Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Generically extendible cardinals

In this paper, we study generically extendible cardinal, which is a generic version of extendible cardinal. We prove that the generic extendibility of $\omega_1$ or $\omega_2$ has small consistency strength, but of a cardinal $>\omega_2$ is not. We also consider some results concerning with generic extendible cardinals, such as indestructibility, generic absoluteness of the reals, and Boolean valued second order logic

A doctrinal approach to modal/temporal Heyting logic and non-determinism in processes

The study of algebraic modelling of labelled non-deterministic concurrent processes leads us to consider a category LB , obtained from a complete meet-semilattice B and from B-valued equivalence relations. We prove that, if B has enough properties, then LB presents a two-fold internal logical structure, induced by two doctrines definable on it: one related to its families of subobjects and one to its families of regular subobjects. The first doctrine is Heyting and makes LB a Heyting category, the second one is Boolean. We will see that the difference between these two logical structures, namely the different behaviour of the negation operator, can be interpreted in terms of a distinction between non-deterministic and deterministic behaviours of agents able to perform computations in the context of the same process. Moreover, the sorted first-order logic naturally associated with LB can be extended to a modal/temporal logic, again using the doctrinal setting. Relations are also drawn to other computational model

Control Synthesis for Multi-Agent Systems under Metric Interval Temporal Logic Specifications

This paper presents a framework for automatic synthesis of a control sequence for multi-agent systems governed by continuous linear dynamics under timed constraints. First, the motion of the agents in the workspace is abstracted into individual Transition Systems (TS). Second, each agent is assigned with an individual formula given in Metric Interval Temporal Logic (MITL) and in parallel, the team of agents is assigned with a collaborative team formula. The proposed method is based on a correct-by-construction control synthesis method, and hence guarantees that the resulting closed-loop system will satisfy the specifications. The specifications considers boolean-valued properties under real-time. Extended simulations has been performed in order to demonstrate the efficiency of the proposed controllers.Comment: 8 pages version of the accepted paper to IFAC World Congres

Constraint Programming viewed as Rule-based Programming

We study here a natural situation when constraint programming can be entirely reduced to rule-based programming. To this end we explain first how one can compute on constraint satisfaction problems using rules represented by simple first-order formulas. Then we consider constraint satisfaction problems that are based on predefined, explicitly given constraints. To solve them we first derive rules from these explicitly given constraints and limit the computation process to a repeated application of these rules, combined with labeling.We consider here two types of rules. The first type, that we call equality rules, leads to a new notion of local consistency, called {\em rule consistency} that turns out to be weaker than arc consistency for constraints of arbitrary arity (called hyper-arc consistency in \cite{MS98b}). For Boolean constraints rule consistency coincides with the closure under the well-known propagation rules for Boolean constraints. The second type of rules, that we call membership rules, yields a rule-based characterization of arc consistency. To show feasibility of this rule-based approach to constraint programming we show how both types of rules can be automatically generated, as {\tt CHR} rules of \cite{fruhwirth-constraint-95}. This yields an implementation of this approach to programming by means of constraint logic programming. We illustrate the usefulness of this approach to constraint programming by discussing various examples, including Boolean constraints, two typical examples of many valued logics, constraints dealing with Waltz's language for describing polyhedral scenes, and Allen's qualitative approach to temporal logic.Comment: 39 pages. To appear in Theory and Practice of Logic Programming Journa

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of Ω-logic in Zermelo-Fraenkel set theory with choice (ZFC). The dual isomorphism between algebra and coalgebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of Ω-logical validity can then be countenanced within a coalgebraic logic, and Ω-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of Ω-logical validity correspond to those of second-order logical consequence, Ω-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Применение диаграмм решений не полностью определенных функций k-значной логики при синтезе логических схем

Objectives. The problem of circuit implementation of incompletely specified (partial) k-valued logic functions given by tabular representations is considered. The stage of technologically independent optimization is studied to obtain minimized representations of systems of completely specified Boolean functions from tabular representations of partial functions of k-valued logic. According to these representations of Boolean functions, technological mapping is performed at the second stage of the synthesis of logic circuits.Methods. Using additional definitions of Multi-valued Decision Diagrams (MDD) representing partial functions of k-valued logic, and Binary Decision Diagrams (BDD) representing partial systems of Boolean functions at the stage of technologically independent optimization is proposed. The task of additional definition of MDD is oriented to reducing the number of vertices of the MDD graph that correspond to the cofactors of the Shannon expansion of a multi-valued function.Results. The MDD minimization problem is reduced to solving the problems of coloring undirected graphs of incompatibility of cofactors by minimum number of colors. Encoding of multi-valued values of arguments and values of functions of k-valued logic by binary codes leads to systems of partial Boolean functions, which are also further defined in order to minimize their multi-level BDD representations.Conclusion. The proposed approach makes it possible to define partial multi-valued functions to fully defined Boolean functions in two stages. At the second stage, well-known and effective methods are used to redefine BDD representing systems of partial Boolean functions. As a result of this two-step approach, minimized BDD representations of systems of completely defined functions are obtained. According to completely defined Boolean functions, a technological mapping into a given library of logical elements is performed, i.e. the optimized descriptions of Boolean function systems are covered with descriptions of logical elementsЦели. Рассматривается проблема схемной реализации не полностью определенных функций k-значной логики, заданных табличными представлениями. Изучается этап технологически независимой оптимизации. Целью этого этапа является получение по табличным представлениям не полностью определенных функций k-значной логики минимизированных представлений систем полностью определенных булевых функций, по которым выполняется технологическое отображение (technology mapping) – второй этап синтеза логических схем.Методы. При синтезе логических схем на этапе технологически независимой оптимизации предлагается использовать доопределения многозначных диаграмм решений (Reduced Ordered Multi-valued Decision Diagrams, ROMDD), которые далее называются MDD, и доопределения бинарных диаграмм решений (Binary Decision Diagram, BDD), задающих не полностью определенные системы булевых функций. Доопределение MDD ориентировано на уменьшение числа вершин графа MDD, соответствующих кофакторам разложения Шеннона многозначной функции.Результаты. Задача минимизации MDD сведена к решению задач минимальной раскраски неориентированных графов несовместимости кофакторов. Кодирование многозначных значений аргументов и значений функций k-значной логики двоичными кодами приводит к системам не полностью определенных булевых функций, которые также доопределяются с целью минимизации их многоуровневых BDD-представлений.Заключение. Предложенный подход позволяет в два этапа провести доопределение частичных многозначных функций до полностью определенных булевых функций. На втором этапе используются известные и эффективные методы доопределения BDD, задающих системы не полностью определенных булевых функций. В результате такого двухэтапного подхода получаются минимизированные BDD-представления систем полностью определенных функций. По полностью определенным булевым функциям выполняется технологическое отображение в заданную библиотеку логических элементов, т. е. покрытие оптимизированных описаний систем булевых функций описаниями логических элементов