AspectOriented Programming AOP methodology

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Two Formal Models of Interactive Machine

Professor Hao, Kegang hkg_at_nwu.edu.cn
Department of Computer ScienceNorthwest
University
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Background

• Professor Peter Wegner was doubtful of the
  Church-Turing thesis and considered interactive
  computing beyond Turing Machines 1-3.
• Peter Wegner Dina GoldinComputation Beyond
  Turing Machines Seeking appropriate methods to
  model computing and human thought. COMMUNICATIONS
  OF THE ACM 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

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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.

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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.

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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

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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
AO Active Output
O
outer IO-Port
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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.

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Turing Machine ? Interactive Turing Machine
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Definations of Turing Machine Interactive
Turing Machine

• An Infinite tape which is divided in to cells
• A Finite Control with 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

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Definations (Continue)

• Interactive Turing Machine
• Inner IO-Port PI pi1,,pil
• PO po1,,pol
• Outer IO-Port AIai1,,aik
• AOao1,,aok
• Transition rules
• s,q,pispi,aisai ? s,q,m,pospo,
  aosao .
• s,s,spi,sai ,spo,sao ?S q,q ?Q m ?M
• pi ? PI,ai ? AI po ? PO,ao ? AO.
• represents optional item.

• Turing Machine
• Transition rules
• s, q ? s, q, m
• s , s ?S q,q ?Q
• m ?M

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Synchronization

• s,q,pispi,(aisai) ? s,q,m, pospo, ( aosao )
  
• 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 (sao in
  ao),

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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.

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Word and Interactive Word

• A set of Symbols S s1,s2,,sn
• Word a string of symbols.
• For example, ws2s3s2s5
• Interactive word a string of events
• event ltInteractive portsymbolgt
• For example,
• iwltpi1s2gt ltpi1s3gt ltpi1s2gt ltpo1s5gt

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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

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Theorem 1

• Definition
• 1. For a word ws1s2sk ,where si ? S,we call
  the fallowing I-word is. its corresponding I-word
• iwlt pi1s1 gt lt pi1s2 gt lt pi1sk gt lt pi1send
  gt
• where send is a special symbol
• 2. Corresponding I-language of a language
  is a set of its elements 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.

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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 Interactive
  Turing Machine is not Beyond Turing Machine in
  the above sense.

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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.
• ??????(?????)1997. 5.

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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

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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)
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An example for Petri Net
place, transition
thinking
fork
eating
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An example for Open Net(philosophies eating)
place,transition
t T
0
01
t T
PF
P T
PF
PE


02
PF
0
t E

t E
outer place
outer transition
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Five philosophies eating and thinking
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An example of Process
01
PF PT PF
TE
PE
TT
02
PF
PF
0
PT
TE
0
TT
01
PF
PF
TE
PE
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Word concurrent stream

• A concurrent stream on the alphabet S
  cwltv,pgt,where v is a partial order set,p is a
  mapping pv?S

• Alphabet S
• An word on the alphabet S wltu,pgt,where u is a
  order set, p is a mapping
• pu?S

good good,gobd,gobc,gabd,g
abc,gac
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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.

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Suitability(continue)

• A set of concurrent stream is called suitable
  with an Open Net N, if for each its element Cs
  there exist a execution of N so that the
  concurrent stream Cs is suitable with the N
  executing.

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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.

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Chomsky Hierarchy

• finite automata ? regular grammars
• pushdown automata ? context free grammars
• linear bounded automata ? context-sensitive
  grammars
• Turing machines ? unrestricted grammars
  (recursively enumerable sets)

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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
  various sub classes ?
• Are they really Beyond Turing Machines ?

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Thank you!
Professor Hao, Kegang hkg_at_nwu.edu.cn
Department of Computer ScienceNorthwest
University