EXPRESS'01 1
Turing Machines,Transition Systems,
and Interaction
Dina Goldin, U.ConnecticutScott Smolka, SUNY at Stony Brook
Peter Wegner, Brown University
EXPRESS'01 2
Algorithmic vs. Interactive Computation
computation: finite transformation of input to outputinput: 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
EXPRESS'01 3
• Persistent Turing Machines (PTMs)an interactive extension of the TM model
• Interactive Transition Systems (ITSs)effective transition systems induced by PTMs
• Unbounded non-determinismexhibited by ITSs
• It pays to be persistentexpressiveness of persistent vs. amnesic computation
Summary
EXPRESS'01 4
Nondeterministic 3-tape TMs
s - current state
w1 - contents of input tape
w2 - contents of work tape
w3 - contents of output tape
n1 , n2 , n3 - tape head posns
321321 ,,,,,, nnnwwws
'CC |
• Configurations:
input
work
outputS
• Computation is a sequence of transitions:
EXPRESS'01 5
N3TM macrosteps
win, wNotation:
win
So
w
win
Sh
w’
wout
M
|< s0, win, w, , 1, 1, 1 > < sh, win, w’, wout, 1, 1, 1 >
w’, wout
EXPRESS'01 6
Divergent Computation
win, w M
< s0, win, w, , 1, 1, 1 >
sdiv,
If computation diverges starting in configuration
corresponding macrostep notation is:
For all win *,
win, sdiv Msdiv,
EXPRESS'01 7
• Persistent Stream Language of a PTM: set of streams
• Conductive stream semantics:
Persistent Stream Languages
,...},,,{ 2211 outinoutin
SS *)*(
in1
S0
Shout1
w1
in1 in2
S0
w1
Shout2
w2
in2
...
Persistent Turing Machine (PTM): N3TM with persistent stream-based computational semantics
EXPRESS'01 8
Formal Definition
))}'(('
,',
wMPSL
wwww oi
))(()( MPSLMPSL
}3an is )({ TMNMMPSL |PSL
21 MM PSL:eequivalenc PSL
(Coinductive definition, relative to N3TM M and memory w)
PSL(M(w)) = { (wi, wo), ’ S | w’*:
EXPRESS'01 9
• inputs in1; outputs 1
• inputs in2; outputs 1st bit of in1
• inputs in3; outputs 1st bit of in2
• ...
• Example:
PTM Example: LatchM
),...}0,1(),0,0(),1,0(),1,1{(io
#
1
0
(1*,1)
(0*,1)
(0*,1) (1*,0)
(1*,1)
(0*,0)
)( LatchMPSL
EXPRESS'01 10
Interactive Transition Systemsover
• S is set of states• r is initial state (root)• m is transition relation
** SSmRequired to be recursively enumerable
< S, m, r >
EXPRESS'01 11
From PTMs to ITSs
reach(M), m,ξ(M)
oMiwssw ,',m
oiwsws ,',, iff
Reachable memories of a PTM M:
Set of words (work-tape contents) w encountered after zero or more macrosteps.
where
*reach(M)
EXPRESS'01 12
ITS Bisimulation
iiii rmST ,,,
21 rRr
Let be ITSs, i=1,2
21 SSR is a (strong) interactive bisimulation if:
'',',,s.t.',',,
2
1
tRsmwtwttmwswstRs
oi
oi
1.
2.
3. Clause 2. with roles of s and t reversed
T1 =bisim T2 if an interactive bisim. between them
EXPRESS'01 13
Theorem:
isomorphic are andstructures The isoms ,, TM
Proof:
)()( 21
21
MMMM
iso
ms
iff:preserving-structure
) obvious the choose recursive; is( :onto
MmMTMT )(:, MT
) for the to for the (set msiso
EXPRESS'01 14
Infinite Equivalence Hierarchy
• Lk(M) = stream prefix language of PTM Mset 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 )
==2=1 ...
EXPRESS'01 15
Equivalence Hierarchy Gap
• Proof: construct PTMs M1 and M2 where L(M1) = L (M2 ) but PSL (M1 ) = PSL (M2 )
• Note: M2 exhibits unbounded non-determinism
/
=PSL==2=1 ...
EXPRESS'01 16
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’
EXPRESS'01 17
ITS for MUD
(*, 1)
n = 0 n = 1 n = 2 n = 3(*, 1)(*, 1) (*, 1) (*, 1)
(*, 1) (*, 1) (*, 1)(*, 1)
(*, 0)
...
(*, )sdiv
(*, )
...
EXPRESS'01 18
Amnesic PTM Computation:
stream-based but not persistent
:*'),,({)( wwwMASL oi |S
))}'(('
,
wMPSL
wi
w', wo
}3an is )({ TMNMMASL |ASL
21 MM ASL:eequivalenc ASL
EXPRESS'01 19
Amnesic PTM Computation
in1
S0
Shout1
w1
in1 in2
S0
Shout2
w2
in2
Example: outi = ini2
PTM M is amnesic if PSL(M) ASL
...
EXPRESS'01 20
Proof:: Given an N3TM M, construct M’ such that
PSL(M') = ASL(M)
: Consider 3rd elem. (0,0) of io for Mlatch!For any M with io 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).
It pays to be Persistent
ASL PSL
EXPRESS'01 22
• Reactive and embedded systems
• Dataflow, process algebra, I/O automata, synchronous languages, finite/pushdown automata over infinite words, interaction games, online algorithms
• Sequential Interaction Machines [Wegner & Goldin]
Modeling Interactive Computation: Related Work
EXPRESS'01 23
• Interactive computability
• Universal PTM
• Interactive complexity
• Where are the ports?
http://www.cse.uconn.edu/~dqg/papers/
Future Work
EXPRESS'01 24
• 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
Interactive Computability
EXPRESS'01 25
ITS Isomorphism
s.t.bijectionif : 2121 SSTT iso
21)( rr
2
1
1
),'(,),(,',,
:',*,,
mwswsmwsws
Sssww
oi
oi
oi
iff
1.
2.
iiii rmST ,,,Let be ITSs, i=1,2