Anticipation in constructing and interrogating Natural Information Systems:

1. Anticipation in constructing and interrogating Natural Information Systems: From Classical through Nano to Quantum Computing

2. Authors • B. Nick Rossiter, Informatics, Northumbria University, Newcastle upon Tyne, UK, NE1 8ST, nick.rossiter@unn.ac.uk • M. A. Heather, Sutherland Building, Northumbria University, NE1 8ST, m.heather@unn.ac.uk

3. Weak Anticipation in Classical Databases • Today’s classical information systems anticipate the real world. • Databases store, organise and search collections of real-world 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

4. 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 anticipatory.

5. Classical Database Techniques • The multilevel ANSI/SPARC architecture. • Three layers and the mappings between them: views derived for end-users from the conceptual schema External Schema Conceptual Schema describing the data in terms of types addressing the layout of data on disk Internal Schema The mappings between schemas are algebraic or calculus expressions.

6. C CI E I E 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 conceptual schema I Internal schema E external schema

7. 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.

8. X + f* 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

9. 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.

10. 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.

11. Pullback of S and M in Context of IMG S = source, M = medium, IMG = image, W = world

12. 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

13. 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.

14. 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

15. 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)

16. 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

17. 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.

18. 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

19. 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.

20. 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).