1. Turing Machines, Transition Systems, and Interaction
Dina Goldin, U.Connecticut
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2. Algorithmic vs. Interactive Computation
computation: finite transformation of input to output
input: finite-size (string or number)
closed system: all input available at start, all output generated at end
Church-Turing thesis: captures this notion of computation
computation: ongoing process which performs a task or delivers a service
dynamically generated stream of input tokens (requests, percepts, messages)
later inputs depend on earlier outputs (lack of modularity) and vice versa (history dependence)
objects, processes, components, control devices, reactive systems, intelligent agents
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3. Example: Driving home from work
Output: a sequence of pairs of #s (time-series data) - for turning the wheel - for pressing gas/break (similar to classical AI search/planning problems)
Algorithmic input: a description of the world (a static “map”)
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4. Driving home from work (cont.)
But… the output depends on every grain of sand in the road (chaotic behavior).
Can we possibly have a map that’s detailed enough?
Worse yet… the domain is dynamic. The output depends on weather conditions, and on other drivers and pedestrians.
We can’t possibly be expected to predict that in advance!
Nevertheless the problem is solvable – interactively!
Interactive input: stream of video camera (eye) images. ?
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5. Outline
Persistent Turing Machines (PTMs) an interactive extension of the TM model
Interactive Transition Systems (ITSs) effective transition systems induced by PTMs
Unbounded non-determinism exhibited by ITSs
It pays to be persistent expressiveness of persistent vs. amnesic computation
Summary and future work
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6. Nondeterministic 3-tape TMs
Configurations:
s - current state
w1 - contents of input tape
w2 - contents of work tape
w3 - contents of output tape
n1 , n2 , n3 - tape head posns
input
work
output S
Computation is a sequence of transitions:
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7. < win, w > Þ < w’, wout > N3TM macrosteps ® |
< s0, win, w, e, 1, 1, 1 >
< sh, win, w’, wout, 1, 1, 1 >
win
win w w’ So e Sh
wout
< win, w > Þ
< w’, wout >
Notation: M
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8. Divergent Computation
If computation diverges starting in configuration
corresponding macrostep notation is:
For all win  S*,
< s0, win, w, e, 1, 1, 1 >
< win, w > Þ
< sdiv, t > M
< win, sdiv > Þ
< sdiv, t > M
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9. Extending N3TM computations
Inputs are dynamic streams of tokens (strings). For each input token, there is an N3TM computation generating a corresponding output token.
The contents w of the work tape at the beginning of each N3TM computation is the same as at the end of the previous one.
fM (inputk, wk-1) = (outputk, wk)
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10. Persistent Stream Languages
Persistent Turing Machine (PTM): N3TM with persistent stream-based computational semantics
in1 S0 e Sh
out1 w1
in2
in2 w1 w2
... e
out2 S0 Sh
Persistent Stream Language of a PTM: set of streams
Conductive stream semantics:
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11. Formal Definition < w , w > Þ < w ' , w > Ù s ' Î PSL ( M
(Coinductive definition, relative to N3TM M and memory w)
PSL(M(w)) = { (wi, wo), s’  S | $w’S*:
< w , w
> Þ
< w ' , w
> Ù i o s ' Î
PSL ( M ( w '
))}
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12. PTM Example: ) ( M PSL Î inputs in1; outputs 1
inputs in2; outputs 1st bit of in1
inputs in3; outputs 1st bit of in2
...
Example: # 1
(1*,1)
(0*,1)
(1*,0)
(0*,0) ) (
Latch M
PSL Î
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13. Interactive Transition Systems over S
< S, m, r >
S is set of states
r is initial state (root)
m is transition relation
Required to be recursively enumerable
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14. From PTMs to ITSs > < Þ w s , ' = < reach(M), m, e > ξ(M)
Reachable memories of a PTM M:
Set of words (work-tape contents) w encountered
after zero or more macrosteps.
= < reach(M), m, e >
ξ(M)
where
>
< Þ o M i w s , '
>Îm
< o i w s , '
iff
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15. ITS Isomorphism
Let
be ITSs, i=1,2 1. 2.
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16. ITS Bisimulation T1 =bisim T2 if $ an interactive bisim. between them
Let
be ITSs, i=1,2
is a (strong) interactive bisimulation if: 1. 2.
3. Clause 2. with roles of s and t reversed
T1 =bisim T2 if $ an interactive bisim. between them
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17. Interactive Stream Equivalence
Infinite sequences of input/output token-pairs emanating from a particular ITS state
For an ITS T and state s, ISL(T(s)) [and ISL(T)] are defined similarly to PSL(M(s)) [and PSL(M)]
T1 =ISL T2 if ISL(T1) = ISL(T2)
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18. Theorem:
Proof:
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19. Equivalence Relations for PTMs vs. ITSs
=iso
=bisim
=ISL
=PSL
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20. Outline
Persistent Turing Machines (PTMs) an interactive extension of the TM model
Interactive Transition Systems (ITSs) effective transition systems induced by PTMs
Unbounded non-determinism exhibited by ITSs
It pays to be persistent expressiveness of persistent vs. amnesic computation
Summary and future work
UCONN HYDRA 12/4/1

21. Infinite Equivalence Hierarchy
Lk(M) = stream prefix language of PTM M set of prefixes of length  k for streams in PSL(M).
L (M) = Uk Lk(M)
Corresponding notion of equivalence:
M1 =k M2 : Lk(M1) = Lk ( M2 ) =1 =2 =
...
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22. Equivalence Hierarchy Gap
=PSL = =2 =1
...
Proof: construct PTMs M1 and M2 where L(M1) = L (M2 ) but PSL (M1 ) = PSL (M2 )
Note: M2 exhibits unbounded non-determinism /
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23. Example of Unbounded Nondeterminism
MUD ignores inputs, output 0 or 1 with each macrostep. On 1st macrostep, initializes a persistent string n of 1’s:
while true do
write ‘1’ on the work tape, move head to the right;
nondeterministically choose to exit loop or continue
The output at every macrostep is determined as follows:
if n > 0
then decrement n by 1 and output ‘1’;
else output ‘0’
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24. ITS for MUD ... ... sdiv n = 0 (S*, t) (S*, t) e (S*, 1) (S*, 1)
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25. Outline
Persistent Turing Machines (PTMs) an interactive extension of the TM model
Interactive Transition Systems (ITSs) effective transition systems induced by PTMs
Unbounded non-determinism exhibited by ITSs
It pays to be persistent expressiveness of persistent vs. amnesic computation
Summary and future work
UCONN HYDRA 12/4/1

26. Amnesic PTM Computation: stream-based but not persistent
<
> Þ < w', wo > Ù w , e i s Î '
PSL ( M ( w '
))}
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27. Amnesic PTM Computation
in1 S0 e Sh
out1 w1
in2
in2 e w2
... e
out2 S0 Sh
Example: outi = ini2
PTM M is amnesic if PSL(M)  ASL
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28. It pays to be Persistent
ASL
PSL
Proof:
Given an N3TM M, construct M’ such that
PSL(M') = ASL(M)
Consider 3rd elem. (0,0) of sio for Mlatch! For any M with sio in ASL(M), there will also be a stream in ASL(M) with (0,0) as 1st element. Therefore, for all M, ASL(M)  PSL(Mlatch).
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29. Functions or objects?
Functions (side-effect-free) or objects: does it matter for modeling programs?
Objects contain persistent values:
x1 = foo(args 1) y1 = cntr(add 1)
x2 = foo(args 2) y2 = cntr(get ttl)
x3 = foo(args 3) y3 = cntr(add 2)
x4 = foo(args 2) y4 = cntr(get ttl)
_________________________________________________
x2 = x4 y2  y4
History dependence (emerges in the context of multiple invocations).
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30. Summary of Results =ASL =PSL =1 =2 = =ms =ISL =bisim =iso = PTMs ITSs
... =
=ms
PTMs
ITSs
=ISL
=bisim
=iso
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31. Modeling Interactive Computation: Related Work
Reactive and embedded systems
Dataflow, process algebra, I/O automata, synchronous languages, finite/pushdown automata over infinite words, interaction games, online algorithms
Concurrency theory
Sequential Interaction Machines [Wegner&Goldin]
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32. Future Work Interactive computability Interactive complexity
Where are the ports?
Scott Smolka, SUNY at Stony Brook
Paul Attie, Northeastern Univ. Peter Wegner, Brown Univ.
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33. Interactive Computability
A stream language L is interactively computable if L  PSL
(properties of L expressed in Temporal Logic)
A behavior B is interactively computable if B is interaction bisimilar to an ITS T  T
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34. Systems of Concurrent PTMs
in3
out3 M3 t1
in1
in4 M1 M4
out1
out4 M2 t2
out2
in2
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