1. EXPRESS'011 Turing Machines, Transition Systems, and Interaction Dina Goldin, U.Connecticut Scott Smolka, SUNY at Stony Brook Peter Wegner, Brown University

2. EXPRESS'012 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

3. EXPRESS'013 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

4. EXPRESS'014 Nondeterministic 3-tape TMs s - current state w 1 - contents of input tape w 2 - contents of work tape w 3 - contents of output tape n 1, n 2, n 3 - tape head posns Configurations: input work output S Computation is a sequence of transitions:

5. EXPRESS'015 N3TM macrosteps   w in, w  Notation: w in SoSo w  ShSh w’ w out M  |  w’, w out 

6. EXPRESS'016 Divergent Computation   w in, w  M  s div,   If computation diverges starting in configuration corresponding macrostep notation is: For all w in   *,   w in, s div  M  s div,  

7. EXPRESS'017 Persistent Stream Language of a PTM: set of streams Conductive stream semantics: Persistent Stream Languages in 1 S0S0   ShSh out 1 w1w1 in 1 in 2 S0S0 w1w1  ShSh out 2 w2w2 in 2... Persistent Turing Machine (PTM): N3TM with persistent stream-based computational semantics

8. EXPRESS'018 Formal Definition ))}'((',', wMPSL www w o i      (Coinductive definition, relative to N3TM M and memory w) PSL(M(w)) = {  (w i, w o ),  ’   S |  w ’   *:

9. EXPRESS'019 inputs in 1 ; outputs 1 inputs in 2 ; outputs 1 st bit of in 1 inputs in 3 ; outputs 1 st bit of in 2... Example: PTM Example : # 1 0 (1*,1) (0*,1) (1*,0) (1*,1) (0*,0) )( Latch MPSL 

10. EXPRESS'0110 Interactive Transition Systems over  S is set of states r is initial state (root) m is transition relation Required to be recursively enumerable

11. EXPRESS'0111 From PTMs to ITSs  reach(M), m,  ξ(M)   o M i ws sw,',  m  oi wsws,',, iff Reachable memories of a PTM M: Set of words (work-tape contents) w encountered after zero or more macrosteps. where

12. EXPRESS'0112 ITS Bisimulation Letbe ITSs, i=1,2 is a (strong) interactive bisimulation if: 1. 2. 3. Clause 2. with roles of s and t reversed T 1 = bisim T 2 if  an interactive bisim. between them

13. EXPRESS'0113 Theorem: Proof:

14. EXPRESS'0114 Infinite Equivalence Hierarchy L k (M) = stream prefix language of PTM M set of prefixes of length  k for streams in PSL(M). L  (M) = U k L k (M) Corresponding notion of equivalence: M 1 = k M 2 : L k (M 1 ) = L k ( M 2 ) == =2=2 =1=1...

15. EXPRESS'0115 Equivalence Hierarchy Gap Proof: construct PTMs M 1 and M 2 where L  (M 1 ) = L  (M 2 ) but PSL (M 1 ) = PSL (M 2 ) Note: M 2 exhibits unbounded non-determinism / = PSL == =2=2 =1=1...

16. EXPRESS'0116 Example of Unbounded Nondeterminism M UD 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’

17. EXPRESS'0117 ITS for M UD  (  *, 1) n = 0 n = 1n = 2n = 3 (  *, 1) (  *, 0)... (  *,  ) s div (  *,  )...

18. EXPRESS'0118 Amnesic PTM Computation: stream-based but not persistent ))}'((', wMPSL w i    w', w o    

19. EXPRESS'0119 Amnesic PTM Computation in 1 S0S0   ShSh out 1 w1w1 in 1 in 2 S0S0   ShSh out 2 w2w2 in 2 Example: out i = in i 2 PTM M is amnesic if PSL(M)  ASL...

20. EXPRESS'0120 Proof : Given an N3TM M, construct M ’ such that PSL(M') = ASL(M) Consider 3 rd elem. (0,0) of  io for M latch ! For any M with  io in ASL(M), there will also be a stream in ASL(M) with (0,0) as 1 st element. Therefore, for all M, ASL(M)  PSL(M latch ). It pays to be Persistent ASLPSL

21. EXPRESS'0121 Summary of Results PTMs ITSs = == =2=2 =1=1 = ms = iso = bisim = ISL = PSL... = ASL

22. EXPRESS'0122 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

23. EXPRESS'0123 Interactive computability Universal PTM Interactive complexity Where are the ports? http://www.cse.uconn.edu/~dqg/papers/ Future Work

24. EXPRESS'0124 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

25. EXPRESS'0125 ITS Isomorphism 1. 2. Letbe ITSs, i=1,2

26. EXPRESS'0126 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 T 1 = ISL T 2 if ISL(T 1 ) = ISL(T 2 )