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  • 1
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    Oxford University Press
    Publication Date: 2018-03-06
    Description: We study the structural properties of $Q$-degrees and prove that every non-computable c.e. $Q$-degree contains a perfect set. Using this result and Batyrshin’s theorem [ 4 ] we have that there is a non-computable c.e. $Q$-degree containing a single c.e. $1$-degree. We show that if $K$ is a creative set, then there is a $\Sigma^0_2\setminus\Delta^0_2$ set $B$ which is $Q$-incomparable with $K$ and for all c.e. sets $W$, if $W\leq_{Q} B$ then $W\leq_{Q}\varnothing$.
    Print ISSN: 1367-0751
    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 2
    Publication Date: 2018-03-06
    Description: The aim of this article is to describe an algorithm for the automatic generation of proofs in an axiomatic system for Classical Propositional Logic. The idea of the algorithm was taken from a book by Helena Rasiowa and Roman Sikorski [ 31 ], where the authors suggest using the method of diagrams to automatically obtain proofs in Classical Propositional Logic. However, in this article the method of diagrams developed by Rasiowa and Sikorski was replaced by a right-sided erotetic calculus developed by Andrzej Wiśniewski. Erotetic calculi have been used before in designing similar algorithms. The proofs are presented together with the estimations of their lengths, widths and sizes — measures introduced for the purposes of the article. The estimations are then used to derive the conclusion that the axiomatic system simulates polynomially the erotetic calculus.
    Print ISSN: 1367-0751
    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 3
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    Oxford University Press
    Publication Date: 2018-03-06
    Description: We extend classical Propositional Logic (PL) by adding a new primitive binary connective $\varphi|\psi$, intended to represent the ‘superposition’ of sentences $\varphi$ and $\psi$, an operation motivated by the corresponding notion of quantum mechanics, but not intended to capture all aspects of the latter as they appear in physics. To interpret the new connective, we extend the classical Boolean semantics by employing models of the form $\langle M,f\rangle$, where $M$ is an ordinary two-valued assignment for the sentences of PL and $f$ is a choice function for all pairs of classical sentences. In the new semantics $\varphi|\psi$ is strictly interpolated between $\varphi\wedge\psi$ and $\varphi\vee\psi$. By imposing several constraints on the choice functions, we obtain corresponding notions of logical consequence relations and corresponding systems of tautologies, with respect to which $|$ satisfies some natural algebraic properties such as associativity, closedness under logical equivalence and distributivity over its dual connective. Thus various systems of Propositional Superposition Logic (PLS) arise as extensions of PL. Axiomatizations for these systems of tautologies are presented and soundness is shown for all of them. Completeness is proved for the weakest of these systems. For the other systems completeness holds if and only if every consistent set of sentences is extendible to a consistent and complete one, a condition whose truth is closely related to the validity of the deduction theorem.
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    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 4
    Publication Date: 2018-03-06
    Description: In this article, we address a variant of the vehicle routing problem which occurs in real-life applications, called the clustered vehicle routing problem. The clustered vehicle routing problem (CluVRP) looks for an optimal collection of routes, with respect to cost minimization, for a fleet of vehicles, fulfilling all customers’ requirements which have a particular structure being divided into clusters, the capacity constraints of the vehicles and with the additional constraint that all the customers belonging to the same cluster must be visited consecutively before leaving the cluster. We propose a novel approach for solving the problem using a decomposition-based method which splits the CluVRP into two easier subproblems. The results of the computational experiments on three sets of benchmark instances from the literature show that our approach is competitive in comparison with the known solution approaches published to date.
    Print ISSN: 1367-0751
    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 5
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    Oxford University Press
    Publication Date: 2018-03-06
    Description: An ongoing debate about the differences between formal provability in an axiomatic system and informal provability of mathematical claims in mathematics as a whole resulted in the construction of various logics whose main purpose is to capture the inferential behaviour of the notion of informal provability, just as multiple logics of formal provability capture the behaviour of the concept of formal provability. Known logics of informal provability, based on classical logic, are unable to incorporate all intuitive principles of informal provability (most notably, reflection, which says that whatever is provable is true). One solution to this problem is to treat informal provability as an operator (Shapiro, 1985, North Holland; Reinhardt, 1986, J. Philos. Log. , 15 , 427–74; Koellner, 2016, Oxford University Press). Another solution is to weaken some of the intuitively adequate principles (Horsten, 2002, Hansel-Hohenhausen). Recently, in yet another approach to the issue, two three-valued non-deterministic logics of informal provability have been developed (Pawlowski and R. Urbaniak, 2016, Rev. Symbo. Log. ) to overcome this difficulty. Alas, the logics have been characterized semantically and no proof systems for them are available. The purpose of this article is to define tree-like proof systems for those logics and to prove the corresponding soundness and completeness theorems.
    Print ISSN: 1367-0751
    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 6
    Publication Date: 2018-03-06
    Description: This article discusses classification strategies in rule-based classifiers, reveals how often induced rules did not lead to unambiguous classification and emphasizes a major role that classification strategies play in classification of unknown examples. Five selected popular classification strategies proposed by Michalski $et al$, Grzymała-Busse and Zou, An, Stefanowski, Sulzmann and Fürnkranz are reviewed and compared experimentally. Additionally, a new strategy that exploits $\chi^{2}$ statistic to measure the association between the rule coverage and the indicated class is proposed. The experiment was conducted on 30 UCI datasets using MODLEM and modified RIPPER classifiers.
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    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 7
    Publication Date: 2018-03-06
    Description: In a previous article by two of the present authors and S. Bonzio, Łukasiewicz near semirings were introduced and it was proven that basic algebras can be represented (precisely, are term equivalent to) as near semirings. In the same work it has been shown that the variety of Łukasiewicz near semirings is congruence regular. In other words, every congruence is uniquely determined by its 0-coset. Thus, it seems natural to wonder whether it could be possible to provide a set-theoretical characterization of these cosets. This article addresses this question and shows that kernels can be neatly described in terms of two simple conditions. As an application, we obtain a concise characterization of ideals in Łukasiewicz semirings. Finally, we close this article with a rather general Cantor–Bernstein type theorem for the variety of involutive idempotent integral near semirings.
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    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 8
    Publication Date: 2018-03-06
    Description: Since the pioneering work of Birkhoff and von Neumann, quantum logic has been interpreted as the logic of (closed) subspaces of a Hilbert space. There is a progression from the usual Boolean logic of subsets to the ‘quantum logic’ of subspaces of a general vector space—which is then specialized to the closed subspaces of a Hilbert space. But there is a ‘dual’ progression. The set notion of a partition (or quotient set or equivalence relation) is dual (in a category-theoretic sense) to the notion of a subset. Hence the Boolean logic of subsets has a dual logic of partitions. Then the dual progression is from that logic of set partitions to the quantum logic of direct-sum decompositions (i.e. the vector space version of a set partition) of a general vector space—which can then be specialized to the direct-sum decompositions of a Hilbert space. This allows the quantum logic of direct-sum decompositions to express measurement by any self-adjoint operators. The quantum logic of direct-sum decompositions is dual to the usual quantum logic of subspaces in the same sense that the logic of partitions is dual to the usual Boolean logic of subsets.
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    Electronic ISSN: 1368-9894
    Topics: Mathematics
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  • 9
    Publication Date: 2018-03-06
    Description: Hemi-implicative semilattices (lattices), originally defined under the name of weak implicative semilattices (lattices), were introduced by the second author of the present article. A hemi-implicative semilattice is an algebra $(H,{\wedge},{\rightarrow},1)$ of type $(2,2,0)$ such that $(H,{\wedge})$ is a meet semilattice, $1$ is the greatest element with respect to the order, $a{\rightarrow} a = 1$ for every $a\in H$ and for every $a$, $b$, $c\in H$, if $a\leq b{\rightarrow} c$ then $a{\wedge} b \leq c$. A bounded hemi-implicative semilattice is an algebra $(H,{\wedge},{\rightarrow},0,1)$ of type $(2,2,0,0)$ such that $(H,{\wedge},{\rightarrow},1)$ is a hemi-implicative semilattice and $0$ is the first element with respect to the order. A hemi-implicative lattice is an algebra $(H,{\wedge},\vee,{\rightarrow},0,1)$ of type $(2,2,2,0,0)$ such that $(H,{\wedge},\vee,0,1)$ is a bounded distributive lattice and the reduct algebra $(H,{\wedge},{\rightarrow},1)$ is a hemi-implicative semilattice. In this article, we introduce an equivalence for the categories of bounded hemi-implicative semilattices and hemi-implicative lattices, respectively, which is motivated by an old construction due J. Kalman that relates bounded distributive lattices and Kleene algebras.
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    Topics: Mathematics
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  • 10
    Publication Date: 2017-01-06
    Print ISSN: 1367-0751
    Electronic ISSN: 1368-9894
    Topics: Mathematics
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