1. Carlos Eduardo Maldonado Research Professor Universidad del Rosario INNOVATION AND COMPLEXITY

2. INNOVATION ENTAILS COMPLEXITY Complex systems contain and lead to surprise (emergence) They are unpredictable (chaotic, catastrophic) They do not have centrality or hierarchy (local control) (self-organization) They are essentially open systems (complex networks) (NET)

3. INNOVATION AND PROBLEM SOLVING Innovation and problem solving: two faces of one and the same token They root in biology, not just in culture

4. INNOVATION AND/AS RESEARCH Basic Research Experimental Research Applied Research  All depends on de the mode and degree of innovation Incremental Innovation Radical Innovation  Targets-based Research  Research grounded on habilities and skills

5. Two kind of problems DecidibleIndecidible Cannot be solved algorithmically, not even with unlimited or infinite time and space resources P N-P N-P Complete N-P Hard Easy/Irrevelevant Problems Hyper- computation Simulation Metaheurístics Difficult Relevant Problems

6. MODEL REAL SYSTEM (REAL WORLD ) COMPUTER MODELING SIMULATION

7. OPTIMIZATION (COMBINATORIAL COMPLEXITY) Local Optimization (or partial) Global Optimization

8. P and N-P: COMPLEXITY  It is easier to find a solution than verifying it: P: It is necessary that a problem admits a method to find a solution in a P time. N-P: It is sufficient that a problem admits a method to verify the solution in a P time.

9. P, N-P and OPTIMIZATION Problems: P = N-P P ≠ N-P P ≤ N-P P C N-P

10. MODERN METHODS OF HEURISTICS Fuzzy Systems Neural Networks Genetic Programming Agents (multi-agents)- based Systems

11. TECHNIQUES FOR LOCAL OPTIMIZATION (Stochastic) Hill climbing Simulated Annealing Taboo Search Evolutionary Algorithms Constraint Handling

12. METHODS OF GLOBAL OPTIMIZATION Problems of combinatorial complexity Heuristics: Algorithm that looks for good solutions at a reasonable computational cost, without though guarantee of optimality (or even feasibility). Usually works with specific problems Metaheuristics: They are heuristics in a larger and deeper scope  Bio-inspired Computation

13. MODELING, SIMULATION, OPTIMIZATION Data mining Optimization Metaheuristics Evolutive Computation Swarm Intelligence Artificial Life Sciences of Complexity. Other Prediction Multi-Agent Models Cellular Automata Artificial Chemistry. Other

14. METAHEURISTICS Single-Solution Based Population-Based Metaheuristics for Multiobjective Optimization Hybrid Metaheuristics Parallel Metaheuristics  Distinction between Decidable and Indecidable Problems (Computationally)

15. COMPLEXITY OF ALGORITHMS AND PROBLEMS DECIDIBLE PROBLEMS INDECIDIBLE PROBLEMS Ej.: The Halting Problem (Turing)

16. COMPLEXITY OF ALGORITHMS An algorithm needs two important resources to solve a problem: space and time The time complexity of an algorithm is the number of steps required to solve a problem of size n

17. ALGORITHM AND TIME Polynomial-time algorithm p(n) = a k. n k + … + a j. n j + … + a l. n + a o Exponential-time algorithm Its complexity is: O(c n ), where c is a real constant superior to 1

18. COMPLEXITY OF PROBLEMS The complexity of a problem is equivalent to the complexity of the best algorithm solving that problem A problem is tractable (or easy) if there exists a P- time algorithm to solve it A problem is intractable (or difficult) if no P-time algorithm exists to solve the problem C/A complexity theory of problems deals with decision problems. A decision problem always has a yes or no answer

19. Optimization Methods Exact Methods Approximate Methods Branch and x Restricted Programming Dynamic Programming A*, IDA* Heuristic Algorithms and Approximate Algorithms MetaheuristicsSpecific heuristic problems Single-based solutions Metaheuristics Population-based Metaheuristics

20. METAHEURISTICS Metaheuristics P Metaheuristics Hybrid Metaheuristics Parallel Metaheuristics

21. WHAT IS COMPUTABLE? That we can know That we can say That we can decide upon  That we can solve

22. NEW PROBLEMS IN COMPUTATION Conversations Numbering Proves Finite Time Infinite Time Continuous Time Discrete Time New Computational Paradigms. Changing Conceptions of What is Computable. S. Barry Cooper, B. Löwe, A. Sorbi (Eds.), Springer Verlag, 2008

23. LOGICS AND COMPUTATION Intuition Bubbles Non-Classical Logics: Paraconsistent Logics Relevant Logics Quantum Logics Time Logics Many-Valued Logics Epistemic Logics Fuzzy Logics Computational Complexity Algorithmic Complexity

24. INNOVATION AND KNOWLEDGE Innovating and solving problems as a matter of pushing-back the frontiers of knowledge Making life every time more possible Gaining degrees of freedom Pushing-back cenral controls and rigid hierarchies Trusting in local controls and dynamic centers Working in a small-world: complex networks

25. INNOVATION AND AESTHETICS Spearhead science does not pretend to control or predict, any longer Science distrusts conclusive arguments and yet strives for them Science assesses harmony