the power of correction queries cristina bibire research group on mathematical linguistics, rovira i...

The Power of Correction The Power of Correction QueriesQueries

Cristina BibireCristina BibireResearch Group on Mathematical Linguistics, Rovira i Research Group on Mathematical Linguistics, Rovira i

Virgili UniversityVirgili University Pl. Imperial Tarraco 1, 43005, Tarragona, SpainPl. Imperial Tarraco 1, 43005, Tarragona, Spain

E-mail: cristina.bibire@estudiants.urv.esE-mail: cristina.bibire@estudiants.urv.es

99thth of February 2006 of February 2006Seminar VSeminar V

Outline

Introduction

A class which makes the difference

Practical results

The injectivity property

Future Work

Introduction Introduction The correcting string of s in the language L is the smallest string s' (in lex-length order) such that s.s' belongs to L.

The answer to a correction query for a string consists of its correcting string.

Myhill-Nerode theorem:

The number of states in the smallest DFA accepting L is equal to the number of equivalence classes in .L

*if iffLx y z xz L yz L

IntroductionIntroductionDefinition By the number of MQs a language require in order to be identified we understand how many different strings are submitted by the learner L* to the teacher until it outputs the target language.

Note that each string is counted only once: even if the algorithm reaches a point where the learner should submit to the teacher a string which was previously submitted we will not count this as another MQ.

Definition By the number of CQs a language require in order to be identified we understand how many different strings are submitted by the learner LCA to the teacher until it outputs the target language.

In this case, not only we do not count twice the same string, but the learner LCA can also obtain some implicit answers. Therefore, the algorithm does not ask these questions and we do not count them as new CQs.

A class which makes the A class which makes the differencedifferenceTheorem There exists an infinite class of languages which requires a

polynomial number of MQs but a linear number of CQs in order to be identified.

A class which makes the A class which makes the differencedifferenceTheorem There exists an infinite class of languages which requires a

polynomial number of MQs but a linear number of CQs in order to be identified.Let us consider the class of singletons over .S

A class which makes the A class which makes the differencedifferenceTheorem There exists an infinite class of languages which requires a

polynomial number of MQs but a linear number of CQs in order to be identified.Let us consider the class of singletons over .Lemma 1. For any fixed alphabet of length and any language L in , the number of MQs needed by L* in order to identify L is:

where m is the size of the minimal DFA accepting L.

Lemma 2. For any fixed alphabet of length and any language L in , the number of CQs needed by LCA in order to identify L is:

where m is the size of the minimal DFA accepting L.

1k S

S

22 1 (4 7) 2 5,MQ m k m k m k

1k S

1 ( 3),CQ m k m k

A class which makes the A class which makes the differencedifferenceIf n is the length of the unique string in L, then:

22 1 4 1 2 1

1 1

MQ n k n k n k

CQ n k n k

2 :k

22 7 5

3

MQ n n n

CQ n n

A class which makes the A class which makes the differencedifferenceIf n is the length of the unique string in L, then:

22 1 4 1 2 1

1 1

MQ n k n k n k

CQ n k n k

2 :k

22 7 5

3

MQ n n n

CQ n n

1:k

A class which makes the A class which makes the differencedifferenceIf n is the length of the unique string in L, then:

22 1 4 1 2 1

1 1

MQ n k n k n k

CQ n k n k

2 :k

1:k

22 7 5

3

MQ n n n

CQ n n

3 3

2

MQ n n

CQ n

Practical ResultsPractical Results

Practical ResultsPractical Results

The injectivity propertyThe injectivity propertyFor any regular language L, we say that it has the injectivity property if for every two strings s and s',

Ls s C s C s

The injectivity propertyThe injectivity propertyFor any regular language L, we say that it has the injectivity property if for every two strings s and s',

Theorem Let L be a regular language. If L has the injectivity property then the number of CQs is at most meanwhile the number of MQs is at least , where k is the size of the alphabet and n the number of states of the minimal DFA accepting L.

Ls s C s C s

21 logn k n n 1nk

The injectivity propertyThe injectivity propertyFor any regular language L, we say that it has the injectivity property if for every two strings s and s',

Theorem Let L be a regular language. If L has the injectivity property then the number of CQs is at most meanwhile the number of MQs is at least , where k is the size of the alphabet and n the number of states of the minimal DFA accepting L.Proof

Ls s C s C s

21 logn k n n 1nk

MQ S S S E

The injectivity propertyThe injectivity propertyFor any regular language L, we say that it has the injectivity property if for every two strings s and s',

Theorem Let L be a regular language. If L has the injectivity property then the number of CQs is at most meanwhile the number of MQs is at least , where k is the size of the alphabet and n the number of states of the minimal DFA accepting L.Proof

Ls s C s C s

21 logn k n n 1kn

MQ S S S E

2

1 1

log

S n

S k S S S k S k n

E n

The injectivity propertyThe injectivity propertyProof

λλ

ss11

ss22

......

ssnn

cc11

cc22

ccnn

ss11aa

11

......

ss11aa

kk

......

ssnnaa

kk

S

SΣ-S

E

The injectivity propertyThe injectivity propertyProof

λλ

ss11

ss22

......

ssnn

cc11

cc22

ccnn

ss11aa

11

......

ss11aa

kk

......

ssnnaa

kk

S

SΣ-S

E

CQ S S S

S n

S S k S kn

The injectivity propertyThe injectivity propertyProof

λλ

ss11

ss22

......

ssnn

cc11

cc22

ccnn

ss11aa

11

......

ss11aa

kk

......

ssnnaa

kk

S

SΣ-S

E

CQ S S S

S n

S S k S kn

CQ n kn

The injectivity propertyThe injectivity propertyProof

λλ

ss11

ss22

......

ssnn

cc11

cc22

ccnn

ss11aa

11

......

ss11aa

kk

......

ssnnaa

kk

S

SΣ-S

E

CQ S S S

S n

S S k S kn

CQ n kn 1n

1CQ kn

Future WorkFuture Work The injectivity property:

- Find a proper upper bound for the number of MQs (polynomial?)

- Find a mathematical relation between the degree of injectivity and the reducing factor (MQ/CQ)

Other classes of languages:

- Classes of languages learnable from positive examples (finite thickness, finite elasticity, characteristic sample) or not (finite tell tale)

- The class of k-reversible languages

- Others ?