uconn hydra 12/4/11 turing machines, transition systems, and interaction dina goldin, u.connecticut

UCONN HYDRA 12/4/1 1

Turing Machines,Transition Systems,

and Interaction

Dina Goldin, U.Connecticut


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


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”)


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.

Driving home from work (cont.)

?


• 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 and future work

Outline


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:


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


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,


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)


• 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


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’*:


• 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


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 >


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)


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


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


• 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)


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


PTMs

ITSs

=ms

=iso=bisim=ISL

=PSL

Equivalence Relationsfor PTMs vs. ITSs







Outline


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 ...


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 ...


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’


ITS for MUD

(*, 1)

n = 0 n = 1 n = 2 n = 3(*, 1)(*, 1) (*, 1) (*, 1)

(*, 1) (*, 1) (*, 1)(*, 1)

(*, 0)

...

(*, )sdiv

(*, )

...







Outline


Amnesic PTM Computation:

stream-based but not persistent

:*'),,({)( wwwMASL oi |S

))}'(('

,

wMPSL

wi

w', wo

}3an is )({ TMNMMASL |ASL

21 MM ASL:eequivalenc ASL


Amnesic PTM Computation

in1

S0

Shout1

w1

in1 in2

S0

Shout2

w2

in2

Example: outi = ini2

PTM M is amnesic if PSL(M) ASL

...


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


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).


Summary of Results

ASL PSL

PTMs

ITSs

=

==2=1 =ms

=iso=bisim=ISL

=PSL ...

=ASL


• 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]

Modeling Interactive Computation: Related Work


• Interactive computability

• Interactive complexity

• Where are the ports?

http://www.cse.uconn.edu/~dqg/papers/

Scott Smolka, SUNY at Stony BrookPaul Attie, Northeastern Univ.Peter Wegner, Brown Univ.

Future Work


• 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


M1

M2

M4

M3t1

t2

in1 in4

in3

in2

out2

out1

out3

out4

Systems of Concurrent PTMs

uconn hydra 12/4/11 turing machines, transition systems, and interaction dina goldin, u.connecticut

Documents

soso w shsh w w

contents w

w notation

uconn hydra

divergent computation

contents of input tape

n3tm macrosteps w

contents of work tape