Set-theoretic Analysis of Nominative Data

In the paper we investigate the notion of nominative data that can be considered as a general mathematical model of data used in computing systems. The main attention is paid to flat nominative data called nominative sets. The structure of the partially-ordered set of nominative sets is investigated in terms of set theory, lattice theory, and algebraic systems theory. To achieve this aim the correct transferring of basic set-theoretic operations to nominative sets is proposed. We investigate a lower semilattice of nominative sets in terms of lower and upper cones, closed and maximal closed intervals of nominative sets. The obtained results can be used in formal software development

Towards a Computational Model of Actor-Based Language Comprehension

Neurophysiological data from a range of typologically diverse languages provide evidence for a cross-linguistically valid, actor-based strategy of understanding sentence-level meaning. This strategy seeks to identify the participant primarily responsible for the state of affairs (the actor) as quickly and unambiguously as possible, thus resulting in competition for the actor role when there are multiple candidates. Due to its applicability across languages with vastly different characteristics, we have proposed that the actor strategy may derive from more basic cognitive or neurobiological organizational principles, though it is also shaped by distributional properties of the linguistic input (e.g. the morphosyntactic coding strategies for actors in a given language). Here, we describe an initial computational model of the actor strategy and how it interacts with language-specific properties. Specifically, we contrast two distance metrics derived from the output of the computational model (one weighted and one unweighted) as potential measures of the degree of competition for actorhood by testing how well they predict modulations of electrophysiological activity engendered by language processing. To this end, we present an EEG study on word order processing in German and use linear mixed-effects models to assess the effect of the various distance metrics. Our results show that a weighted metric, which takes into account the weighting of an actor-identifying feature in the language under consideration outperforms an unweighted distance measure. We conclude that actor competition effects cannot be reduced to feature overlap between multiple sentence participants and thereby to the notion of similarity-based interference, which is prominent in current memory-based models of language processing. Finally, we argue that, in addition to illuminating the underlying neurocognitive mechanisms of actor competition, the present model can form the basis for a more comprehensive, neurobiologically plausible computational model of constructing sentence-level meaning

Quantity and Quality: Not a Zero-Sum Game

Quantification of existing theories is a great challenge but also a great chance for the study of language in the brain. While quantification is necessary for the development of precise theories, it demands new methods and new perspectives. In light of this, four complementary methods were introduced to provide a quantitative and computational account of the extended Argument Dependency Model from Bornkessel-Schlesewsky and Schlesewsky. First, a computational model of human language comprehension was introduced on the basis of dependency parsing. This model provided an initial comparison of two potential mechanisms for human language processing, the traditional "subject" strategy, based on grammatical relations, and the "actor" strategy based on prominence and adopted from the eADM. Initial results showed an advantage for the traditional subject" model in a restricted context; however, the "actor" model demonstrated behavior in a test run that was more similar to human behavior than that of the "subject" model. Next, a computational-quantitative implementation of the "actor" strategy as weighted feature comparison between memory units was used to compare it to other memory-based models from the literature on the basis of EEG data. The "actor" strategy clearly provided the best model, showing a better global fit as well as better match in all details. Building upon the success modeling EEG data, the feasibility of estimating free parameters from empirical data was demonstrated. Both the procedure for doing so and the necessary software were introduced and applied at the level of individual participants. Using empirically estimated parameters, the models from the previous EEG experiment were calculated again and yielded similar results, thus reinforcing the previous work. In a final experiment, the feasibility of analyzing EEG data from a naturalistic auditory stimulus was demonstrated, which conventional wisdom says is not possible. The analysis suggested a new perspective on the nature of event-related potentials (ERPs), which does not contradict existing theory yet nonetheless goes against previous intuition. Using this new perspective as a basis, a preliminary attempt at a parsimonious neurocomputational theory of cognitive ERP components was developed

Kleene Algebra of Partial Predicates

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 [17], forms a Kleene algebra. A Kleene algebra is a De Morgan algebra [3] (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 [8]. The term “Kleene algebra” was introduced by A. Monteiro and D. Brignole in [3]. A similar notion of a “normal i-lattice” had been previously studied by J.A. Kalman [16]. More details about the origin of this notion and its relation to other notions can be found in [24, 4, 1, 2]. It should be noted that there is a different widely known class of algebras, also called Kleene algebras [22, 6], 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 [17, 5, 18, 32, 29, 30]. A partial predicate also corresponds to the notion of a partial set [26] 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) [25]. In [27] 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 [31, 28, 33, 15], for formalizing extensions of the Floyd-Hoare logic [7, 9] which allow reasoning about properties of programs in the case of partial pre- and postconditions [23, 20, 19, 21], for formalizing dynamical models with partial behaviors in the context of the mathematical systems theory [11, 13, 14, 12, 10].Korniłowicz Artur - Institute of Informatics, University of Białystok, PolandIvanov Ievgen - Taras Shevchenko National University, Kyiv, UkraineNikitchenko Mykola - Taras Shevchenko National University, Kyiv, UkraineRaymond Balbes and Philip Dwinger. Distributive Lattices. University of Missouri Press, 1975.T.S. Blyth and J. Varlet. Ockham Algebras. Oxford science publications. Oxford University Press, 1994.Diana Brignole and Antonio Monteiro. Caractérisation des algèbres de Nelson par des egalités. Instituto de Matemática, Universidad Nacional del Sur, Argentina, 1964.Roberto Cignoli. Injective de Morgan and Kleene algebras. Proceedings of the American Mathematical Society, 47(2):269–278, 1975.J.P. Cleave. A Study of Logics. Oxford logic guides. Clarendon Press, 1991. ISBN 9780198532118.J.H. Conway. Regular algebra and finite machines. Chapman and Hall mathematics series. Chapman and Hall, 1971.R.W. Floyd. Assigning meanings to programs. Mathematical aspects of computer science, 19(19–32), 1967.Adam Grabowski. Robbins algebras vs. Boolean algebras. Formalized Mathematics, 9(4): 681–690, 2001.C.A.R. Hoare. An axiomatic basis for computer programming. Commun. ACM, 12(10): 576–580, 1969.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.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.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.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.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.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.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.S.C. Kleene. Introduction to Metamathematics. North-Holland Publishing Co., Amsterdam, and P. Noordhoff, Groningen, 1952.S. Körner. Experience and Theory: An Essay in the Philosophy of Science. International library of philosophy and scientific method. Routledge & Kegan Paul, 1966.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.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.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.Dexter Kozen. On Kleene algebras and closed semirings, pages 26–47. Springer Berlin Heidelberg, 1990. doi:10.1007/BFb0029594.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.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.Maurizio Negri. Three valued semantics and DMF-algebras. Boll. Un. Mat. Ital. B (7), 10(3):733–760, 1996.Maurizio Negri. DMF-algebras: representation and topological characterization. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1(2):369–390, 1998.Maurizio Negri. Partial probability and Kleene logic, 2013.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.M.S. Nikitchenko and S.S. Shkilniak. Applied logic. Publishing house of Taras Shevchenko National University of Kyiv, Ukraine (in Ukrainian), 2013.Mykola Nikitchenko and Stepan Shkilniak. Algebras and logics of partial quasiary predicates. Algebra and Discrete Mathematics, 23(2):263–278, 2017.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.Helena Rasiowa. An Algebraic Approach to Non-Classical Logics. North Holland, 1974.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_6.261112

Semantic networks

AbstractA semantic network is a graph of the structure of meaning. This article introduces semantic network systems and their importance in Artificial Intelligence, followed by I. the early background; II. a summary of the basic ideas and issues including link types, frame systems, case relations, link valence, abstraction, inheritance hierarchies and logic extensions; and III. a survey of ‘world-structuring’ systems including ontologies, causal link models, continuous models, relevance, formal dictionaries, semantic primitives and intersecting inference hierarchies. Speed and practical implementation are briefly discussed. The conclusion argues for a synthesis of relational graph theory, graph-grammar theory and order theory based on semantic primitives and multiple intersecting inference hierarchies

Investigation dans des systèmes abstraits avec entrées et sorties comme fonctions partielles de temps

Cette thèse est consacrée à l'investigation de propriétés de systèmes où les entrées et sorties sont des fonctions partielles sur le domaine temporel. Dans nos travaux, des systèmes de ce genre sont mappés vers des abstractions appelées "blocs". La notion de bloc peut être considérée comme une extension spécifique des notions de systèmes avec entrées et sorties qui ont été étudiés, en plusieurs variantes, en théorie des systèmes. Les aspects essentiels des blocs sont leurs non-déterminisme; partialité des entrées - sorties; et le domaine temps-réel. Les résultats originaux suivants ont été établis dans cette thèse: (1) Les notions de non-anticipation faible et forte considérées dans les travaux de la théorie des systèmes de T. Windeknecht, M. Mesarovic, Y. Takahara pour différentes classes de systèmes ont été comparées et étendues aux blocs. (2) Un théorème de représentation de blocs fortement non-anticipatifs a été prouvé. Il a été montré que de tels blocs peuvent être représentés par une classe de systèmes abstraits dynamiques appelés Systèmes Markoviens Non-déterministes Complets (NCMS). Ces derniers s'appuient sur la notion de système de solution introduit dans la Théorie des Processus de O. Hájek. (3) Des critères généraux pour l'existence de couples d'entrées - sorties totaux de blocs fortement non-anticipatifs et l'existence de sorties totales pour des entrées totales d'un bloc fortement non-anticipatif. Les résultats obtenus sont utiles pour la formalisation et l'analyse de langages de spécification basés sur des diagrammes de blocs, ainsi que pour des langages de développement pour des systèmes cyber-physiques et des systèmes de traitement de données temps-réel.The thesis is devoted to investigation of properties of systems with inputs and outputs as partial functions on the real time domain. In our work systems of this kind are mapped to abstractions called blocks. The notion of a block can be considered as a specific extension of the notions of a system with inputs and outputs which were studied in various variants of mathematical systems theory. The main aspects of blocks are nondeterminism, partiality of inputs/outputs, real time domain. The following novel results concerning blocks were obtained in the thesis: (1) Weak and strong notions of nonanticipation considered in the works on mathematical systems theory by T. Windeknecht, M. Mesarovic, Y. Takahara for different classes of systems were extended to blocks and compared. (2) A representation theorem for strongly nonanticipative blocks was proved. It was shown that such blocks can be represented using an introduced class of abstract dynamical systems called Nondeterministic Complete Markovian Systems (NCMS) which is based on the notion of a solution system introduced in the Theory of Processes by O. Hájek. (3) General criteria for the existence of total input-output pairs of a strongly nonanticipative block and the existence of a total output for a given total input of a strongly nonanticipative block. The obtained results are useful in formalization and analysis of block diagram-based specification and development languages for cyber-physical systems and real-time information processing systems