Rudy's Diamond Strategies

This complementary Blog to the Chinese Challenge Blog is presenting studies to a mathematical theory of Diamonds. Diamond theory is studying for the first time, tabular categories as an interaction of categories and saltatories.

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Wednesday, January 23, 2008

Double Cross Playing Diamonds

The paper "Double Cross Playing Diamonds" is a further development of the Diamond Way of Thinking, which is inspired by the Chinese writing system and applications of polycontextural logics to category theory. A new understanding of interactivity is proposed. It is introduced as a comparison between Robin Milner's model of interaction and the diamond strategies to interactionality.

PREVIEW of Double Cross Playing Diamonds, which will be published in:

Seifert, Uwe/Jin Hyun Kim/Anthony Moore (Eds.):
Paradoxes of Interactivity

Perspectives for Media Theory, Human-Computer Interaction,
and Artistic Investigations,
Bielefeld: Transcript 2008 (in editing).

1. Models of Interactivity between flows and salti
“Interactivity is all there is to write about: It is the Paradox and the Horizon of Realization."

Grammatologically, the Western notational system is not offering space in itself to place sameness and otherness necessary to realize interaction/ality.
Alphabetism is not prepared to challenge the dynamics of interaction directly.
The Chinese writing system in its scriptural structuration is able to place complex differences into itself, necessary for the development and design of formal systems and programming languages of interaction.

The challenge of interactionality to Western thinking, modeling and design interactivity has to be confronted with the decline of the scientific power of alphanumeric notational systems as media of living in a complex world.

The challenge I see for media artists is not only to develop interactional media constellations but also to intervene between the structures and dynamics of interactional systems as international corporations, governments, military and academia force them on us.

1.1 Comparison of two approaches to interactivity
This paper takes the risk to compare two fundamentally different approaches to interaction and reflection in computational systems:
Milner's bigraphs and diamond theory.
Milner's bigraph model and theory of interaction is highly developed, while the diamond model applied to this interactional scenario and confronted with the bigraphs model is presented here for the first time.

The Milner model is presupposing a world-view (ontology, epistemology) of homogeneity and openness. Its basic operation is composition in the sense of category theory. Composition is associative and open for infinite iterability.
The Milner model is a model of interaction in a global sense but it is not yet thematizing formally the chiastic interplay of local and global aspects of interaction.

Its merits is to have developed a strict separation of topography (locality) and connectivity for a unifying theory of global and mobile interaction (ubiquitous computing) surpassing, in principle, the limits of Turing computability.

In contrast, the diamond model, which is just emerging, is based on an antidromic and parallactic structure of combination of events in an open/closed world of a multitude of discontextural universes. In such a pluri-versal world model, each composition is having its complementary combination. With that, iterability for diamonds is not an abstract iterativity but interwoven in the concrete situations to be thematized, and determined by iterative and accretive repetitions, involving their complementary counterparts, without a privileged conceptual initial/final object.

This leads to a theory of diamonds as a complementary interplay of categories and saltatories (jumpoids) with the main rules, globally, of complementarity and locally, of bridging. Diamonds are involving bi-objects belonging at once to categories and to saltatories, ruled by composition and saltisition (jump-operation).

1.2 Interactionality as interplays between categories and saltatories
In less technical terms, the polycontextural approach of diamond theory is supporting three new features:

First, it supports the idea of irreducible multi-medial contextures and their qualitative incomparability. That is, different media like sound, video, picture, text, graphics, etc., are conceived as logically different and as organized and distributed conceptually in a heterarchical sense.
To thematize media as a digital contexture is not more than to emphasize their informatical and physical aspect, which is as such a contexture, too.

Second, it supports the possibility of mapping the (outer) environment of a contexture (media) in itself, i.e., to offer an inner environment for reflectionality. Contextures, to be different from systems, have to reflect their environment into their own domain. Hence, a contexture has to be understood as being involved into interplays of inner and outer environments.

Third, it supports the possibility of realizing simultaneously movements (actions) and complementary counter-movements on a basic level of conceptualization and formalization.

If composition of events inside a contexture and mediation of different contextures to a compound-contexture, polycontexturality, are characterized by the rules of combination, i.e., identity, commutativity and associativity, a new feature of composition is discovered by the diamond approach, which is antidromic and parallax, corresponding structurally the otherness of the categorical system.

Therefore, the questions of interactionality in a diamond framework are not primarily, how do we globally move, physically and informatically, from one topographic place to another, but how do we move by interaction from one medium to another medial system of a complex knowledge space.

With the appearance of the semantic web and knowledge grid such developments are unavoidable. Obviously, the polycontextural diamond approach is not opting for a principally homogeneous global field of informatical and physical events but for a discontexturality of different media, situations, contexts of meaning.

The Milner Model is well based, principally, on category theory, the diamond model has to develop its own new formalism, risked here as a diamondization of category theory. Hence, both theories are in a constellation, which offers a reasonable possibility for comparisons.

Because the bigraph model is based on category theory and its concept of composition with its abstract iterability, the diamond model has to develop a distinct concept of composition (combination), one which involves a complementarity of, at least, two different concepts of composition; technically, the categorical and the saltatorical composition, and which is opening up the operativity of an open/closed concept of iter/alterability.

Even if only metaphorically and still vague, what is common to both models is there dichotomous, dual, complementary and orthogonal approach to interaction and interactionality.

The Milner model is focused on message passing, flow of informatic objects, the diamond model on agents and their reflectional/interactional activities with an emphasis on intervention.

Double Cross Playing Diamonds

Understanding interactivity in/between bigraphs and diamonds

1 Models of interactivity between flows and salti
1.1 Comparision of two approaches to interactivity 1
1.2 Interaction as interplays of categories and saltatories 1

2 Milner’s bi-graph model of interaction
2.1 Locality and connectivity 3
2.2 Strategies 5
2.3 Orthogonality of topography and connectivity 7

3 Diamond theory of interactionality
3.1 Diamond Strategy 9
3.2 Diamond Theory 10
3.3 Diamond Structuration 13
3.4 Interactionality as interplays in Diamonds 17

4 Bigraphs in diamond webs
4.1 Disseminated Diamonds 20
4.2 Towards a diamond web of bigraphs 20

5 Bibliography