Anticipation as Prediction in the Predication of Data Types Michael Heather and Nick Rossiter CEIS, Northumbria University, Newcastle NE1 8ST /

SYMPOSIUM 5: SOFT COMPUTING, COMPUTATIONAL INTELLIGENCE, FUZZY SYSTEMS, NEURAL NETWORKS, LEARNING INVITED SESSION 5.1. MONDAY AUGUST 6, 15:00-18:00, ROOM 126 Computational Intelligence and Experimental Design Within Dynamical Anticipatory Systems and Networks Chairman: Stefan Pickl – Co-Chairman: Jörg Schütze Intelligent Network Structures and Max-Min Paths Problems, Dmitrii Lozovanu (Moldava), Stefan Pickl (Germany) A new Concept for Meeting the Demands for Services in Communities, Jörg Schütze (Austria), Markus Hill (Germany) Anticipation as Prediction in the Predication of Data Types, Michael Heather and Nick Rossiter (United Kingdom) Interactive Resource Planning - An Anticipative Concept in the Simulation Based Decision Support System EXPOSIM, Ulrike Leopold-Wildburger (Austria), Stefan Pickl (Germany)

Typing is Essential Typing is: –an essential feature of modern systems theory –the essence of anticipation in anticipatory systems Typing appears in many different guises –basic type of matter –fundamental particles of this universe (hadron) –computer systems Typing remains the burning question

Viewpoints on Typing Information systems have highlighted a number of problems –relationship between physical systems and corresponding information systems –huge variability in form of types (superficially) fundamental particle types in physics the operational information of genome types (biology and medicine)

Types - Aristotle Information concerned with language and logic Realised by Aristotle The first to apply to a system of types the word categories –in Greek originally referred to a legal indictment a statement –drafted with full formal specification –to stand up to rigorous argument in court.

Types -- Latin Equivalent word in Latin for a legal indictment is predicamentum Words categories and predicates have continued in this sense in English Usually –Predicate is language-oriented –Category is logic-related

Typing Pre-Aristotle Typing –traced back to Sanskrit literature –did not completely begin with Aristotle Pre-socratic attempts at classification relevant to anticipatory systems are: –well known fundamental types of Parmenides (everything stays the same) –Heraclitus (all is flux) These receive considerable attention in philosophy.

Parmenides Constancy of Parmenides –At fist glance little relevance to anticipatory systems –As need to predict their own future states May be true for weak anticipation –an anticipatory system embodies a model of itself But for the intensional form –Parmedian aspect of anticipatory systems the prediction can be ascertained from the typing of the system independently of time.

Heraclitus Heraclitus emphasises the extensional form Anticipation lies in semantics Extension may include time –if time is independent variable still only weak anticipation –if time is part of the data then strong anticipation

Global Types Most theoretical work on anticipatory systems –weak anticipation –local dynamical systems methods may be quite adequate Now problems of –globalisation –very complex subjects: biology and medicine Call for solutions with strong anticipation

Strong Anticipation Troubles arise in typing –lack of powerful tools –cannot produce results with strong anticipation Twentieth century mathematics –dominated by a logic which is Parmedian in extension as well as in its intension.

Three Viewpoints Weaknesses are apparent from the work of three eminent mathematicians – the undecidability of Gödel – the paradox of Russell – the impredication of Poincaré

Gödel 1 Gödel has famously shown That while first-order predicate calculus is complete and decidable

Gödel 2 that for higher-order systems e.g. relation between intension and extension –not possible to determine whether any system based on number and relying on axioms is true or false –general result that makes the goal of ultimate consistency within set theory unattainable –Russell's paradox and Poincaré's impredication are particular manifestations of Gödel's undecidability

Russell Russell was acutely aware of the inadequacy of set membership –the set of all sets could not be a member of itself From his study of denotational predication –explored a number of advanced theories of typing –on his own admission these did not succeed.

Recalling Russell's connection with the theory of types, it was with some trepidation that I approached him in 1967 with the proof that it was unnecessary. To my relief he was delighted. The Theory was, he said, the most arbitrary he and Whitehead had ever had to do, not really a theory but a stopgap, and he was glad to have lived long enough to see the matter resolved. G. Spencer Brown, Laws of Form, p. xiii, 1972 Russell’s view in 1967

Poincaré’s view Poincaré had already pointed out that the crux of the problem lies in the scoping of the predicate.

Predicates Three relevant strands: –logical system of Whitehead & Russell's Principia Mathematica allows solely for simple predicate giving rise only to weak anticipation –a predicate has to be coextensive with its subject to give certainty –a predicate needs to be variable to allow for a varying context

Consequences Real problems arise in closed worlds of information systems –provide very striking examples of the difficulties that result from simplification and normalisation of predicates. There is a demand for strong anticipation if information is to be reliably exchanged through open interoperable systems

Way Forward Need the implementation of formal systems that can avoid the undecidability of Gödel Category Theory (CT) has its focus and strengths in higher order logic e.g. functors –Pure CT is though axiomatic –n-categories rely on number –so both offend Gödel –Applied CT, based on a process view and of composition, appears to not offend Gödel

Composition in CT a) with Gödel; b) against Gödel a) (g o f) = gf b) (g o f)´ ≠ gf

Adjointness between two Composition Triangles F ┤G

Composition Triangles in Detail a) unit of adjunction  ; b) co-unit of adjunction 

Architecture for Predication Topos T involving categories L, R and context category C for scoping predicate

Summary Implications of Gödel –First-order predicate calculus is complete and decidable –Set theory, as defined with axiom and number, is not complete and decidable for higher order –Anticipation requires higher order logic –Applied category theory, without axiom or number, seems appropriate Example architecture given in category theory for predication with topos and composition.