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• W2341014258 abstract "IS THERE A GENERAL DEFINITION OF STRUCTURE? WHAT DOES KNOWING STRUCTURAL COMPLEXITY GIVE YOU? OPEN QUESTIONS, CURRENT RESEARCH, WHERE LEADING Martin Zwick Systems Science Ph.D. Program Portland State University Portland OR 97202-0751 zwick@sysc.pdx.edu http://www.sysc.pdx.edu/Faculty/Zwick prepared for: Session on Measures of Structural Complexity New England Complex Systems Institute Sept. 21-26, 1997 ABSTRACT A discrete multivariate relation, defined set-theoretically, is a subset of a cartesian product of sets which specify the possible values for two or more variables. Where three or more variables are involved, the highest order relation, namely the relation between all of the variables, may or may not be decomposable without loss into sets of lower order relations which involve subsets of the variables. In a completely parallel manner, a relation defined information-theoretically, namely a joint probability distribution involving all the variables, may or may not be decomposed without loss into lower-order distributions involving subsets of the variables. Decomposability also called reconstructability analysis, is the specification of the losses suffered by all possible decompositions. The decomposability of relations, defined either setor information-theoretically, offers a fundamental approach to the idea of complexity and bears on all of the themes prominent in both the new and the old sciences of complexity. Decomposability analysis gives precise meaning to the idea of structure, i.e., to the interrelationship between a whole and its parts, where these are conceived either statically or dynamically. It specifies the structuring and distribution and the amount of information needed to describe complex systems. It sheds specific light on chaotic versus non-chaotic dynamics in discrete dynamic systems. It provides a framework for characterizing the dual processes of integration and~ differentiation which govern the diachronics of self-organization.A discrete multivariate relation, defined set-theoretically, is a subset of a cartesian product of sets which specify the possible values for two or more variables. Where three or more variables are involved, the highest order relation, namely the relation between all of the variables, may or may not be decomposable without loss into sets of lower order relations which involve subsets of the variables. In a completely parallel manner, a relation defined information-theoretically, namely a joint probability distribution involving all the variables, may or may not be decomposed without loss into lower-order distributions involving subsets of the variables. Decomposability also called reconstructability analysis, is the specification of the losses suffered by all possible decompositions. The decomposability of relations, defined either setor information-theoretically, offers a fundamental approach to the idea of complexity and bears on all of the themes prominent in both the new and the old sciences of complexity. Decomposability analysis gives precise meaning to the idea of structure, i.e., to the interrelationship between a whole and its parts, where these are conceived either statically or dynamically. It specifies the structuring and distribution and the amount of information needed to describe complex systems. It sheds specific light on chaotic versus non-chaotic dynamics in discrete dynamic systems. It provides a framework for characterizing the dual processes of integration and~ differentiation which govern the diachronics of self-organization. IS THERE A Gt:NERAL DEFINITION OF STRUCTURE? ANSWER: Yes, a structure is a set of relations. 1. Relation =a constraint linking entities, e.g., variables. 2. Variables can be nominal~ discipline-general; can be dynamic. 3. Constraint defined, e.g., (a) setor (b) info.-theoretically, i.e., (a) subset of cartesian product or (b) multivariate probability distribution. 4. Projections of relation define lattice of relations (LOR). 5. Structure= cut through LOR= decomposition of a relation. 6. Complexity1*(structure) =#degrees of freedom (info.-theor.) = # parameters needed to specify it 7. Represents topology, not strength, of constraints. 8. Resolution-dependent; data-independent. 9. Lattice of structures (LOS) = all possible decompositions. *Other definitions possible in this framework. RELATION: R = { (ab bj, cb d1)} c A® B ® C ® D SET-THEOR. R = { p( ab bj, ck, d1) } INFO.-THEOR. PROJECTION: R = RABcD => RABc call Rx simply X AB LATTICE OF RELATIONS (LOR) ABCD" @default.
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• W2341014258 date "1997-01-01" @default.
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• W2341014258 title "Complexity and Decomposability of Relations" @default.
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