repeated forever. They are persistent in the sense that the computation they perform is allowed to depend on the \state" the PTM or ITS was in after p

Transcription

1 Turing Machines, Transition Systems, and Interaction Dina Q. Goldin Scott A. Smolka Peter Wegner Math & Computer Science Dept. Computer Science Dept. Computer Science Dept. U. Mass/Boston SUNY at Stony Brown University Boston, MA Stony Brook, NY Providence, RI October 4, 2000 Abstract We present Persistent Turing Machines (PTMs), an interactive extension of the classical Turing-machine model of computation. A PTM repeatedly receives an input token from the environment, computes for a while, and then outputs the result. The new twist, however, is that the PTM is allowed to \remember" its previous state (work-tape contents) upon commencing a new computation. The Persistent Stream Language (PSL) of a PTM is a set of innite sequences of pairs of the form (w i ; w o ), recording for each interaction with the environment, the input token received by the PTM and the corresponding output token. The Amnesic Stream Language (ASL) of a PTM is dened similarly, although, in this case, the PTM begins each new computation with a blank work tape. Our main results are that the class of ASLs is strictly contained in the class of PSLs, indicating that the extension of Turing machines with persistence is a nontrivial one, and that the class of PTMs is isomorphic to Interactive Transition Systems, a very general class of eective transition systems. 1 Introduction Numerous researchers have examined the relationship between Turing machines and transition systems, including [Bou85, ds85, BBK87, BIM88, Dar90, Vaa93]. For example, Vaandrager [Vaa93] introduces the notion of eective process graph, a transition system with a countable number of states such that in each state the outgoing transitions can be computed. He also notes that such graphs, and indeed most process calculi that have been proposed in the literature, are Turing powerful. That is, the expressiveness of these formalisms allows one to simulate each Turing machine in lock step. But what about the other direction? In other words, what extensions to the basic Turingmachine model are necessary to allow them to simulate transition systems? This is essentially the problem we address in this paper. In particular, we show, in a precise sense, that by viewing Turing machines as interactive and by introducing a notion of persistent computation, we get a class of objects that is isomorphic to a very general class of eective transition systems. We refer to our extended Turing-machine model as Persistent Turing Machines (PTMs), and to the class of transition systems we consider as Interactive Transition Systems (ITSs). PTMs and ITSs are interactive in the sense that, upon receiving an input from their environment, they compute for a while and then output the result to their environment, and this process is 1

2 repeated forever. They are persistent in the sense that the computation they perform is allowed to depend on the \state" the PTM or ITS was in after producing its last output and prior to receiving its next input. The main contributions of this paper are the following: We formalize the notion of a persistent Turing machine in terms of the persistent stream language (PSL) of a nondeterministic 3-tape Turing machine (N3TM). The PSL of an N3TM is a set of interaction streams, where an interaction stream is a coinductively dened, innite sequence of pairs of the form (w i ; w o ). Each such pair represents a computation performed by the N3TM, producing output token w o, in response to receiving input token w i from the environment. The new twist, however, is that the N3TM is allowed to \remember" its previous state (work-tape contents) upon commencing a new computation. This provides the impetus for calling such Turing machines \persistent", as in persistent Turing machines or PTMs. We also dene the amnesic stream language (ASL) of an N3TM; in this case, the N3TM begins each new computation with a blank work tape. Our main result about PSLs and ASLs is that the class of ASLs is strictly contained in the class of PSLs. One may consequently conclude that, in a stream-based setting, the extension of the Turing-machine model with persistence is a nontrivial one. We then dene interactive transition systems (ITSs), a new kind of eective transition system. ITS transitions are 4-tuples of the form hs; w i ; s 0 ; w o i meaning that the ITS, while in state s and having received input string w i from its environment, transits to state s 0 and outputs w o. Moreover, such transitions are eective. We further equip ITSs with three notions of behavioral equivalence ITS isomorphism, interactive bisimulation and interactive stream equivalence and show that ITS isomorphism renes interactive bisimulation, and interactive bisimulation renes interactive stream equivalence. Finally, we show that (natural partitionings of) the class of ITSs and the class of PTMs are isomorphic. This is the main result of the paper in that it shows that a persistent, stream-based extension of the classical Turing-machine model is on equal footing with eective transition systems. Related Work The notion of interaction embodied in PTMs and ITSs can be found in various dataow models [Kah74, Den75, KM77, BCLGH94, LP95, Lee97] and process calculi such as CCS [Mil89] and the -calculus [MPW92]. Such models of computation are typically purely functional in nature, and, therefore, the notion of persistency or \state" present in PTMs is absent. However, persistency can be captured in dataow models by \feedback loops" and in process calculi by explicitly modeling the data store. Persistent Turing machines, and the closely related Sequential Interaction Machines, have been introduced in earlier papers by Wegner and Goldin [Weg96, Weg97, Weg98, WG99, GST00, Gol00]. A major emphasis of this body of work is to show how such a computational framework can be used as a basis for modeling various forms of interactive computing, such as object-oriented, agent-based, dynamical systems, and others. An alternative approach to extending the Turing-machine model to interactive computation is put forth in [LW00]. 2

3 The rest of this paper is structured along the following lines. Section 2 presents nondeterministic 3-tape Turing machines (N3TMs) and denes how computation proceeds on such a machine. Section 3 denes the notions of persistent and amnesic stream languages (PSLs and ASLs, respectively) and contains the proof that the class of PSLs strictly contains the class of ASLs. Section 4 considers interactive transition systems (ITSs) and the accompanying notions of interactive bisimulation and interactive stream equivalence. Section 5 contains our main result, viz. the isomorphism of ITSs and PTMs. Section 6 oers some concluding remarks. 2 Nondeterministic 3-tape Turing Machines In this section, we dene the notion of a 3-tape Turing machine. The denition is standard as far as Turing machines go (see e.g. [HU79]), modulo the fact that a 3-tape machine comes equipped with three tapes rather than one. We subsequently dene how computation proceeds on a 3-tape machine. Denition 2.1 A nondeterministic 3-tape Turing machine (N3TM) is a quadruple hk; ; ; s 0 i where: K is a nite set of states. is a nite alphabet containing the blank symbol #, but not containing L (left) and R (right). K (K [ fhg) ( [ fl; Rg) ( [ fl; Rg) ( [ fl; Rg) is the transition relation. s 0 2 K is the initial state. h =2 K is the halting state. A 3-tape Turing machine is deterministic (D3TM) if is a function : K! (K [ fhg) ( [ fl; Rg) ( [ fl; Rg) ( [ fl; Rg). 2 An N3TM has three tapes: an input tape, output tape, and work tape. Each of these tapes has an associated tape head and corresponding tape-head position. An N3TM makes a transition from its current state based on the (possibly blank) symbols found on the tapes at the current tape-head positions. Such a transition will take it to a new state (possibly the halt state h) and for each of the three tapes, either a new symbol will be written at the current head position or the position of the head will be shifted by one location to the left (L) or right (R). The following denition of an N3TM conguration is also standard. N is the natural numbers. Denition 2.2 Let M = hk; ; ; s 0 i be an N3TM. A conguration of M is a septuple hs; w 1 ; w 2 ; w 3 ; n 1 ; n 2 ; n 3 i, where s 2 K is the state of the conguration. w 1 2 is the contents of the input tape of M. w 2 2 is the contents of the work tape of M. 3

4 w 3 2 is the contents of the output tape of M. n 1 ; n 2 ; n 3 2 N are the tape head positions for M's three tapes, respectively. 2 Let w be a word in, n 2 N a natural number, and c 2 an arbitrary character in. Then w[n] denotes the n th character in the word w## (w appended with an innite string of blanks), and w[c=n] denotes the word w## with its n th character replaced by c. The intuition behind this notation, which we use in the following denition, is that w represents the contents of an N3TM tape and the rest of the tape (i.e. to the right of w) is assumed to be blank. Denition 2.3 Let M be an N3TM and C, C 0 two congurations of M: C = hs; w 1 ; w 2 ; w 3 ; n 1 ; n 2 ; n 3 i; C 0 = hs 0 ; w 0 1 ; w0 2 ; w0 3 ; n0 1 ; n0 2 ; n0 3i We say that Cj?! C 0 (yields in one microstep) if hs; w 1 [n 1 ]; w 2 [n 2 ]; w 3 [n 3 ]; s 0 ; c 1 ; c 2 ; c 3 i 2 and, for i = 1; 2; 3: n 0 = n i i + 1 if c i = R w 0 = w i i if c i = L or c i = R n 0 = n i i? 1 if c i = L and n i 6= 1 w 0 = w i i[c i =n i ] otherwise n 0 = n i i otherwise Denition 2.4 Let M be an N3TM and C, C 0 two congurations of M. We say that Cj?! C 0 (yields in zero or more microsteps) if there exist congurations C 0 ; : : : ; C n for some n 0 such that C = C 0 ; C 0 = C n, and C i j?! C i+1 for 0 i < n. j?! is the reexive transitive closure of j?!. 2 It is well known that N3TMs are equivalent to single-tape TMs. That is, given an N3TM M accepting some language L, there exists a single-tape TM accepting L [HU79]. However, distinguishing the input tape from the output tape from the work tape will facilitate our subsequent development of N3TM-based stream computation (Section 3). Next, we dene an N3TM macrostep. Our choice of terms \microstep" (Denition 2.3) and \macrostep" is inspired by the treatment of Statechart semantics in [PS91]. Denition 2.5 Let M be an N3TM. We say that hw 1 ; w 2 ; w 3 i j=) M hw 0 ; 1 w0 ; 2 w0 3i (yields in one macrostep) if hs 0 ; w 1 ; w 2 ; w 3 ; 1; 1; 1ij?! hh; w 0 1 ; w0 2 ; w0 3 ; 1; 1; 1i 2 2 A macrostep is a shorthand notation for a halting computation of a Turing machine. In the classical setting, w 2 and w 3 are (the empty string); i.e., computation begins with the work tape and output tape blank. This extension of the classical notion of computation is essential for our development of persistent, stream-based computation, the topic of Section 3. Note that an even more general denition of a macrostep, where the tape head positions at the end of the macrostep are not necessarily ones, is not necessary: for any N3TM M that halts with the heads in non-unit positions, there is one that halts in exactly the same conguration, but with the heads in unit positions. 4

5 3 Stream Computation for N3TMs We introduce the notion of stream-based computation for N3TMs. Given an enumerable set of action tokens A, S A is the class of streams over A given by S A = A S A. 1 Intuitively, a stream over A is an innite sequence of elements of A. Fixing the alphabet of an N3TM to be, we shall be particularly interested in the class of streams S, the members of which we shall refer to as interaction streams. An interaction stream, which is a pair of the form h(w i ; w o ); 0 i with (w i ; w o ) 2 and 0 2 S, can be thought of as a recording (history) of an innite \session" between an N3TM and its environment. As formalized in the following denition of a persistent stream language, each element (w i ; w o ) of an interaction stream represents a computation performed by the N3TM, producing output tape contents w o, in response to w i being placed on its input tape by the environment. The new twist, however, is that the N3TM is allowed to \remember" its previous state (work-tape contents) upon commencing a new computation. Denition 3.1 Given an N3TM M and some w 2, P SL(M(w)) (the persistent stream language of M with memory w) is dened as follows: P SL(M(w)) = fh(w i ; w o ); 0 i 2 S j 9w0 2 : hw i ; w; ij=) M hw i ; w 0 ; w o i and 0 2 P SL(M(w 0 ))g P SL(M), the persistent stream language of M, is dened as P SL(M()). We also have that PSL = fp SL(M) j M is an N3TMg. 2 The stream language of an N3TM is said to be amnesic if each computation of the N3TM begins with a blank work tape; i.e., the N3TM \forgets" the state it was in when the previous computation ended. This idea is formalized in the following denition. Denition 3.2 Given an N3TM M, ASL(M) (the amnesic stream language of M) is dened as follows: ASL(M) = fh(w i ; w o ); 0 i 2 S j 9w0 2 : hw i ; ; ij=) M hw i ; w 0 ; w o i and 0 2 ASL(M)g We also have that ASL = fasl(m) j M is an N3TMg. 2 Example 3.1 Consider the N3TM M Latch = hk; ; ; s 0 i where K = fs 0 g, = f0; 1; #g, and = fhs 0 ; 0; #; #; h; 0; 0; 1i; hs 0 ; 1; #; #; h; 1; 1; 1i; hs 0 ; 0; 0; #; h; 0; 0; 0i; hs 0 ; 0; 1; #; h; 0; 0; 1i; hs 0 ; 1; 0; #; h; 1; 1; 0i; hs 0 ; 1; 1; #; h; 1; 1; 1ig In terms of the interaction streams in P SL(M Latch ), M Latch outputs the rst bit of the input token it received from the environment in conjunction with its previous interaction with the environment (except for the rst interaction where it outputs a 1). Hence its name. Note that M Latch is deterministic. On the other hand, the interaction streams in ASL(M Latch ) are of the form f(w 1 ; 1); (w 2 ; 1); (w 3 ; 1); : : :g. 1 We have dened streams coinductively; see, e.g, [BM96]. This style of denition will allow us to apply coinduction as a proof technique later in the paper. 5

6 For example, assume, for simplicity, that the input tokens M Latch receives from the environment are single bits and that the rst four of these form the bit sequence Then the corresponding interaction stream io 2 P SL(M Latch ) is of the form: io = f(1; 1); (0; 1); (0; 0); (1; 0); : : :g It is easy to see that no interaction stream in P SL(M Latch ) starts with the pair (0; 0). 2 One might argue that the interaction between M Latch and its environment is not essential; rather its behavior could be modeled by a machine that receives its entire (innite) stream of input tokens prior to computation and then proceeds to output (the rst bit of each element of) prepended with a 1. The problem with this approach is that, in general, the elements of are generated dynamically and therefore cannot be known in advance. The following proposition is used in the proofs of Propositions 3.2 and 3.4. Proposition 3.1 Given an N3TM M, let L(M) be dened as follows: f(w i ; w o ) 2 j 9w 0 2 : hw i ; ; ij=) M hw i ; w 0 ; w o ig Then, ASL(M) = S L(M ), the set of all streams over L(M). Proof: Follows from Denition Proposition 3.2 Let M 1 and M 2 be N3TMs. P SL(M 1 ) = P SL(M 2 ) implies ASL(M 1 ) = ASL(M 2 ) but not vice versa. Proof: We rst show that P SL-equivalence renes ASL-equivalence. Consider L(M) of Proposition 3.1. From Denition 3.1, it follows that L(M) = fa 2 j 9 2 P SL(M) : head() = Ag where, given a stream, head() returns the rst element of. It is therefore the case that whenever P SL(M 1 ) = P SL(M 2 ), L(M 1 ) = L(M 2 ). Therefore, S L(M1 ) = S L(M2 ), and ASL(M 1 ) = ASL(M 2 ). To show that the renement is strict, we refer to M Latch of Example 3.1 and consider an N3TM M F lip where ASL(M F lip ) = ASL(M Latch ), yet P SL(M F lip ) 6= P SL(M Latch ). M F lip is the (deterministic) N3TM hk; ; ; s 0 i where K = fs 0 g, = f0; 1; #g, and = fhs 0 ; 0; #; #; h; 0; 0; 1i; hs 0 ; 1; #; #; h; 1; 1; 1i; hs 0 ; 0; 0; #; h; 0; 0; 1i; hs 0 ; 0; 1; #; h; 0; 0; 0i; hs 0 ; 1; 0; #; h; 1; 1; 1i; hs 0 ; 1; 1; #; h; 1; 1; 0ig The interaction streams of P SL(M F lip ) are the \negation" of those of M Latch. That is, M F lip outputs :b whenever M Latch outputs b, for bit b. On the other hand, the interaction streams of ASL(M F lip ) are exactly those of ASL(M Latch ). 2 Proposition 3.3 ASL PSL Proof: It suces to show that, given an N3TM M, we can construct an N3TM M 0 such that P SL(M 0 ) = ASL(M). The construction is as follows: 6

7 M 0 always starts its computation by erasing the contents of its work tape and moving the work-tape head back to position 0; it then proceeds just like M. From Denitions 3.1 and 3.2, it follows that P SL(M 0 ) = ASL(M). 2 Proposition 3.4 The inclusion of ASL in PSL is strict. Proof: We refer to M Latch and io 2 P SL(M Latch ) dened in Example 3.1 to show that there does not exist an N3TM M such that ASL(M) = P SL(M Latch ). Assume such an N3TM M exists; then, io 2 ASL(M). Therefore, by Proposition 3.1, (0; 0), the third element of io, is in L(M). This in turn implies that there are interaction streams in ASL(M) starting with (0; 0). Contradiction (see Example 3.1)! Therefore, no such M exists. 2 In the sequel, our focus will be on persistent stream computation. We shall therefore refer to N3TMs as persistent TMs (PTMs) to underscore this point. We say that a PTM M is amnesic if P SL(M) 2 ASL. Example 3.2 M Latch is not amnesic. On the other hand, the squaring machine of [PR98, Figure 1]), which repeatedly accepts an integer n from its environment and outputs n 2, is amnesic. 2 4 Interactive Transition Systems In this section, we introduce a kind of \eective" transition system (see, for example, [Vaa93]) that we shall refer to as an \interactive transition system." Our main result about interactive transition systems is that they are isomorphic to PTMs (Theorem 5.1). Denition 4.1 An interactive transition system (ITS) is a quadruple hs; ; m; ri where: S is an enumerable set of states. is a nite alphabet. m S S is the transition relation; m is required to be recursive, i.e., its interpretation as the function m : S! 2 S is recursive. r 2 S is the initial state (root). An ITS is deterministic if m is a recursive function m : S! S. 2 Intuitively, a transition hs; w i ; s 0 ; w o i of an ITS T means that T, while in state s and having received input string w i from its environment, transits to state s 0 and outputs w o. Moreover, such transitions are eective. We assume that all states in S are reachable from the root. We now dene three notions of equivalence for ITSs, each of which is successively coarser than the previous one. Denition 4.2 Two ITSs T 1 = hs 1 ; ; m 1 ; r 1 i and T 2 = hs 2 ; ; m 2 ; r 2 i are isomorphic, notation T 1 = iso T 2, if there exists a bijection : S 1! S 2 such that: 7

8 1. (r 1 ) = r w i ; w o 2 ; s; s 0 2 S : hs; w i ; s 0 ; w o i 2 m 1 i h (s); w i ; (s 0 ); w o i 2 m 2 Note that = iso is an equivalence relation. 2 Denition 4.3 Let T 1 = hs 1 ; ; m 1 ; r 1 i and T 2 = hs 2 ; ; m 2 ; r 2 i be ITSs. A relation R S 1 S 2 is a (strong) interactive bisimulation between T 1 and T 2 if it satises: 1. r 1 Rr 2 2. if srt and hs; w i ; s 0 ; w o i 2 m 1, then there exists t 0 2 S 2 with ht; w i ; t 0 ; w o i 2 m 2 and s 0 Rt 0 ; 3. if srt and ht; w i ; t 0 ; w o i 2 m 2, then there exists s 0 2 S 1 with hs; w i ; s 0 ; w o i 2 m 1 and s 0 Rt 0. T 1 and T 2 are interactively bisimilar, notation T 1 T 2, if there exists an interactive bisimulation between them. Note that is an equivalence relation. 2 The third and nal notion of equivalence is based on the notion of the interactive stream language of an ITS. Denition 4.4 Given an ITS T = hs; ; m; ri and a state s 2 S, ISL(T (s)) (the interactive stream language of T in state s) is dened as follows: ISL(T (s)) = fh(w i ; w o ); 0 i 2 S j 9s0 2 S : hs; w i ; s 0 ; w o i 2 m and 0 2 ISL(T (s 0 ))g: ISL(T ), the interactive stream language of T, is dened as ISL(T (r)). Two ITSs T 1 and T 2 are interactive stream equivalent, notation T 1 T 2, if ISL(T 1 ) = ISL(T 2 ). 2 Proposition 4.1 Let T 1 = hs 1 ; ; m 1 ; r 1 i and T 2 = hs 2 ; ; m 2 ; r 2 i be ITSs. T 1 = iso T 2 implies T 1 T 2 (but not vice versa) and T 1 T 2 implies T 1 T 2 (but not vice versa). Proof: The proof that ITS isomorphism (strictly) renes interactive bisimilarity is straightforward. To show that interactive bisimilarity renes interactive stream equivalence, suppose T 1 T 2. Then there exists an interactive bisimulation R between T 1 and T 2 such that r 1 Rr 2. Now let h(w i ; w o ); 1 i be an arbitrary interaction stream in ISL(T 1 ). In this case, 9s 1 2 S 1 such that hr 1 ; w i ; s 1 ; w o i 2 m 1 and 1 2 ISL(T 1 (s 1 )). Since r 1 Rr 2, 9s 2 2 S 2 such that hr 2 ; w i ; s 2 ; w o i 2 m 2 and s 1 Rs 2. This in turn implies that there exists an interaction stream h(w i ; w o ); 2 i 2 ISL(T 2 ) with 2 2 ISL(T 2 (s 2 )). By coinduction, we have that ISL(T 1 (s 1 )) = ISL(T 2 (s 2 )) and, since the interaction stream in ISL(T 1 ) we considered above was arbitrary, ISL(T 1 ) = ISL(T 2 ). This yields T 1 T 2 as desired. To show that interactive bisimilarity strictly renes interactive stream equivalence, consider the following pair of ITSs over alphabet = f0; 1g: T 1 = hfr 1 ; s 1 ; t 1 g; ; m 1 ; r 1 i and T 2 = hfr 2 ; s 2 g; ; m 2 ; r 2 i, where: m 1 = fhr 1 ; 0; s 1 ; 1i; hr 1 ; 0; t 1 ; 1i; hs 1 ; 0; r 1 ; 1i; ht 1 ; 1; r 1 ; 0ig m 2 = fhr 2 ; 0; s 2 ; 1i; hs 2 ; 0; r 2 ; 1i; hs 2 ; 1; r 2 ; 0ig It is easy to see that T 1 T 2 but T 1 6 T

9 5 Equivalence of ITS and PTM In this section, we show that (natural partitionings of) the class of PTMs and the class of ITSs are isomorphic. For this purpose, we assume a xed alphabet, denote the class of PTMs with alphabet by M, and denote the class of ITSs with alphabet by T. To obtain the desired result, we begin by dening the \reachable memories" of a PTM. Recalling (Denition 4.1) that the states of an ITS are assumed to be reachable from the ITS's root, reachable memories provide us with an analogous concept for PTMs. Denition 5.1 Let M 2 M be a PTM with alphabet. Then reach(m), the reachable memories of M, is dened as: reach(m) = fw 2 j9k 0; 9w 1 i ; : : : ; w k i ; w 1 o; : : : ; w k o ; s 1 ; : : : ; s k 2 : hw 1 i ; ; ij=) M hw 1 i ; s1 ; w 1 oi; hw 2 i ; s1 ; ij=) M hw 2 i ; s2 ; w 2 oi; : : : ; hw k i ; sk?1 ; ij=) M hw k i ; sk ; w k oi and w = s k g 2 As noted above, we will show that natural partitionings of M and T are isomorphic. For T, the partition in question will be the one induced by ITS isomorphism (Denition 4.2), and for M, it will be the partition induced by \macrostep equivalence", which we now dene. Denition 5.2 Two PTMs M 1 ; M 2 are macrostep equivalent, notation M 1 = ms M 2, if there exists a bijection : reach(m 1 )! reach(m 2 ) such that: 1. () = 2. 8w i ; w o 2 ; s; s 0 2 reach(m 1 ) : hw i ; s; ij=) M1 hw i ; s 0 ; w o i i hw i ; (s); ij=) M2 hw i ; (s 0 ); w o i Note that = ms is an equivalence relation over M. 2 The mapping : M! T is given by (M) = hreach(m); ; m; i, where hs; w i ; s 0 ; w o i 2 m 1 i hw i ; s; ij=) M hw i ; s 0 ; w o i. Note that (M) is indeed an ITS, as reach(m) is enumerable, m is eective, and the set of states of (M) is reachable from its root. For any M, (M) lockstepsimulates M and vice versa. Furthermore, P SL(M) = ISL((M)). Example 5.1 The ITS of Figure 1 depicts the image, under, of the PTM M Latch of Example 3.1. Transitions such as (1; 0) represent the innite family of transitions where, upon receiving a bit string starting with 1 as input, the ITS outputs a 0. 2 The following proposition shows that maps equivalent PTMs to equivalent ITSs. Proposition 5.1 For all M 1 ; M 2 2 M; M 1 = ms M 2 i (M 1 ) = iso (M 2 ). Proof: Set in the denition of = iso (Denition 4.2) to the in the denition of = ms (Definition 5.2), for the )-direction of the proof, and vice versa for the (-direction of the proof. 2 The following proposition shows that the mapping is surjective (up to = iso ). 9

10 (1*, 1) 1 (1*, 1) # (0*, 1) (1*, 0) (0*, 1) 0 (0*, 0) Figure 1: (M Latch ) Proposition 5.2 For all T 2 T, there exists M 0 2 M such that T = iso (M 0 ). Proof: Let T = hs; ; m; i. Since m is recursive, there exists a bijective encoding! of S into strings over and a PTM M 2 M such that!(r) = and Let T 0 = hs 0 ; ; m 0 ; i, where hs; w i ; s 0 ; w o i 2 m i hw i ;!(s); ij=) M hw i ;!(s 0 ); w o i S 0 = f!(s) j s 2 Sg; m 0 = fh!(s); w i ;!(s 0 ); w o i j hs; w i ; s 0 ; w o i 2 mg It is clear that (M) = T 0. Also, T 0 = iso T, where! is the desired mapping. 2 Figure 2 illustrates Propositions 5.1 and 5.2. The class M of PTMs, partitioned into equivalence classes up to = ms The class T of ITSs, partitioned into equivalence classes up to = iso M 1 ξ M 2 M 0 T (1) for any M 1 and M 2, M 1 = ms M 2 iff ξ(m 1 ) = iso ξ(m 2 ); (2) for any T, there exists M 0 s.t. T = iso ξ(m 0 ) Figure 2: Properties of the mapping : M! T The main result of this section, which essentially allows one to view persistent Turing machines and interactive transition systems as one and the same, now follows. Theorem 5.1 M= = ms and T = = iso are isomorphic. Proof: Let : 2 M! 2 T be the mapping given by for any X M; (X) = ft j 9M 2 X : T = iso (M)g. It follows from Propositions 5.1 and 5.2 that is the desired isomorphism. 2 10

11 6 Conclusions We have presented Persistent Turing Machines (PTMs), an extension of the classical Turingmachine model with appropriate notions of interaction and persistency. Our main results are that the class of PTM-based persistent stream languages strictly contains the class of non-persistent (amnesic) stream languages, and that (natural quotients of) the space of PTMs and the space of ITSs are isomorphic. For future work, we would like to extend the PTM and ITS models to networks of the same and develop a corresponding process algebra. It would also be interesting to compare the expressive power of such networks to that of traditional dataow models of computation such as [Kah74, Den75, KM77, Lee97]. Acknowledgements: We would like to thank Peter Fejer for his valuable technical feedback on an earlier draft of this manuscript, and Jan van Leeuwen for interesting exchanges on interactive computing. References [BBK87] J.C.M. Baeten, J.A. Bergstra, and J.W. Klop. On the consistency of Koomen's fair abstraction rule. Theoretical Computer Science, 51(1/2):129{176, [BCLGH94] A. Benveniste, P. Caspi, P. Le Guernic, and N. Halbwachs. Data-ow synchronous languages. In J. W. de Bakker, W.-P. de Roever, and G. Rozenberg, editors, A Decade of Concurrency Reections and Perspectives. LNCS 803, Springer-Verlag, [BIM88] [BM96] B. S. Bloom, S. Istrail, and A. R. Meyer. Bisimulation can't be traced. In Proceedings of the 15th ACM Symposium on Principles of Programming Languages, J. Barwise and L. Moss. Vicious Circles. CSLI Lecture Notes #60. Cambridge University Press, [Bou85] G. Boudol. Notes on algebraic calculi of processes. In K. Apt, editor, Logics and Models of Concurrent Systems, pages 261{303. LNCS, Springer-Verlag, [Dar90] [Den75] [ds85] P. Darondeau. Concurrency and computability. In I. Guessarian, editor, Semantics of Systems of Concurrent Processes, Proceedings LITP Spring School on Theoretical Computer Science, La Roche Posay, France, LNCS 469, Springer-Verlag. J. B. Dennis. First Version Data Flow Procedure Language. Technical Memo MAC TM61, MIT, Laboratory for Computer Science, May R. de Simone. Higher-level synchronizing devices in meije-sccs. Theoretical Computer Science, 37:245{267, [Gol00] D. Goldin. Persistent Turing machines as a model of interactive computation. In Proceedings of FoIKS 2000, Burg, Germany, February [GST00] D. Goldin, S. Srinivasa, and B. Thalheim. Information systems = databases + interaction: Towards principles of information system design. In Proceedings of ER 2000, Salt Lake City, Utah,

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