Two Formal Models of Interactive Machine

1. Two Formal Models of Interactive Machine Professor Hao, Kegang hkg@nwu.edu.cn Department of Computer ScienceNorthwest University Institute of Software Engineering, Northwest University

2. Background • Professor Peter Wegner was doubtful of the Church-Turing thesis and considered interactive computing beyond Turing Machines [1-3]. • Peter Wegner & Dina Goldin:Computation Beyond Turing Machines Seeking appropriate methods to model computing and human thought. COMMUNICATIONS OF THEACM April 2003 / Vol.46, No.4 • Peter Wegner, Interactive Foundations of Computing April 1997 • Peter Wegner and Dina Goldin, Mathematical Models of Interactive Computing, January 1 1999 Institute of Software Engineering, Northwest University

3. Need a theoretical framework • However, in order to prove this assertion, a rigor definition of the interactive machine model should be given. • Then it is possible base on this foundation to discuss whither or not the interaction extension is beyond Turing Machine. • They wrote:” our concept of interactive models was questioned because we originally failed to provide a theoretical framework comparable to that for Turing machines.” Institute of Software Engineering, Northwest University

4. Two Formal Models of Interactive Machine Abstract • One of the formal models of interactive machine is named Interactive Turing Machine that extends Turing Machine by adding certain interactive mechanism. • Another is Open Net that is an extension of the Petri Net, • Both of them are given by formal definition, so we could base on this foundation to study capabilities of the different interactive machines. • we define a concept named concurrent stream that is basically an annotated partial order set. • there is an issue analogic to Chomsky Hierarchy : what class of concurrent stream is corresponded with what kind of interactive machines? It is still an open problem. Institute of Software Engineering, Northwest University

5. Turing Machine Tape divided in to Infinite cells R/W Head Finite Control Symbols S = {s1,s2,…,sn} States Q = {q0,q1,q2,…,qm} q0 Q, F Q, Move M= { L,R,N } Institute of Software Engineering, Northwest University

6. 4 Styles of interaction environment System Inner IO-Port PI: Passive Input Sk I PO: Passive Output Sk O Sk AI: Active Input I Sk O AO: Active Output outer IO-Port Institute of Software Engineering, Northwest University

7. Synchronization • Environment can send a symbol to inner I-Port Only when inner I-port is empty . • Environment can get a symbol from inner O-Port Only when inner O-port is not empty and inner O-port is restored to empty after the environment got the symbol. • System can send a symbol to outer O-Port Only when outer O-port is empty . • System can get a symbol from outer I-Port Only when outer I-port is not empty and outer I-port is restored to empty after the environment got the symbol. Institute of Software Engineering, Northwest University

8. System P I O A environment I O Turing Machine  InteractiveTuring Machine Institute of Software Engineering, Northwest University

9. Definations of Turing Machine& InteractiveTuring Machine • An Infinite tape which is divided in to cells • A Finite Controlwith a R/W Head • A set of Symbols: S = {s1,s2,…,sn} B: blank • A set of States: Q = {q0,q1,q2,…,qm} F  Q • 3 moves: M= { L,R,N } Institute of Software Engineering, Northwest University

10. Definations (Continue) InteractiveTuring Machine • Inner IO-Port PI= {pi1,…,pil} PO= {po1,…,pol} • Outer IO-Port AI={ai1,…,aik} AO={ao1,…,aok} • Transitionrules： s,q,[pi:spi,][ai:sai] s’,q’,m,[po:spo,][ao:sao]. s,s’,spi,sai ,spo,saoS; q,q’ Q; m M; pi  PI,ai AIpo  PO,ao  AO. […] represents optional item. Turing Machine • Transition rules： s, q  s’, q’, m • s ,s’S; q,q’ Q; • m M Institute of Software Engineering, Northwest University

11. Synchronization s,q,pi:spi,(ai:sai) s’,q’,m, po:spo, ( ao:sao ) For Input: • Before the rule is used spi is in pi (sai in ai), • and after the rule is used pi,(ai) is empty For Output: • Before the rule is used po(ao) are empty • and after the rule is used spo is in po (saoin ao), Institute of Software Engineering, Northwest University

12. Execution of ITM • A ITM cannot execute alone. • To execute a ITM, it needs cooperate with the environment. • During the execution of ITM the environment forms a sequence of I/O events. Institute of Software Engineering, Northwest University

13. Word and Interactive Word • A set of Symbols: S ={s1,s2,…,sn} • Word: a string of symbols. For example, w=s2s3s2s5 • Interactive word: a string of events event: <Interactive port:symbol> For example, iw=<pi1:s2> <pi1:s3> <pi1:s2> <po1:s5> Institute of Software Engineering, Northwest University

14. Acceptability and Suitability • The language accepted by a TM is the set of words which cause the TM enter a final state when placed on the tape with it in state q0. • The language suitable with a ITM is the set of interactive words which could as the environment cooperate with the ITM executing and cause the ITM enter a final state eventually after all events in the I-word occurred Institute of Software Engineering, Northwest University

15. Theorem 1 • Definition 1. For a word w=s1s2…sk ,where si S,we call the fallowing I-word is. its corresponding I-word iw=< pi1:s1 >< pi1:s2 >… < pi1:sk >< pi1:send > where send is a special symbol 2. Corresponding I-language of a language is a set of its element’s corresponding I-words • Theorem 1 For every language accepted by a TM, then there exists a ITM which is suitable with its corresponding I-language. Institute of Software Engineering, Northwest University

16. Theorem 2 For every language suitable with a ITM, then there exists a TM by which the language is accepted. • Conclusion: • The Acceptability ( Suitability) of InteractiveTuring Machineis not Beyond Turing Machine in the above sense. Institute of Software Engineering, Northwest University

17. Open Nets- a model for interactive concurrent system Hao, Kegang: Open Net - a model for interactive concurrent system Software Engineering Institute Technical Report , Northwest University,1996.3. Journal of Northwest University, 1997.5. 西北大学学报》（自然科学版）1997. 5. Institute of Software Engineering, Northwest University

18. Content of the article • Static structure of Open Net • Process - Dynamic behavior of Open Net 。 • Composition and decomposition of Open Net (and its process) • Outside view of Open Net, Black box Theory • Hierarchical structure of Open Net Institute of Software Engineering, Northwest University

19. Main Points of the article Petri Nets Open Nets Closed System Open System Initial - result model interactive model nonhierarchical hierarchical white box theory black box theory Word-String(order set)  concurrent stream(partial order set) Institute of Software Engineering, Northwest University

20. t T PF PE P T · · t E An example for PetriNet place, transition thinking fork eating Institute of Software Engineering, Northwest University

21. An example for Open Net（philosophies eating） place,transition t T 0 01 t T PF PF PE P T · · 02 0 PF t E · t E outer place outer transition Institute of Software Engineering, Northwest University

22. Five philosophies eating and thinking Institute of Software Engineering, Northwest University

23. An example of Process 01 PF PT PF TE PE TT 02 PF PF 0 TE PT 0 TT PF 01 PF TE PE Institute of Software Engineering, Northwest University

24. o g o o d o d b g a c Word & concurrent stream • A concurrent stream on the alphabet S ：cw=<v,p>，where v is a partial order set，pis a mapping: p:vS • Alphabet: S • An word on the alphabet S ：w=<u,p>，where u is a order set， p is a mapping: p:uS {good} {good,gobd,gobc,gabd,gabc,gac} Institute of Software Engineering, Northwest University

25. Suitability • A concurrent stream is suitable with an Open Net execution means that it could as the environment cooperate with the Open Net executing and cause the Net enter a certain state. • In other word, the concurrent stream suitable with a Open Net execution is the subset of a execution process that consists of only IO events in the process. Institute of Software Engineering, Northwest University

26. Suitability(continue) • A set of concurrent stream is called suitable with an Open Net N, if for each it’s element Cs there exist a execution of N so that the concurrent stream Cs is suitable with the N executing. Institute of Software Engineering, Northwest University

27. Problem to be solved: • Is the suitability of ITM equal to Open Net ? Whither or not for every I-language L suitable with a ITM, there exists an Open Net N that L as a set of concurrent steam is suitable with N and vice versa ? • Note that TM has infinite cells tape, but Petri Net has only finite places, even though the capacity of places may not be limited. Institute of Software Engineering, Northwest University

28. Chomsky Hierarchy: • finite automata  regular grammars • pushdown automata  context free grammars • linear bounded automata context-sensitive grammars • Turing machines  unrestricted grammars (recursively enumerable sets) Institute of Software Engineering, Northwest University

29. Some open problems: • Do we need define interactive concurrent Turing Machine and its sub classes? • How to classify the sets of concurrent steams? • What is the class, which is suitable with interactive concurrent Turing Machine and its varioussub classes? • Are they really Beyond Turing Machines ? Institute of Software Engineering, Northwest University

30. Thank you！ Professor Hao, Kegang hkg@nwu.edu.cn Department of Computer ScienceNorthwest University Institute of Software Engineering, Northwest University