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 21

Summary of Results

ASL PSL

PTMs

ITSs

=

==2=1 =ms

=iso=bisim=ISL

=PSL ...

=ASL

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

EXPRESS'01 26

• 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)]

Interactive Stream Equivalence

T1 =ISL T2 if ISL(T1) = ISL(T2)