Fully Evaluated Left-Sequential Logics (2403.14576v1)
Abstract: We consider a family of two-valued "fully evaluated left-sequential logics" (FELs), of which Free FEL (defined by Staudt in 2012) is most distinguishing (weakest) and immune to atomic side effects. Next is Memorising FEL, in which evaluations of subexpressions are memorised. The following stronger logic is Conditional FEL (inspired by Guzm\'an and Squier's Conditional logic, 1990). The strongest FEL is static FEL, a sequential version of propositional logic. We use evaluation trees as a simple, intuitive semantics and provide complete axiomatisations for closed terms (left-sequential propositional expressions). For each FEL except Static FEL, we also define its three-valued version, with a constant U for "undefinedness" and again provide complete, independent aziomatisations, each one containing two additional axioms for U on top of the axiomatisations of the two-valued case. In this setting, the strongest FEL is equivalent to Bochvar's strict logic.
- https://doi.org/10.48550/arXiv.1504.08321 [cs.LO].
- https://doi.org/10.48550/arXiv.1010.3674 [cs.LO,math.LO].
- https://doi.org/10.48550/arXiv.1810.02142 [cs.LO,math.LO].
- Non-commutative propositional logic with short-circuit evaluation. Journal of Applied Non-Classical Logics, 31(3-4):234-278 (online available). https://doi.org/10.1080/11663081.2021.2010954
- A Course in Universal Algebra - The Millennium Edition. Available at http://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html.
- Church, A. (1956). Introduction to Mathematical Logic. Princeton University Press.
- Cornets de Groot, S.H. (2012). Logical systems with left-sequential versions of NAND and XOR. MSc. thesis Logic, University of Amsterdam (June 2020). https://eprints.illc.uva.nl/id/eprint/1743/1/MoL-2020-02.text.pdf.
- Hoare, C.A.R. (1985). A couple of novelties in the propositional calculus. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 31(2):173-178. https://doi.org/10.1002/malq.19850310905
- Kleene, S.C. (1938). On a notation for ordinal numbers. Journal of Symbolic Logic, 3:150-155.
- McCarthy, J. (1963). A basis for a mathematical theory of computation. In P. Braffort and D. Hirschberg (eds.), Computer Programming and Formal Systems, pages 33–70, North-Holland, Amsterdam.
- An independent axiomatisation for free short-circuit logic. Journal of Applied Non-Classical Logics, 28(1), 35-71 (online available). https://doi.org/10.1080/11663081.2018.1448637
- Staudt, D.J.C. (2012). Completeness for two left-sequential logics. MSc. thesis Logic, University of Amsterdam (May 2012). https://doi.org/10.48550/arXiv.1206.1936 [cs.LO].