We show that the set of all partial predicates over a set D together with the disjunction, conjunction, and negation operations, defined in accordance with the truth tables of S.C. Kleene’s strong logic of indeterminacy [], forms a Kleene algebra. A Kleene algebra is a De Morgan algebra [] (also called quasi-Boolean algebra) which satisfies the condition x ∧¬:x ⩽ y ∨¬ :y (sometimes called the normality axiom). We use the formalization of De Morgan algebras from [].
The term “Kleene algebra” was introduced by A. Monteiro and D. Brignole in []. A similar notion of a “normal i-lattice” had been previously studied by J.A. Kalman []. More details about the origin of this notion and its relation to other notions can be found in [, , , ]. It should be noted that there is a different widely known class of algebras, also called Kleene algebras [, ], which generalize the algebra of regular expressions, however, the term “Kleene algebra” used in this paper does not refer to them.
Algebras of partial predicates naturally arise in computability theory in the study on partial recursive predicates. They were studied in connection with non-classical logics [, , , , , ]. A partial predicate also corresponds to the notion of a partial set [] on a given domain, which represents a (partial) property which for any given element of this domain may hold, not hold, or neither hold nor not hold. The field of all partial sets on a given domain is an algebra with generalized operations of union, intersection, complement, and three constants (0, 1, n which is the fixed point of complement) which can be generalized to an equational class of algebras called DMF-algebras (De Morgan algebras with a single fixed point of involution) []. In [] partial sets and DMF-algebras were considered as a basis for unification of set-theoretic and linguistic approaches to probability.
Partial predicates over classes of mathematical models of data were used for formalizing semantics of computer programs in the composition-nominative approach to program formalization [, , , ], for formalizing extensions of the Floyd-Hoare logic [, ] which allow reasoning about properties of programs in the case of partial pre- and postconditions [, , , ], for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory [, , , , ].
[1] Raymond Balbes and Philip Dwinger. Distributive Lattices. University of Missouri Press, 1975.
[2] T.S. Blyth and J. Varlet. Ockham Algebras. Oxford science publications. Oxford University Press, 1994.
[3] Diana Brignole and Antonio Monteiro. Caractérisation des algèbres de Nelson par des egalités. Instituto de Matemática, Universidad Nacional del Sur, Argentina, 1964.
[4] Roberto Cignoli. Injective de Morgan and Kleene algebras. Proceedings of the American Mathematical Society, 47(2):269–278, 1975.
[5] J.P. Cleave. A Study of Logics. Oxford logic guides. Clarendon Press, 1991. ISBN 9780198532118.
[6] J.H. Conway. Regular algebra and finite machines. Chapman and Hall mathematics series. Chapman and Hall, 1971.
[7] R.W. Floyd. Assigning meanings to programs. Mathematical aspects of computer science, 19(19–32), 1967.
[8] Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4): 681–690, 2001.
[9] C.A.R. Hoare. An axiomatic basis for computer programming. Commun. ACM, 12(10): 576–580, 1969.
[10] Ievgen Ivanov. On the underapproximation of reach sets of abstract continuous-time systems. In Erika Ábrahám and Sergiy Bogomolov, editors, Proceedings 3rd International Workshop on Symbolic and Numerical Methods for Reachability Analysis, SNR@ETAPS 2017, Uppsala, Sweden, 22nd April 2017, volume 247 of EPTCS, pages 46–51, 2017. doi:10.4204/EPTCS.247.4.
[11] Ievgen Ivanov. On representations of abstract systems with partial inputs and outputs. In T. V. Gopal, Manindra Agrawal, Angsheng Li, and S. Barry Cooper, editors, Theory and Applications of Models of Computation – 11th Annual Conference, TAMC 2014, Chennai, India, April 11–13, 2014. Proceedings, volume 8402 of Lecture Notes in Computer Science, pages 104–123. Springer, 2014. ISBN 978-3-319-06088-0. doi:10.1007/978-3-319-06089-7_8.
[12] Ievgen Ivanov. On local characterization of global timed bisimulation for abstract continuous-time systems. In Ichiro Hasuo, editor, Coalgebraic Methods in Computer Science – 13th IFIP WG 1.3 International Workshop, CMCS 2016, Colocated with ETAPS 2016, Eindhoven, The Netherlands, April 2–3, 2016, Revised Selected Papers, volume 9608 of Lecture Notes in Computer Science, pages 216–234. Springer, 2016. ISBN 978-3-319-40369-4. doi:10.1007/978-3-319-40370-0_13.
[13] Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. On a Decidable Formal Theory for Abstract Continuous-Time Dynamical Systems, pages 78–99. Springer International Publishing, 2014. ISBN 978-3-319-13206-8. doi:10.1007/978-3-319-13206-8_4.
[14] Ievgen Ivanov, Mykola Nikitchenko, and Uri Abraham. Event-based proof of the mutual exclusion property of Peterson’s algorithm. Formalized Mathematics, 23(4):325–331, 2015. doi:10.1515/forma-2015-0026.
[15] Ievgen Ivanov, Mykola Nikitchenko, and Volodymyr G. Skobelev. Proving properties of programs on hierarchical nominative data. The Computer Science Journal of Moldova, 24(3):371–398, 2016.
[16] J. A. Kalman. Lattices with involution. Transactions of the American Mathematical Society, 87(2):485–485, February 1958. doi:10.1090/s0002-9947-1958-0095135-x.
[17] S.C. Kleene. Introduction to Metamathematics. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen, 1952.
[18] S. Körner. Experience and Theory: An Essay in the Philosophy of Science. International library of philosophy and scientific method. Routledge & Kegan Paul, 1966.
[19] Artur Kornilowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the algebra of nominative data in Mizar. In Maria Ganzha, Leszek A. Maciaszek, and Marcin Paprzycki, editors, Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, FedCSIS 2017, Prague, Czech Republic, September 3–6, 2017., pages 237–244, 2017. ISBN 978-83-946253-7-5. doi:10.15439/2017F301.
[20] Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. An approach to formalization of an extension of Floyd-Hoare logic. In Vadim Ermolayev, Nick Bassiliades, Hans-Georg Fill, Vitaliy Yakovyna, Heinrich C. Mayr, Vyacheslav Kharchenko, Vladimir Peschanenko, Mariya Shyshkina, Mykola Nikitchenko, and Aleksander Spivakovsky, editors, Proceedings of the 13th International Conference on ICT in Education, Research and Industrial Applications. Integration, Harmonization and Knowledge Transfer, Kyiv, Ukraine, May 15–18, 2017, volume 1844 of CEUR Workshop Proceedings, pages 504–523. CEUR-WS.org, 2017.
[21] Artur Korniłowicz, Andrii Kryvolap, Mykola Nikitchenko, and Ievgen Ivanov. Formalization of the Nominative Algorithmic Algebra in Mizar, pages 176–186. Springer International Publishing, 2018. ISBN 978-3-319-67229-8. doi:10.1007/978-3-319-67229-8_16.
[22] Dexter Kozen. On Kleene algebras and closed semirings, pages 26–47. Springer Berlin Heidelberg, 1990. doi:10.1007/BFb0029594.
[23] Andrii Kryvolap, Mykola Nikitchenko, and Wolfgang Schreiner. Extending Floyd-Hoare Logic for Partial Pre- and Postconditions, pages 355–378. Springer International Publishing, 2013. ISBN 978-3-319-03998-5. doi:10.1007/978-3-319-03998-5_18.
[24] Antonio Monteiro and Luiz Monteiro. Axiomes indépendants pour les algèbres de Nelson, de Łukasiewicz trivalentes, de de Morgan et de Kleene. Notas de lógica matemática, (40): 1–11, 1996.
[25] Maurizio Negri. Three valued semantics and DMF-algebras. Boll. Un. Mat. Ital. B (7), 10(3):733–760, 1996.
[26] Maurizio Negri. DMF-algebras: representation and topological characterization. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(2):369–390, 1998.
[27] Maurizio Negri. Partial probability and Kleene logic, 2013.
[28] M.S. Nikitchenko and S.S. Shkilniak. Mathematical logic and theory of algorithms. Publishing house of Taras Shevchenko National University of Kyiv, Ukraine (in Ukrainian), 2008.
[29] M.S. Nikitchenko and S.S. Shkilniak. Applied logic. Publishing house of Taras Shevchenko National University of Kyiv, Ukraine (in Ukrainian), 2013.
[30] Mykola Nikitchenko and Stepan Shkilniak. Algebras and logics of partial quasiary predicates. Algebra and Discrete Mathematics, 23(2):263–278, 2017.
[31] Nikolaj S. Nikitchenko. A composition nominative approach to program semantics. Technical Report IT-TR 1998-020, Department of Information Technology, Technical University of Denmark, 1998.
[32] Helena Rasiowa. An Algebraic Approach to Non-Classical Logics. North Holland, 1974.
[33] Volodymyr G. Skobelev, Mykola Nikitchenko, and Ievgen Ivanov. On algebraic properties of nominative data and functions. In Vadim Ermolayev, Heinrich C. Mayr, Mykola Nikitchenko, Aleksander Spivakovsky, and Grygoriy Zholtkevych, editors, Information and Communication Technologies in Education, Research, and Industrial Applications – 10th International Conference, ICTERI 2014, Kherson, Ukraine, June 9–12, 2014, Revised Selected Papers, volume 469 of Communications in Computer and Information Science, pages 117–138. Springer, 2014. ISBN 978-3-319-13205-1. doi:10.1007/978-3-319-13206-8_6.