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Anticipation in constructing and interrogating Natural Information Systems: From Classical through Nano to Quantum Computing Authors • B. Nick Rossiter, Informatics, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, [email protected] • M. A. Heather, Sutherland Building, Northumbria University, NE1 8ST, [email protected] Weak Anticipation in Classical Databases • Today’s classical information systems anticipate the real world. • Databases store, organise and search collections of realworld data. • In terms of anticipatory systems, – Reductionism and normalisation are needed with von Neumann architectures consisting of fixed instructions between bit cells. – Hence weak anticipatory as information systems (databases) are constructed using classical methods Developing non-classical Areas • The new developing areas are: – quantum computation, exploiting quantum mechanics principles in physics, – nanoscale chemistry, – bio- and molecular-computing processing as in genetics • Natural computing is: – Real-world processing • does not rely on any model. – Data can be input neat • without any reductionist pre-processing. • The corresponding information system or database using natural computing should therefore be strong Classical Database Techniques • The multilevel ANSI/SPARC architecture. • Three layers and the mappings between them: External Schema Conceptual Schema Internal Schema views derived for end-users from the conceptual schema describing the data in terms of types addressing the layout of data on disk The mappings between schemas are algebraic or calculus expressions. Non-classical Architecture • Not layered (Theory of Categories) • Use Dolittle approach (push me-pull you creature of High Lofting): • Actually matches SQL standard approach where layering is not enforced C CI E I E C conceptual schema I Internal schema E external schema Classical Relationships • Relationships are often performed in a separate process: – Entity-Relationship Modelling – Unified Modelling (UML) • Normalisation is needed to verify schema design: – particularly to relate key and non-key attributes. • The levels, mappings and relationships all have to be integrated in a consistent database design. Non-classical Database Design • Formally a database design is a topos and representable as a Dolittle diagram subsuming the pullback/pushout relationships as: Cartesian Closed Category X + f* What is f*? • f* is an examination and re-indexing functor – organises the data into a key for storage and applies a query for interrogation of the database. – puts together a key by concatenation as in the relational model. – looks up information for retrieval by inspecting the key. • In quantum theory: – the key (X) is entanglement, – the colimit (+) is superposition, • In genetics it is a DNA strand. Enriched Pullback • In terms of the Dolittle diagram: – f* is the same operation in classical and natural computation. • What then corresponds to the database schema in natural computing? • The pullback diagram contains many more arrows than in the Dolittle diagram. • This enriched diagram satisfies our needs. Pullback of S and M in Context of IMG ls x m S s ls x m S XIMG M rs x m s x m *m M s (s)-1 m (m)-1 W/IMG Figure 2: Pullback showing fuller collection of arrows S = source, M = medium, IMG = image, W = world Contents of Enriched Dolittle • The Dolittle diagram relates binary categorial limits (X) and colimits (+) for types • Includes Heyting implication in intuitionistic logic. • The pullback functor (f*) looks for: – a particular sub-limit to represent a relationship, – emulating the join operation of databases (*). • Other arrows represent: – projection – existential quantification – universal quantification Higher-level Arrows • Originality with the unit of adjunction – Example: gives properties of relationship onto limit – = 0, no creativity, mapping S to S XIMG M is 1:1 – = 1, maximum creativity, mapping S to S IMG M is from S to cartesian product of S M. • Style with the counit of adjunction. – Example: gives properties of relationship onto colimit – = 1, preservation of style, each S is found exactly once in S XIMG M – = 0, loss of style, each S occurs in S IMG M maximum number of times (S M). • The Dolittle diagram is the universal relation with all possible relationships in parallel. Work with Databases and Categories • Michael Johnson, Robert Rosebrugh and RJ Wood, Entity-Relationship-Attribute Designs and Sketches, TAC 10(3) 94-111. – sketches for design (class structure) – models for states (objects) where model is used in categorical sense – lextensive category (finite limits, stable disjoint finite sums) for query language More details for Sketches • 12 different types: – See Wells, C, (1993), Sketches: Outline with References at http://www.cwru.edu/artsci/math/wells/pub/pdf/sketch.pdf • Employed also in databases by: – Zinovy Diskin, Boris Cadish: Algebraic Graph-Based Approach to Management of Multidatabase Systems, NGITS’95 69-79 (1995). • Sketch idea originally from Charles Ehresmann. – Finite Discrete (FD) sketch D = (E, L, R, S) • finite graph E (data structure) • set of diagrams L in E (constraints) • Finite set R of discrete cones in D (relationships) • Finite set S of discrete cocones in D (attributes) Models from Sketches • Model (M) – sketch homomorphism • maps any E to category V where V is a database state • L commutative diagrams, R limit cones, S colimit cocones • preserve products • the more detailed the sketch, the less nice the model • strength of anticipation depends on nature of model • In FD sketches in Johnson et al: • finite sums satisfy the lextensive axiom • sums are well-behaved View on Sketches • Intuitively appealing as match categorial concepts with those of information systems • In sketches: – The cones match X in Dolittle for design giving relationships – The cocones match + in Dolittle.for design giving attributes – The graph matches C in Dolittle for schema – The set of diagrams that will commute is not in Dolittle • all our diagrams commute. • Good equivalence but our constructions are more general in further mappings as in interoperability. Other Recent Work • Grover has developed the idea of using quantum algorithms for faster searching of databases • Selinger has produced a collection of operations at such a level that they could form the basis of a quantum programming language. • Both offer potential for the development of quantum databases. • However, databases in general require a conceptual level for the representation, querying and updating of data. • The present approach is not yet at the f* conceptual level: – relies on low-level operations analogous to classical methods – like the CNOT gate (controlled NOT gate) where two input qubits, control and target, are xor'd and stored in a target qubit B + A. • As a kind of f*, Grover makes use of the 'oracle', treated as a black box and used for collapsing the wave function, that is to determine when a solution has been derived. – However, this form of the oracle lacks non-locality Discussion • What is the status for natural computing as a strong anticipatory system? • Nano-computation may be a separate transitional phase between and distinct from classical and quantum processors. • Issue remains whether natural, nano and quantum computing are all facets of the same operation. • Anticipatory systems theory latent in the Dolittle diagram suggests they are all the same. • If the Dolittle diagram is much closer to reality than most models, it is a strong anticipatory system. References [1] Adleman, L, On constructing a molecular computer, DNA based computers, Lipton, R, and Baum, E, (edd), DIMACS, American Mathematical Society 1-21 (1996). [2] Grover L K, A fast quantum mechanical algorithm for database search, Proc 28th Annual ACM Symposium Theory of Computing 212-219 (1996). [3] Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing I. Non-locality in Quantum Theory, International Journal Computing Anticipatory Systems 13 307-326 (2002). [4] Heather, M A, & Rossiter, B N, Locality, Weak or Strong Anticipation and Quantum Computing. II. Constructivism with Category Theory, International Journal Computing Anticipatory Systems 13 327-339 (2002). [5] Selinger, P, Towards a Quantum Programming Language, 42pp, http://quasar.mathstat.uottawa.ca/~selinger/papers.html#qpl (2002).