1. 1 2.05.2011 New Models of Computation Vadim Pesonen

2. 2 New Models of Computation New... in what respect? Computation beyond Turing Machines Super-Turing computation a.k.a. hypercomputation „Turing’s Ideas and Models of Computation“, book chapter from „Alan Turing: Life and Legacy of a Great Thinker“ by Eugene Eberbach, Dina Goldin, Peter Wegner, Springer 2004 2.05.2011

3. 3 Outline Algorithmic computation What is algorithmic computation? Examples Turing Machine Beyond algorithms – Super-Turing computation What is super-Turing computation? Examples Models 2.05.2011

4. 4 The Entscheidungsproblem, 1928 The decision problem asks for: an algorithm that will take as input a description of a mathematical statement and produce as output whether the statement is true or false But what is an algorithm? 2.05.2011 David Hilbert

5. 5 Algorithm and Algorithmic Computation Algorithm: systematic procedure that produces – in a finite number of steps – the answer to a question or the solution to a problem 2.05.2011 al-Khwārazmī Algorithmic computation: the computation is performed in a closed-box fashion, transfering a finite input, determined by the start of the computation, to a finite output, available at the end of the computation, in a finite amount of time

6. 6 The Automatic Machines (a-machines) Can compute any algorithm Follow the properties of algorithmic computation a-machine = Turing machine (TM) 2.05.2011 Alan Turing computation is closed resources are finite behaviour is fixed

7. 7 The Turing Machine (architecture) The Turing machine consists of: a tape divided into cells, each stores one symbol of finite alphabet a read/write head, which can move left of right by a single cell a control mechanism, which can be in one of a finite amount of states a transition table, which defines the next action and new state 2.05.2011 Several TM models exist, but in any of them: The set of states, symbols and actions is finite The tape is infinite (vs. endless)

8. 8 The Turing Machine (properties) The TM models closed computations – all inputs are given in advance The TM has unbounded, but finite amount of time and memory to perform computation Every TM computation starts in an identical initial configuation – for a given input the behaviour does not depend on time 2.05.2011

9. 9 The Finite State Machines Turing Machines don’t exist – the limited tape reduces them to Finite State Machines (FSMs) TMs are formal models rather than actual machines All digital sequential circuits today are FSMs 2.05.2011

10. 10 Common Misinterpretation of Turing Turing thesis: Whenever there is an effective method for obtaining the values of a mathematical functions, the function can be computed by a Turing Machine Strong Turing thesis: Whenever there is an effective method for solving a problem with a computer, it can be computed by a Turing Machine One should not equate computers and algorithmic computation 2.05.2011

11. 11 The Turing Machine (properties) The TM models closed computations – all inputs are given in advance The TM has unbounded, but finite amount of time and memory to perform computation Every TM computation starts in an identical initial configuation – for a given input the behaviour does not depend on time 2.05.2011

12. 12 The Turing Machine (limitations) The TM models closed computations – all inputs are given in advance The TM has unbounded, but finite amount of time and memory to perform computation Every TM computation starts in an identical initial configuation – for a given input the behaviour does not depend on time 2.05.2011

13. 13 Beyond Turing Machines The TM models closed computations – all inputs are given in advance Some modern computing systems are expected to process infinite amount of dynamically generated input requests 2.05.2011

14. 14 Beyond Turing Machines (contd.) The TM has unbounded, but finite amount of time and memory to perform computation Some modern computers are expected to continue computing indefinitely without halting 2.05.2011

15. 15 Beyond Turing Machines (contd.) Every TM computation starts in an identical initial configuation – for a given input the behaviour does not depend on time For some systems we expect a history-dependant behaviour, with the output determined both by the current input and the system’s computation history 2.05.2011

16. 16 Example – Driving Home from Work The DHV problem: Consider an automatic car whose task is to drive you home from work. The output for this problem should be a time-series plot of signals to the car’s controls. How can we copute this signal? 2.05.2011

17. 17 Example – Driving Home from Work (contd.) In an algorithmic scenario all inputs are provided a priori, including: exact map of the city exact car condition exact road conditions (each pothole and grain of sand) exact whether conditions We can remain optimisitc until we think of humans Computational tasks situated in the real world which includes human agents are not solvable algorithmically 2.05.2011

18. 18 Example – Driving Home from Work (contd.) But this problem is computable – interactively The DHV example proves that there exist problems that cannot be solved algorithmically, but are nevertheless computable 2.05.2011

19. 19 Super-Turing Computation Super-Turing computation: all computation, including that which cannot be carried out by a Turing Machine The 3 principles that allow us to derive a more expressive model of computation: interaction with the world infinity of resources evolution of the system 2.05.2011

20. 20 Interaction with the Environment Interactive computation involves interaction with an external world during the computation, rather than before and after it The 2 types of interaction: sequential interaction distributed interaction Distributed interaction is more expressive than sequential, just as interactive computation is more expressive that algorithmic computation 2.05.2011 interaction with the world infinity of resources evolution of the system

21. 21 Infinity of Resources The TM can be extended by removing any a priori bounds on its resources, resulting in: infinite initial configuration infinite architecture infinite time infinite alphabet Impracticality of infinite resources is not an obstacle here, as we are dealing with a mathematical model - TMs have infinite tape and cellular automata have infinite amount of cells 2.05.2011 interaction with the world infinity of resources evolution of the system

22. 22 Evolution of the System Extention by evolution allows the computing device to adapt over time 2.05.2011 interaction with the world infinity of resources evolution of the system Evolution can be done by upgrade of either HW or SW, learning, self-adaptation, self- evolution, self-reprodution... Evolution can be controlled by interaction with the environment, or by some performance measure, such as fitness, utility or cost

23. 23 Example of Super-Turing Computation Distributed client-server computation A typical paradigm used in computer networks Unpredictable concurrent interaction of multiple nodes with a dynamically generated inputs and configuration of communication links (vs closed-box model) The interaction is never-ending, the resources are potentially unlimited (vs finite resources) resources may outgrow any value that we can device for them The input depends on the current state of the world, which in turn depends on the earlier output values of the interactive system (vs fixed behaviour) 2.05.2011

24. 24 Example of Super-Turing Computation Some more trivial examples mobile robotics smart missile / autopilot evolutionary computation real computers 2.05.2011

25. 25 Models of Super-Turing Computation The x-machines: p-machines c-machines o-machines u-machines e-machines a-machines The π -calculus The $-calculus Neural networks Cellular automata... some others 2.05.2011

26. 26 The x-Machines P-machines persistent Turing machines C-machines choice machines O-machines oracle machines U-machines unorganized machines E-machines evolutionary Turing machines A-machines accelerated Turing machines a.k.a. Zeno machines 2.05.2011

27. 27 The π –calculus [Robin Milner et al.] A mathematical model for describing concurrent mobile processes rests upon the primitive notion of interaction Outperforms the TMs in expressivness: interaction of processes no limits on time of process computation evolution of communication topology 2.05.2011

28. 28 The π –calculus (contd.) Is in fact a dynamic extention of the Calculus of Communicating Systems (CCS) No definite metric for measuring the expressiveness of the the π –calculus yet No well defined body of work for the π –calculus yet several variants are developed to accomodate specific appliactions 2.05.2011

29. 29 Applications of the π –calculus Programming languages: Acute Business Process Modeling Language (BPML) Pict occam- π... Spi calculus 2.05.2011

30. 30 The $-calculus A mathematical model of resource-bounded computation a.k.a. flexible computation, anytime algorithm, imprecise computation, design-to-time scheduling rests upon the primitive notion of cost similar to π –calculus in terms of modelling interaction and communication 2.05.2011

31. 31 Applications of the $-calculus General implemenation of adaptive systems robotics neural networks Military applications coordnation of multiple Autonomous Undersea Vehicles (AUVs) Programming languages: Generic Behaviour Message-Passing Lanuage (GBML) Common Control Language (CCL)... 2.05.2011

32. 32 Super-Turing Computation - Summary One should not equate computers and algorithmic computation Super-Turing computation: all computation, including that which cannot be carried out by a Turing Machine The 3 principles of super-Turing computation: interaction with the world infinity of resources evolution of the system 2.05.2011

33. 33 Super-Turing ComputationTHE END Thank you for your attention! Questions? 2.05.2011