School of Mathematics (ETDs)

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Start date: 2024Subject: search.filters.subject.SDG-17: Partnerships for the goals

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    Tableaux and Decision Procedures for Many-Valued Modal Logics
    (University of the Witwatersrand, Johannesburg, 2024) Axelrod, Guy Ross
    The aim of this dissertation is to present results expanding on the work done by Melvin Fitting in [22] and [24]. In [22], Fitting introduces a framework of many-valued modal logics, where modal formulas are interpreted via generalized Kripke models in which both the propositional valuation and the accessibility relation take on values from some Heyting algebra of truth values. For a fixed arbitrary finite Heyting algebra, H, [24] presents a signed semantic tableau system that is sound and complete with respect to all H-frames. We go on to consider the many-valued generalizations of frame properties such as reflexivity and transitivity (as presented in [39]) and give parameterized tableau systems which are sound and complete with respect to classes of H-frames satisfying such properties. Further, a prefixed tableau system is introduced, which allows us to define an intuitive decision procedure deciding the logics of the above- mentioned H-frame classes, as well as logics of H-frames satisfying generalized symmetry properties, which cannot be captured by Fitting’s unprefixed systems. Further, they allow us to derive finite frame properties. Such a decision procedure has been implemented, and is available on GitHub.
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    A symmetry perspective of third-order polynomial evolution equations
    (University of the Witwatersrand, Johannesburg, 2024) Gwaxa, Bongumusa; Jamal, Sameerah
    In this thesis, we analyse the full class of ten Fujimoto-Watanabe equations. In particular, these are highly nonlinear third-order and two fifth-order equations. With the aid of computer algebra software such as Mathematica, we calculate symmetries for these equations and we construct their commutator tables. The one dimensional system of optimal subalgebras is obtained via adjoint operators. Finally, we reduce these higher-order partial differential equations into ordinary differential equations, derive their solutions via a power series solution method and show how convergence may be tested. Lastly, we determine some conservation laws