidalis (Proc Am Math Soc 118:819–821, 1993). The methodological approach we chose consisted in treating each section from a self-contained perspective; so, ultimately, these are independent units of the present exposition.

In this paper, we study to what extent decidability is connected to universality. A natural context for such a study is provided by the axiomatic theory of computability, automata and algorithms. In Section 2, we introduce necessary concepts, constructions, axioms, postulates, conditions and problems. Section 3 contains the main results of the paper. In particular, it is demonstrated (Theorems 1 and 2) that undecidability of algorithmic problems does not depend on existence of universal algorithms and may be caused by weaker conditions. At the same time, results of Theorems 2 and 3 demonstrate that decidability is incompatible with universality. It is also proved (Theorem 5) that sufficiently big classes of total algorithms/automata, such as the class of all primitive recursive functions, cannot have universal algorithms/automata.

A stochastic version of the fusion calculus is presented. The stochastic nature is evident in the labelled transition system providing the operational semantics of stochastic fusion calculus, where labels represent the rates corresponding to exponential distributions. We extend the notion of hyperbisimulation to stochastic fusion calculus, and prove that the stochastic hyperequivalence is a congruence. A complete axiomatic system for the stochastic hyperbisimulation is defined. Some examples inspired by gene regulation illustrate the general patterns of interactions by using stochastic fusion calculus.

As a technique for granular computing, rough sets deal with the vagueness and granularity in information systems. Covering-based rough sets have been proposed to generalize this theory for wider application. Three types of covering-based rough sets have been studied for different situations. To make the theory more complete, this paper proposes a fourth type of covering-based rough sets. Compared with the existing ones, the new type shows its special characteristic in the interdependency between its lower and upper approximations. We carry out a systematical study of this new theory. First, we discuss basic properties such as normality, contraction, and monotone. Then we investigate the conditions for this type of covering-based rough sets to satisfy the properties of Pawlak's rough sets and study the interdependency between the lower and upper approximation operations. In addition, axiomatic systems for the lower and upper approximation operations are established. Lastly, we address the relationships between this type of covering-based rough sets and the three existing ones.

In the present paper, we mainly discuss the (⊥,⊺)-generalized fuzzy rough sets introduced by B.Q. Hu and Z.H. Huang with both constructive approach and axiomatic approach considered. In the former, we started from the investigation of the properties of the ⊥-upper and ⊺-lower approximation operators generated by binary fuzzy relations. In the latter, by defining a pair of fuzzy set-theoretic operators, we show (⊥,⊺)-fuzzy rough approximation operators can be characterized by different sets of axioms.

We revisit the axiomatization of Kolmogorov complexity given by Alexander Shen, currently available only in Russian language. We derive an axiomatization for conditional plain Kolmogorov complexity. Next we show that the axiomatic system given by Shen cannot be weakened (at least in any natural way). In addition we prove that the analogue of Shen's axiomatic system fails to characterize prefix-free Kolmogorov complexity.