## Elementary formal systems, intrinsic complexity, and procrastination (1997)

Venue: | Information and Computation |

Citations: | 13 - 6 self |

### BibTeX

@ARTICLE{Jain97elementaryformal,

author = {Sanjay Jain and Arun Sharma},

title = {Elementary formal systems, intrinsic complexity, and procrastination},

journal = {Information and Computation},

year = {1997},

volume = {132},

pages = {65--84}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recently, rich subclasses of elementary formal systems (EFS) have been shown to be identifiable in the limit from only positive data. Examples of these classes are Angluin’s pattern languages, unions of pattern languages by Wright and Shinohara, and classes of languages definable by length-bounded elementary formal systems studied by Shinohara. The present paper employs two distinct bodies of abstract studies in the inductive inference literature to analyze the learnability of these concrete classes. The first approach, introduced by Freivalds and Smith, uses constructive ordinals to bound the number of mind changes. ω denotes the first limit ordinal. An ordinal mind change bound of ω means that identification can be carried out by a learner that after examining some element(s) of the language announces an upper bound on the number of mind changes it will make before converging; a bound of ω · 2 means that the learner reserves the right to revise this upper bound once; a bound of ω · 3 means the learner reserves the right to revise this upper bound twice, and so on. A bound of ω 2 means that identification can be carried out by a learner that announces an upper bound on the number of times it may revise its conjectured upper bound on the number of mind changes. It is shown in the present paper that the ordinal mind change complexity for identification of languages formed by unions of up to n pattern languages is ω n. It is

### Citations

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Language identification in the limit
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Citation Context ...i (written: M(T ) converges to i) just in case for all but finitely many n, M(T [n]) = i. The following definition introduces Gold’s criterion for successful identification of languages. Definition 3 =-=[Gol67]-=- (a) M TxtEx-identifies a text T just in case M(T ) converges to a grammar for content(T ). (b) M TxtEx-identifies an r.e. language L (written: L ∈ TxtEx(M)) just in case M TxtEx-identifies each text ... |

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Citation Context ...s of EFS, intrinsic complexity, and procrastination. Results are presented in Section 3. 2 Preliminaries N + denotes the set of positive integers. Any unexplained recursion theoretic notation is from =-=[Rog67]-=-. Cardinality of a set S is denoted by card(S). ∅ denotes the empty set. The maximum and minimum of a set are denoted by max(·), min(·), respectively, where max(∅) = 0 and min(∅) = ∞. We let 〈·, ·〉 st... |

208 |
Finding patterns common to a set of strings
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Citation Context ...programs. We consider three subclasses of elementary formal systems in the present paper. The smallest of these classes, the collection of pattern languages (PATTERN), was first introduced by Angluin =-=[Ang80]-=- who showed that this class can be identified in the limit from only positive data. Shinohara [Shi86] showed that the class of pattern languages is not closed under union and many rich concepts can be... |

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Theory of Formal Systems
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(Show Context)
Citation Context ...g, the present paper demonstrates the possibility of using abstract results of inductive inference to gain insights into inductive logic programming. 1 Introduction Arikawa [Ari70] adapted Smullyan’s =-=[Smu61]-=- elementary formal systems (EFS) for investigation of formal languages. Later, Arikawa et al. [ASY92] showed that EFS can also be treated as a logic programming language. Recently various subclasses o... |

34 |
On the role of procrastination in machine learning
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(Show Context)
Citation Context ...n the present paper we employ two distinct bodies of work in the inductive inference literature to analyze the learnability of the above classes. The first approach, introduced by Freivalds and Smith =-=[FS92]-=-, involves the use of constructive ordinals to bound the number of mind changes before the onset of convergence. The second approach ([FKS95, JS94, JS95]) involves the use of reductions to study the i... |

30 | Learning elementary formal systems
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(Show Context)
Citation Context ... gain insights into inductive logic programming. 1 Introduction Arikawa [Ari70] adapted Smullyan’s [Smu61] elementary formal systems (EFS) for investigation of formal languages. Later, Arikawa et al. =-=[ASY92]-=- showed that EFS can also be treated as a logic programming language. Recently various subclasses of EFS have been investigated in the context of learnability. It has been shown that rich classes can ... |

30 |
Identification of unions of languages drawn from an identifiable class, in
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- 1989
(Show Context)
Citation Context ...unions of pattern languages. He also showed that the class of languages formed by union of up to 2 pattern 2slanguages (PATTERN 2 ) is identifiable in the limit from only positive data. Later, Wright =-=[Wri89]-=- generalized this result to show that the classes of languages formed by unions of up to n pattern languages (PATTERN n ) can be identified in the limit from only positive data. Shinohara [Shi94] late... |

26 |
Inductive inference of monotonic formal systems from positive data. New Generation Computing 8 (4):371–384. Special Issue on Algorithmic Learning Theory for ALT’90
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Citation Context ...l logic programs, e.g., the learnability of the class LBEFS n implies the learnability of the class of minimal models of linear Prolog programs consisting of at most n definite clauses (see Shinohara =-=[Shi91]-=- and Arimura [Ari89]). In this respect, these results are also about inductive logic programming. In the present paper we employ two distinct bodies of work in the inductive inference literature to an... |

25 | On the intrinsic complexity of learning
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Citation Context ...EFS and Intrinsic Complexity Recently, a new approach to the study of “intrinsic” complexity of learning has been proposed for identification in the limit of functions by Freivalds, Kinber, and Smith =-=[FKS95]-=- and for identification in the limit of languages by Jain and Sharma [JS94, JS96a]. The main idea of the approach is to introduce reductions between learnable classes of languages. If a collection of ... |

17 | On the intrinsic complexity of language identification
- Jain, Sharma
- 1994
(Show Context)
Citation Context ...ction of finite languages. It was shown that SINGLE is reducible to COINIT but COINIT is not reducible to SINGLE and COINIT is reducible to FIN but FIN is not reducible to COINIT. It was discussed in =-=[JS94]-=- that this reduction captures the intuitive difficulty of learning these classes. SINGLE can be identified by a learning machine that can confirm its success. COINIT cannot be identified by any machin... |

16 |
The correct definition of finite elasticity: Corrigendum to identification of unions
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(Show Context)
Citation Context ...hat if a class has finite thickness, then it is not complete in terms of any of the reductions for intrinsic complexity discussed in this paper. Wright [Wri89] (see also Motoki, Shinohara, and Wright =-=[MSW91]-=-) introduced the notions of finite elasticity and infinite elasticity (to be defined later). Wright [Wri89] showed that if a class L has finite thickness then it has finite elasticity. He also showed ... |

15 | The structure of intrinsic complexity of learning - Jain, Sharma - 1995 |

11 | Inductive inference of Prolog programs with linear data dependency from positive data
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(Show Context)
Citation Context ...ons of the learnability of subclasses of elementary formal systems are important because they yield corresponding results about the learnability of subclasses of logic programs. Arimura and Shinohara =-=[AS94]-=- have used the insight gained from the learnability of EFS subclasses to show that a class of linearly covering logic programs with local variables is identifiable in the limit from only positive data... |

10 | Not-so-nearly-minimal-size program inference - Case, Jain, et al. - 1995 |

9 | A.Yamamota. Algorithmic learning theory with elementary formal systems - Arikawa, Miyano, et al. - 1992 |

7 |
Elementary formal systems and formal languages - Simple formal systems
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(Show Context)
Citation Context ...raditional logic programming, the present paper demonstrates the possibility of using abstract results of inductive inference to gain insights into inductive logic programming. 1 Introduction Arikawa =-=[Ari70]-=- adapted Smullyan’s [Smu61] elementary formal systems (EFS) for investigation of formal languages. Later, Arikawa et al. [ASY92] showed that EFS can also be treated as a logic programming language. Re... |

7 |
Studies on inductive inference from positive data
- Shinohara
- 1986
(Show Context)
Citation Context ...t of these classes, the collection of pattern languages (PATTERN), was first introduced by Angluin [Ang80] who showed that this class can be identified in the limit from only positive data. Shinohara =-=[Shi86]-=- showed that the class of pattern languages is not closed under union and many rich concepts can be represented by unions of pattern languages. He also showed that the class of languages formed by uni... |

6 |
A class of Prolog programs inferable from positive data
- Rao
- 1996
(Show Context)
Citation Context ... from the learnability of EFS subclasses to show that a class of linearly covering logic programs with local variables is identifiable in the limit from only positive data. More recently, Krishna-Rao =-=[KR96]-=- has established the learnability from only positive data of an even larger class of logic programs. We consider three subclasses of elementary formal systems in the present paper. The smallest of the... |

6 |
classes inferable from positive data: Length–bounded elementary formal systems
- Rich
- 1994
(Show Context)
Citation Context ...ight [Wri89] generalized this result to show that the classes of languages formed by unions of up to n pattern languages (PATTERN n ) can be identified in the limit from only positive data. Shinohara =-=[Shi94]-=- later showed that an even richer class, the classes of languages definable by length-bounded elementary formal systems with up to n clauses (LBEFS n ), is identifiable in the limit from only positive... |

3 | More about learning elementary formal systems - Arikawa, Shinohara, et al. - 1991 |

1 |
Completeness of depth-bounded resolution in logic programming
- Arimura
- 1989
(Show Context)
Citation Context ...g., the learnability of the class LBEFS n implies the learnability of the class of minimal models of linear Prolog programs consisting of at most n definite clauses (see Shinohara [Shi91] and Arimura =-=[Ari89]-=-). In this respect, these results are also about inductive logic programming. In the present paper we employ two distinct bodies of work in the inductive inference literature to analyze the learnabili... |