26,033 research outputs found
Modal Ξ©-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of -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 -logical validity can then be countenanced within a coalgebraic logic, and -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 -logical validity correspond to those of second-order logical consequence, -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
or has small consistency strength, but of a cardinal
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 Ξ©-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism
This essay examines the philosophical significance of Ξ©-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 Ξ©-logical validity can then be countenanced within a coalgebraic logic, and Ξ©-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 Ξ©-logical validity correspond to those of second-order logical consequence, Ξ©-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
ΠΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ Π΄ΠΈΠ°Π³ΡΠ°ΠΌΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ ΡΡΠ½ΠΊΡΠΈΠΉ k-Π·Π½Π°ΡΠ½ΠΎΠΉ Π»ΠΎΠ³ΠΈΠΊΠΈ ΠΏΡΠΈ ΡΠΈΠ½ΡΠ΅Π·Π΅ Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ ΡΡ Π΅ΠΌ
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 elementsΠ¦Π΅Π»ΠΈ. Π Π°ΡΡΠΌΠ°ΡΡΠΈΠ²Π°Π΅ΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΠ° ΡΡ
Π΅ΠΌΠ½ΠΎΠΉ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΠΈ Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ k-Π·Π½Π°ΡΠ½ΠΎΠΉ Π»ΠΎΠ³ΠΈΠΊΠΈ, Π·Π°Π΄Π°Π½Π½ΡΡ
ΡΠ°Π±Π»ΠΈΡΠ½ΡΠΌΠΈ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡΠΌΠΈ. ΠΠ·ΡΡΠ°Π΅ΡΡΡ ΡΡΠ°ΠΏ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈ Π½Π΅Π·Π°Π²ΠΈΡΠΈΠΌΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ. Π¦Π΅Π»ΡΡ ΡΡΠΎΠ³ΠΎ ΡΡΠ°ΠΏΠ° ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΏΠΎΠ»ΡΡΠ΅Π½ΠΈΠ΅ ΠΏΠΎ ΡΠ°Π±Π»ΠΈΡΠ½ΡΠΌ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡΠΌ Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ k-Π·Π½Π°ΡΠ½ΠΎΠΉ Π»ΠΎΠ³ΠΈΠΊΠΈ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΠΉ ΡΠΈΡΡΠ΅ΠΌ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ, ΠΏΠΎ ΠΊΠΎΡΠΎΡΡΠΌ Π²ΡΠΏΠΎΠ»Π½ΡΠ΅ΡΡΡ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΎΡΠΎΠ±ΡΠ°ΠΆΠ΅Π½ΠΈΠ΅ (technology mapping) β Π²ΡΠΎΡΠΎΠΉ ΡΡΠ°ΠΏ ΡΠΈΠ½ΡΠ΅Π·Π° Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΡ
Π΅ΠΌ.ΠΠ΅ΡΠΎΠ΄Ρ. ΠΡΠΈ ΡΠΈΠ½ΡΠ΅Π·Π΅ Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΡ
Π΅ΠΌ Π½Π° ΡΡΠ°ΠΏΠ΅ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈ Π½Π΅Π·Π°Π²ΠΈΡΠΈΠΌΠΎΠΉ ΠΎΠΏΡΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΏΡΠ΅Π΄Π»Π°Π³Π°Π΅ΡΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ Π΄ΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
Π΄ΠΈΠ°Π³ΡΠ°ΠΌΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ (Reduced Ordered Multi-valued Decision Diagrams, ROMDD), ΠΊΠΎΡΠΎΡΡΠ΅ Π΄Π°Π»Π΅Π΅ Π½Π°Π·ΡΠ²Π°ΡΡΡΡ MDD, ΠΈ Π΄ΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ Π±ΠΈΠ½Π°ΡΠ½ΡΡ
Π΄ΠΈΠ°Π³ΡΠ°ΠΌΠΌ ΡΠ΅ΡΠ΅Π½ΠΈΠΉ (Binary Decision Diagram, BDD), Π·Π°Π΄Π°ΡΡΠΈΡ
Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠ΅ ΡΠΈΡΡΠ΅ΠΌΡ Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ. ΠΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠ΅ MDD ΠΎΡΠΈΠ΅Π½ΡΠΈΡΠΎΠ²Π°Π½ΠΎ Π½Π° ΡΠΌΠ΅Π½ΡΡΠ΅Π½ΠΈΠ΅ ΡΠΈΡΠ»Π° Π²Π΅ΡΡΠΈΠ½ Π³ΡΠ°ΡΠ° MDD, ΡΠΎΠΎΡΠ²Π΅ΡΡΡΠ²ΡΡΡΠΈΡ
ΠΊΠΎΡΠ°ΠΊΡΠΎΡΠ°ΠΌ ΡΠ°Π·Π»ΠΎΠΆΠ΅Π½ΠΈΡ Π¨Π΅Π½Π½ΠΎΠ½Π° ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΠΎΠΉ ΡΡΠ½ΠΊΡΠΈΠΈ.Π Π΅Π·ΡΠ»ΡΡΠ°ΡΡ. ΠΠ°Π΄Π°ΡΠ° ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ MDD ΡΠ²Π΅Π΄Π΅Π½Π° ΠΊ ΡΠ΅ΡΠ΅Π½ΠΈΡ Π·Π°Π΄Π°Ρ ΠΌΠΈΠ½ΠΈΠΌΠ°Π»ΡΠ½ΠΎΠΉ ΡΠ°ΡΠΊΡΠ°ΡΠΊΠΈ Π½Π΅ΠΎΡΠΈΠ΅Π½ΡΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
Π³ΡΠ°ΡΠΎΠ² Π½Π΅ΡΠΎΠ²ΠΌΠ΅ΡΡΠΈΠΌΠΎΡΡΠΈ ΠΊΠΎΡΠ°ΠΊΡΠΎΡΠΎΠ². ΠΠΎΠ΄ΠΈΡΠΎΠ²Π°Π½ΠΈΠ΅ ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ Π°ΡΠ³ΡΠΌΠ΅Π½ΡΠΎΠ² ΠΈ Π·Π½Π°ΡΠ΅Π½ΠΈΠΉ ΡΡΠ½ΠΊΡΠΈΠΉ k-Π·Π½Π°ΡΠ½ΠΎΠΉ Π»ΠΎΠ³ΠΈΠΊΠΈ Π΄Π²ΠΎΠΈΡΠ½ΡΠΌΠΈ ΠΊΠΎΠ΄Π°ΠΌΠΈ ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡ ΠΊ ΡΠΈΡΡΠ΅ΠΌΠ°ΠΌ Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ, ΠΊΠΎΡΠΎΡΡΠ΅ ΡΠ°ΠΊΠΆΠ΅ Π΄ΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»ΡΡΡΡΡ Ρ ΡΠ΅Π»ΡΡ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·Π°ΡΠΈΠΈ ΠΈΡ
ΠΌΠ½ΠΎΠ³ΠΎΡΡΠΎΠ²Π½Π΅Π²ΡΡ
BDD-ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΠΉ.ΠΠ°ΠΊΠ»ΡΡΠ΅Π½ΠΈΠ΅. ΠΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΡΠΉ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ Π² Π΄Π²Π° ΡΡΠ°ΠΏΠ° ΠΏΡΠΎΠ²Π΅ΡΡΠΈ Π΄ΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΠ΅ ΡΠ°ΡΡΠΈΡΠ½ΡΡ
ΠΌΠ½ΠΎΠ³ΠΎΠ·Π½Π°ΡΠ½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ Π΄ΠΎ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ. ΠΠ° Π²ΡΠΎΡΠΎΠΌ ΡΡΠ°ΠΏΠ΅ ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΡΡΡΡ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠ΅ ΠΈ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΡΠ΅ ΠΌΠ΅ΡΠΎΠ΄Ρ Π΄ΠΎΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½ΠΈΡ BDD, Π·Π°Π΄Π°ΡΡΠΈΡ
ΡΠΈΡΡΠ΅ΠΌΡ Π½Π΅ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ. Π ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠ΅ ΡΠ°ΠΊΠΎΠ³ΠΎ Π΄Π²ΡΡ
ΡΡΠ°ΠΏΠ½ΠΎΠ³ΠΎ ΠΏΠΎΠ΄Ρ
ΠΎΠ΄Π° ΠΏΠΎΠ»ΡΡΠ°ΡΡΡΡ ΠΌΠΈΠ½ΠΈΠΌΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΠ΅ BDD-ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ΠΈΡ ΡΠΈΡΡΠ΅ΠΌ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ. ΠΠΎ ΠΏΠΎΠ»Π½ΠΎΡΡΡΡ ΠΎΠΏΡΠ΅Π΄Π΅Π»Π΅Π½Π½ΡΠΌ Π±ΡΠ»Π΅Π²ΡΠΌ ΡΡΠ½ΠΊΡΠΈΡΠΌ Π²ΡΠΏΠΎΠ»Π½ΡΠ΅ΡΡΡ ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΎΠ΅ ΠΎΡΠΎΠ±ΡΠ°ΠΆΠ΅Π½ΠΈΠ΅ Π² Π·Π°Π΄Π°Π½Π½ΡΡ Π±ΠΈΠ±Π»ΠΈΠΎΡΠ΅ΠΊΡ Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ², Ρ. Π΅. ΠΏΠΎΠΊΡΡΡΠΈΠ΅ ΠΎΠΏΡΠΈΠΌΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π½ΡΡ
ΠΎΠΏΠΈΡΠ°Π½ΠΈΠΉ ΡΠΈΡΡΠ΅ΠΌ Π±ΡΠ»Π΅Π²ΡΡ
ΡΡΠ½ΠΊΡΠΈΠΉ ΠΎΠΏΠΈΡΠ°Π½ΠΈΡΠΌΠΈ Π»ΠΎΠ³ΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠ»Π΅ΠΌΠ΅Π½ΡΠΎΠ²
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