Angluin, Dana. 1982. A note on the number of queries needed to identify regular languages. Information Control, 51:76--87.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of an equivalence oracle, their algorithm learns a homing sequence, which it uses in place of a reset function. It then runs several copies of Angluin s algorithm [5] for learning DFAs given a reset. Angluin has shown that any algorithm for actively learning DFAs requires an equivalence oracle [4]. In this chapter, we introduce a new type of homing sequence for two robots. Because of the strength of the homing sequence, our algorithm does not require an equivalence oracle. For any graph, the expected running time of our algorithm is O(d2s) In practice, our algorithm can use additional ....

....a teacher to provide counterexamples. In fact, two robots on a graph define a DFA whose states are pairs of nodes in G and whose edges correspond to pairs of actions. Since the automata defined in this way form a restricted class of DFAs, our results are not inconsistent with Angluin s work [4] showing that a teacher is necessary for learning general DFAs. Theorem 6 Every strongly connected directed graph has a two robot homing sequence. Proof: The following algorithm (based on that of Kohavi [69, 93] constructs a homing sequence: Initially, let h be empty. As long as there are two ....

Dana Angluin. A note on the number of queries needed to identify regular lan- guages. Itformatiot atd ottrol, 51:76 $7, 1981.

....the signer by asking for his signature on some specially constructed messages is devastating and allows the attacker to determine the factorization of the signer s modulus a total break. The machine learning community has explored similar scenarios, following the pioneering work of Angluin [2, 3]. For example, the learner may be permitted membership queries asking for the value of the unknown function on some specified input or equivalence queries asking if a specified conjectured hypothesis is indeed equivalent to the unknown target hypothesis. If the conjecture is incorrect, ....

....queries asking if a specified conjectured hypothesis is indeed equivalent to the unknown target hypothesis. If the conjecture is incorrect, the learner is given a counterexample an input on which the conjectured and the target functions give di#erent results. For example, Angluin [2] has shown that a polynomial number of membership and equivalence queries are su#cient to exactly identify any regular set (finite automaton) whereas the problem of learning a regular set from random examples is NP complete [1] Even if information is gathered from random examples, ....

Dana Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

....E i (ff) E i Gamma1 (ff) fv i g; else E i (ff) E i Gamma1 (ff) Extract the automaton M for L from the sets E i and T (see text) Fig. 2. Angluin s ID algorithm. The goal of the ID algorithm (Figure 2) is to construct a partition of T 0 that places all the equivalent elements in one state [2]. The equivalence relation is the Nerode relation such that the resulting DFA will be minimal [1] The algorithm starts with an initial partition of one accepting and one non accepting state and refines it successively. In each step i of ID a string v i is drawn such that for any two states q ....

.... to be [1; 2; n] For instance, the array a = 3; 1; 2] of length n = 3 leads to the following instantiation for question (3) int i=2; 2=i 0; T; int j=0; 2=a[0] a[1] 1; T; swap(0,1) j ; 1=j i=2; T; i=2; 2 L The user responds by replacing i=2 by the next step which compares 3 = a[1] a[2] = 2, i.e. the test a[j] a[j 1] Figure 4 depicts the finite state machine for the bubble sort program inferred by the ID algorithm. All states are accepting, and all omitted transitions lead to the dead state d 0 . 4.4 Query Complexity Every affirmatively answered membership question and every ....

D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51(1):76--87, 1981.

....However, most of them are heuristic. They learn faster from some good data, but fail to learn (in the limit) from bad data; little is done to characterize the good data for efficient learning. Some theoretical studies on learning from good examples have been explored. For example, Angluin [ Angluin, 1981 ] defines representatives for live states as good strings for identifying a DFA polynomially. Rivest and Sloan [ Rivest and Sloan, 1988 ] show how to learn an arbitrary concept by first learning its relevant subconcepts. Freivalds et al. [ Freivalds et al. 1989 ] study good examples in ....

....characterizations that are sought in inductive learning. To be successful in learning, the learner must receive data that go through (use or exercise) every live component of one of the finite characterizations 4 . For example, data that go through all live transactions in a DFA (see [ Angluin, 1981 ] all live production rules in a grammar for a language, or all live Horn clauses in a logic program 5 . Clearly such a set of data provides a minimum amount of information about the theory to be identified, and should be supplied in the early stage of learning. We call this set a basic ....

D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51, 1981.

....Related Results In the error free case of learning DFAs, Kearns and Valiant [24] use the prediction preserving reductions of Pitt and Warmuth [28] to show that under cryptographic assumptions, the problem of predicting the class of DFAs is hard when only given access to random examples. Angluin [1] shows that (exact) learning using only membership queries is also hard. She describes a family of automata that cannot be identified in less than exponential time when the learner can only observe the behavior of the machine on inputs of the learner s own choosing. However, as mentioned earlier, ....

....Luby, Michael Kearns, Nati Linial and Joan Feigenbaum for comments and helpful discussions. We would also like to thank the referees of this paper for their careful reading and very helpful comments. 36 DANA RON AND RONITT RUBINFELD Notes 1. 11] actually mention that the adversary argument in [1] showing that DFAs cannot be learned using membership queries only can be modified so that the target DFAs all have distinguishing sequences. This implies that even in the error free case one cannot hope to find a distinguishing sequence in polynomial time by exploration only. This still leaves ....

D. Angluin. (1981). A note on the number of queries needed to identify regular languages. Information and Control, 51 , 76--87.

....are easily extendible to larger alphabets. Our results follow the no reset learning algorithm of Rivest and Schapire [10] which in turn uses Angluin s algorithm [2] for learning automata with reset) as a subroutine. We use a variant of Angluin s algorithm which is similar to the one described in [1]. As in [10] we use a homing sequence to overcome the absence of a reset, only we are able to construct such a sequence without the aid of a teacher, while Rivest and Schapire need a teacher to answer their equivalence queries and supply them with counterexamples for their incorrect hypotheses. ....

....automaton has a distinguishing sequence. Moreover, even if the target automaton is known to have a distinguishing sequence, then there is not necessarily an efficient procedure for finding one such sequence, and this remains true when the automaton has small cover time. Other Related Work Angluin [1] proves that it is hard to exactly learn automata (even with reset) when the learner has access to a membership oracle but not to an equivalence oracle. However, this does not contradict our results, as the adversary argument made by Angluin uses automata which have hard to reach states and hence ....

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Dana Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

....Deutsche Forschungsgemeinschaft (DFG) grant Am 60 9 1. We propose a complexity theoretical approach to strategy learning by restricting our domain to games and trees which are represented by deterministic finite automata [5, 15, 18] and by defining a natural and adequate query learning model [1, 2, 3, 4] for this domain. Thus, we are able to investigate the query and time complexity of branch and strategy learning, where the complexity is measured in the size of the automaton A representing the target tree game, i.e. the number of states of the minimal automaton equivalent to A. The complexity ....

....T is a query of the form uv , which is answered by yes if uv is an infinite branch of T , and by no otherwise. We will also consider the notion of membership query which is widely used in computational learning theory, in particular in the context of learning finite automata, e.g. in [1, 2]. By a membership query the learner can test, whether a finite word is on the tree given by an FST: Definition 2.4 A membership query u 2 Sigma for an FST T is answered by yes , if u 2 L(T ) and by no otherwise. We are looking for algorithms which efficiently learn branches of the form ....

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51(1):76--87, 1981.

....then clearly, the example is positive. When the structurally complete sample set has been acquired, an example that is not accepted by all FSA in G k can be classified as negative. The idea of incremental lattice update was inspired by Hirsh s work on Incremental Version Space Merging [8] Angluin [1] has proposed an algorithm ID to infer the target grammar from a live complete set of examples using a polynomial number of membership queries. The live complete set can be constructed given a structurally complete sample. A trivial upper bound on the number of membership queries needed in our ....

Angluin, D. A Note on the Number of Queries Needed to Identify Regular Languages. Information and control, 51. 1981. pp 76-87.

.... on Incremental Version Space Merging [8] The set G in our algorithm, which represents the set of most general FSA of the lattice that do not accept any negative strings identified by the queries during the inference process is analogous to the border set described by Dupont et al. [4] Angluin [1] has proposed an algorithm (ID) to infer the target grammar from a live complete set of examples (which can be constructed from a structurally complete set) using a polynomial number of membership queries. A trivial upper bound on the number of queries needed in our procedure is exponential in the ....

Angluin, D. A Note on the Number of Queries Needed to Identify Regular Languages. Information and control, 51. '81. pp 76-87.

.... out such important methods as reinforcement learning [8] We propose a complexity theoretical approach to strategy learning by restricting our domain to games and trees which are represented by deterministic finite automata [5, 12, 15] and by defining a natural and adequate query learning model [1 4] for this domain. So we investigate the query and time complexity of branch and strategy learning, where the complexity is measured in the size of the automaton A representing the target tree game, i.e. the number of states of the minimal automaton equivalent to A. Our work is motivated by that ....

....T is a query of the form uv , which is answered by yes if uv is an infinite branch of T , and by no otherwise. We will also consider the notion of membership query which is widely used in computational learning theory, in particular in the context of learning finite automata, e.g. in [1, 2]. By a membership query the learner can test, whether a finite word is on the tree given by an FST: Definition 4. A membership query u 2 Sigma for an FST T is answered by yes , if u 2 L(T ) and by no otherwise. We are looking for algorithms which efficiently learn branches of the form uv ....

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51(1):76--87, 1981.

....the learner with access to sources of additional information, like a knowledgeable teacher who responds to queries generated by the learner. Angluin s ID algorithm learns the target DFA given a live complete sample and a knowledgeable teacher to answer membership queries posed by the learner [Ang81]. The interested reader is referred to [MQ86,Pit89,Lan95,PH98] for recent surveys of different approaches to grammar inference. In many practical learning scenarios, a live complete sample may not be available to the learner at the outset. Instead, a sequence of labeled examples is provided ....

....is not live is called dead. As stated earlier, a canonical DFA can have at most one dead state and we use d 0 to denote this dead state. Given A, a finite set of strings P is said to be live complete if for every live state q i of A there exists a string ff 2 P such that ffi (q 0 ; ff) q i [Ang81]. For example, P = f; a; b; aag is a live complete set for the DFA in Fig. 1. Any superset of a live complete set is also live complete. In order to have a representation for the start state of A we assume that the string is part of any live complete set. The set P 0 = P [fd 0 g represents all ....

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

....situation in automata is much more involved. Two states in an automaton can be indistinguishable by any experiment and still be unmergeable due to the role they play in the infinite behavior of the automaton. As a consequence we can see that the notion of a representative sample introduced by [Angluin (1981)] should be refined in the infinitary case to distinguish between examples of the finite and the infinite behavior of the automaton. In section 2 we introduce the essentials of the theory of regular sets while in section 3 we give a short survey of learnability results for the finitary case. In ....

....of all words up to a given length) Trakhtenbrot and Barzdin (1973) gave a polynomial algorithm for this problem. Angluin (1987) extended the negative results to samples that are almost (up to polynomial proportion of missing entries) uniformcomplete. The positive results were extended by [Angluin (1981)] to representative samples, which include words that excercise every transition in the minimal automaton of the set. A more recent survey of complexity issues appears in [Pitt (1989) The algorithm L , due to [Angluin (1987) is an interactive algorithm which uses two oracles: MEMBER(u) ....

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Angluin, D. (1981), A Note on the Number of Queries Needed to Identify Regular Languages, Inform. and Control 51, 76-87.

....the actions of a learner, selecting actions queries that are statistically expected to minimize its uncertainty and error. 1. 1 Active learning Exploiting the active component of learning typically leads to improved generalization, usually at the cost of additional computation (see Figure 1) [Angluin, 1982; Cohn et al. 1990; Hwang et al. 1991] 1 There are two common situations where this tradeoff is desirable: In many situations the cost of taking an action outweighs the cost of the computation required to incorporate new information into the model. In these cases we wish to select queries ....

D. Angluin. (1982) A note on the number of queries needed to identify regular languages. Inform. Control, 51:76--87.

....from a teacher that can answer both membership queries and equivalence queries. Note that having a teacher which answers membership queries is equivalent to having means of a reset. We use as a subroutine of our algorithm a variant of Angluin s algorithm which is similar to the one described in [1]. As in [19] we use a homing sequence to overcome the absence of a reset, only we are able to construct such a sequence without the aid of a teacher, while Rivest and Schapire s learner needs a teacher to answer its equivalence queries and supply it with counterexamples for its incorrect ....

....is required to output hypotheses that converge in the limit (of the number of examples) to the target DFA. We refer the reader to a survey by Angluin and Smith [5] Here we briefly survey the known efficient learning algorithms for DFAs. We start with the problem of exactly learning DFAs: Angluin [1] proves that it is hard to exactly learn DFA when the learner has access to a membership oracle (which, as noted previously, is equivalent to having means of a reset) but not to an equivalence oracle. However, this does not contradict our results, as the adversary argument made by Angluin uses ....

[Article contains additional citation context not shown here]

D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

.... the string is accepted by the target DFA or not [28] Angluin showed that given a live complete set of examples (that contains a representative string for each live state of the target DFA) and a knowledgeable teacher to answer membership queries it is possible to exactly learn the target DFA [1]. In a later paper, Angluin relaxed the requirement of a live complete set and has designed a polynomial time inference algorithm using both membership and equivalence queries [2] The regular positive and negative inference (RPNI) algorithm is a framework for identifying in polynomial time, a ....

D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

.... by the target DFA or not [Trakhtenbrot Barzdin, 1973] Angluin showed that given a live complete set of examples (that contains a representative string for each live state of the target DFA) and a knowledgeable teacher to answer membership queries it is possible to exactly learn the target DFA [Angluin, 1981]. In a later paper, Angluin relaxed the requirement of a live complete set and has designed a polynomial time inference algorithm using both membership and equivalence queries [Angluin, 1987] The RPNI algorithm is a framework for identifying a DFA consistent with a given sample S in polynomial ....

Angluin, D. (1981). A Note on the Number of Queries Needed to Identify Regular Languages. Information and Control, 51, 76--87.

....whether s 2 L or not. The inference methods for this type of presentation yield a description D of L after making some queries about L. A mixed presentation by both examples and queries is also possible, in which the examples are used to infer an initial description to be refined later by queries [5, 23]. The most part of the work in GI has been devoted to the problem of inferring regular grammars (or finite state automata) 15, 21] Nevertheless, just a few methods have been proposed for regular GI from complete presentation, either in the classical symbolic approach [1, 14, 22] or in ....

D. Angluin, "A note on the number of queries needed to identify regular languages," Information & Control 51, pp.76-87, 1981.

....alencequnce D The extendedeqund alencequnc takes a context free grammar as aninpu while the targetlanguDD muD be only simple deterministic. Therefore a learning algorithm whichu=fl standardequd alence qunce is expected. There is another approach ofqujj learning in polynomial time. AngluD [1] also showed the polynomial time learnability ofregu=B languB W from membership quhip and a representative sample. Here, for a deterministic finiteaute D5F M , a representative sample Q is a finite sute of the targetlangu = L(M) su h that all transitions of M are traced to accept all words in ....

....L, a finite set Q # L is a representative sample i# there exists a 2 standard form simple deterministic grammar G = N,#,P,S)su h that L(G) L and Q is a representative sample of G. # This is anatuSfl extension of the definition of a representative sample for areguF# langu# by AngluG [1]. Example 3: Let G = N,#,P,S) be a simple deterministic grammarsu h that N = S, A, B , # = a, b, c and P = S # aSA, S # bSB, S # c, A # a, B # b . Then abcba is a representative sample of the grammar G, becauS allru # in P are applied in the derivation for abcba ....

D. Angluin, "A note on the number of queries needed to identify regular languages," Inf. Control, vol.51, no.1, pp.76-- 87, Oct. 1981.

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Angluin, Dana. 1982. A note on the number of queries needed to identify regular languages. Information Control, 51:76--87.

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76--87, 1981.

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Dana Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51:76-87, 1981.

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D. Angluin. A note on the number of queries needed to identify regular languages. Inf. & Cntl., 51:76--87, 1981.

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D. Angluin. A note on the number of queries needed to identify regular languages. Information and Control, 51#1#:76#87, 1981.

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