Composability of Infinite-State Activity Automata*
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1 Composability of Infinite-State Activity Automata* Zhe Dang 1, Oscar H. Ibarra 2, Jianwen Su 2 1 Washington State University, Pullman 2 University of California, Santa Barbara Presented by Prof. Hsu-Chun Yen, NTU (Thanks Hsu-Chun!) *Supported in part by NSF Grants IIS and CCR
2 Outline Motivation Modeling by automata Composability results Online delegators Delegators with lookahead Constrained delegation Timed composabilty Summary and references
3 E-Services E-services provide a general framework for discovery, flexible interoperation, and dynamic composition of distributed and heterogeneous processes on the Internet. Some issues: modeling, analysis, composition. Automated composition allows a new e-servicee A to be implemented by composing existing e-services A 1,, A r.
4 Example Existing E-ServicesE Bank Warehouse Online Store Credit Card Company Composability of e-services e asks whether a new service can be created by using existing services. New E-ServiceE
5 Automated Composition Using existing e-services e to construct a new desired e-servicee Models -- Message-oriented modeling -- Activity-based modeling This paper
6 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
7 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
8 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
9 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
10 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
11 Examples R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. (A; A 1, A 2 ) is composable. (A; A 2, A 3 ) is also composable. (A; A 1, A 3 ) is not composable.
12 Modeling by Automata Use of automata to model e-services: e an activity is represented by a symbol in the input alphabet of an automaton. A desired sequence of activities to be performed is a string in the language accepted by the automaton. Mostly finite automata have been studied in the literature.
13 Composability Definition: A system (A;( A 1,, A r ) is composable if there is a composer C such that for every string w = a 1 a n accepted by A, C assigns each symbol in w to one of the A i s In general, C is nondeterministic, and at every step, C may have to guess the index 1 i r to output, i.e., the A i to assign the current input symbol.
14 Notation DFA (NFA): Deterministic (Nondeterministic) Finite Automaton DCM (NCM): A DFA (NFA) augmented with 1-reversal counters.
15 Counters A counter x is a nonnegative integer variable associated with operations: (Increment) x := x + 1 (Decrement) x := x - 1 (Stay) x := x (Test) Is x = 0? 0
16 reversal reversal reversal Note: A counter making r reversals can be simulated by (r + 1) / 2 counters, each making only 1 reversal.
17 Known (Language) containment & equivalence problems are: 1. decidable for DCMs [Ibarra], 2. undecidable for NCMs (even when there is only one 1-reversal counter) [Book&Baker[ Book&Baker]. Membership, emptiness, and disjointness problems are decidable for NCMs [Book&Baker]. If the counters are unrestricted (not reversal-bounded), the machine is equivalent to a Turing machine (even when there are only two counters) [Minsky[ Minsky] hence, all nontrivial problems are undecidable.
18 Theorem: If A is an NCM and A 1,, A r are NFAs,, then composability of (A;( A 1,, A r ) is decidable. Thus, if the target service A has 1-reversal 1 counters and the existing services A i s s are NFAs, it is decidable to determine if every string accepted by A can be composed in terms of strings accepted by the A i s.
19 Theorem: It is undecidable to determine, given a system (A;( A 1, A 2 ), where A and A 1 are DFAs and A 2 is a DCM with only one 1-reversal counter, whether it is composable. Proof (idea): Intricate reduction to the halting problem for two-counter machines (which is undecidable [Minsky]).
20 When the system (A;( A 1,, A r ) is composable,, the composer is a nondeterministic machine, in general. Thus, in general, assignment of symbols of a string w accepted by A to the A i s s can only be done deterministically after the entire string has been processed (i.e., offline) and can not be done online.
21 Online Delegator Definition: Given a system (A;( A 1,, A r ) an online delegator for the system is a deterministic acceptor D with outputs which, knowing: - the current states of A, A 1,, A r and local information of each machine (e.g., whether a counter is zero or non-zero if each machine is a DCM) - current input symbol being processed D can uniquely determine the A i to assign the current symbol.
22 Online Delegation Existing Services New Service Can we provide new service by using existing services
23 Online assignment of activities Existing Services New Service 23
24 Theorem: When the system consists of NCMs,, existence of an online delegator can be decided in 2^( ^(2^( ^(c c m n log n)) n time for some constant c,, where m is the total number of 1-reversal 1 counters in the system and n is the size of the system. However, when the system consists of DFAs,, the time reduces to 2^(c c n log n) n [BC+03] (can be shown to hold for NFAs also [CHIS04]).
25 What if online delegator does not exist? Existing Services New Service Who processes book_taxi? Assume book_air_travel processes. However, if the next activity is book_train, then system gets stuck.
26 Delegator with 1 Lookahead Knowing 1 future activity is enough for composition i.e., assignment of an activity is delayed one step
27 Hierarchy on Lookaheads There exists a hierarchy on lookaheads. (a) 1-lookahead, (b) 2-lookahead, (c) unbounded 27
28 Systems With Lookahead Definition: Given a system (A;( A 1,, A r ) of acceptors in M and a non-negative negative integer k,, a k-lookahead delegator for the system is a deterministic acceptor D in M with outputs (with delay k) which, knowing: - the current states of A, A 1,, A r and local information of each machine - current input symbol being processed - the k lookahead symbols to the right of the current input symbol D can uniquely determine the A i to assign the current symbol. Moreover, for every string w accepted by A,, the subsequence of the string w delegated by D to each A i is accepted by A i. Note: Online delegator corresponds to k = 0.
29 Lookahead Delegators R: registration A: accesses C: credit card payment S: cash payment Online club A offers customers to first register and then pay for their accesses with either cash or credit cards. A accepts (r( (aa( aa*( *(s c)))*. Clearly, (A;( A 2, A 3 ) is not only composable but it has a 1-lookahead 1 delegator. (A; A 1, A 2 ) is composable but has no k- lookahead delegator for any k.
30 k-delegator The k-delegator D starts out by reading the first k input symbols and stores them in a buffer in its finite control. It then simulates A, A 1,, A k and deterministically assigns the symbols processed by A to the A i s s while also updating the buffer as it reads the remaining input symbols. Thus D assigns the symbols processed by A to the A i s s with delay k.. For convenience, so that there is always a k lookahead string to the right of the current symbol, we assume that the input to the machines is suffixed by # s.#
31 Theorem: When the system consists of NCMs, existence of a k-delegator for a given k can be decided in ^(2^( ^(c c m n log n t k )) time for some constant c,, where m is the total number of 1-1 reversal counters in the system, n is the size of the system, and t is size of the input alphabet. However, when the system consists of NFAs,, the time reduces to 2^( ^(c c n log n t k ) [CHIS04].
32 Constrained Delegation (Presburger constraints on delegation of symbols) Load balancing or priority specification: -- Svc 1 processes as many activities as Svc 2 -- Priority of Svc 1 is higher than Svc 2 Cost of delegation: -- Cost for Svc 1 < 3 x Cost for Svc 2 -- Cheapest way to delegate activities -- Total time should be < 20 seconds Algorithms for constrained delegation (e.g., for k-constrained delegation).
33 Open Question A system may be composable but may not have a k-delegator for any k. Open: Is it decidable if a system of DFAs has a k- delegator for some k?
34 Timed Composability A timed automaton can be considered as a finite automaton augmented with a finite number of clocks. The clocks can reset to zero or progress at the same rate, and can be tested against clock constraints [Alur[ Alur/Dill]. Timed automata are widely regarded as a standard model for real-time systems, because of their ability to express quantitative time requirements. Theorem: Composability of discrete timed automata (A;( A 1,, A r ) is decidable.
35 Summary Showed decidable and undecidable results for composability and k-delegation for various types of machines modeling e-services. e Looked at putting additional requirements such as Presburger constraints and timing in composition and k- delegation. We want to study other models of e-services e (e.g., automata with queues, etc.) Open question: Does the system have a k-delegator for some k. Seems quite difficult, even for the case when all the e-services e are DFAs. Characterize the complexity of the decision procedures. E.g., it can be shown that composability of a system of NFAs is at least NP-hard.
36 References [BCG+03] D. Berardi,, D. Calvanese,, G. De Giacomo,, M. Lenzerini,, and M. Mecella, Automatic Composition of E-Services E That Export Their Behavior,, Proc. Int. Conf. On Service Oriented Computing, [GHIS04] C. E. Gerede,, R. Hull, O.H. Ibarra, and J. Su, Automated Composition of E-E Services : Lookaheads,, Proc. Int. Conf. On Service Oriented Computing 2004.
37 Thank You!
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