Efficient Inference of a subclass of Even Linear Languages

1
Efficient Inference of a subclass of Even Linear
Languages

• J. A. Laxminarayana
• G. Nagaraja
• Computer Science and Engg. Deptt.
• Indian Institute of Technology, Bombay
• Powai, Mumbai, INDIA

2
Motivation

• Growing influence of grammatical inference.
• The class of terminal distinguishable languages.
• Well established theoretical framework.
• Scope to improve existing techniques.
• Scope to design efficient algorithms.
• Identification of suitable applications.

3
Grammatical Inference

Sample strings
Parser
Grammar rules
Inference algorithm
Parser Generator
Addtl. Info (if any)
Parser
Test strings
Accept/ Reject
Back
4
Formal Languages

• A formal grammar G has four components.
• A set of symbols ?, called terminals.
• A set of symbols V, called non-terminals with the
  restriction that ? and V are disjoint.
• A special non-terminal symbol S , called a start
  symbol.
• A set of production rules P , where each
  production of the form ? ? ?.

5
Chomsky Hierarchy

• Noam Chomsky defined classes of grammars
• Type 0 Recursively Enumerable Languages
  (Unrestricted Grammars)
• Type 1 Context Sensitive Languages (Context
  Sensitive Grammars)
• Type 2 Context Free Languages (Context Free
  Grammars)
• Type 3 Regular Languages (Regular Grammars)

6
Inductive Inference

• Proposed by Angluin, 1983.
• Deductive and inductive inference.
• Identification by enumeration and identification
  in limit.
• Specifying inference problems.
• Class of rules , hypothesis space ,
• Set of examples, inference methods.
• Criteria for evaluating and comparing inference
  methods.

7
Identification in the limit
G1 G2 Gn
Sample 1 Sample 2 Sample n
G0
Teacher
Learner
8
Golds Results

• The class of phrase structured languages is
  learnable from positive and negative samples.
• Not even the class of regular languages is
  learnable from positive samples alone.
• Any language class which contains all finite
  languages and at least one infinite language
  (super finite language class) is NOT identifiable
  in the limit from positive samples.
• The finite cardinality languages class is
  identifiable from positive samples.

9
Angluins Results

• Angluin1980 proposed that a language class
  that contains some finite languages and some
  infinite languages is identifiable from positive
  samples alone.
• Angluin proposed an efficient characterizable
  method using which one can learn many interesting
  classes of languages. Examples are.
• Parenthesis language, pattern language.
• K-reversible language and TDR language.

10
Terminal Distinguishable Languages

• Based on structural information (skeleton).
• Good algebraic and grammatical characteristics.
• Good incremental behaviour.
• Based on three properties backward determinism,
  terminal completeness, terminal dissimilarity.

11
Example of even linear skeleton

12
Definitions

• Two nodes Nij and Nkl are equivalent and merged
  iff
• SSNF(Nij ) SSNF(Nkl) or
• FS(Nij ) FS(Nkl) or PTF(Nij ) PTF(Nkl)
• An even linear grammar G is TDELG, if f
• B?w and C?w implies B C
• ?A ? N-S and ?x,y ?L(A), Ter(x)Ter(y)
• Let A,B,C ? N, a,b ?? and i) the productions
• S?B and S?C where S is the start symbol
• ii) the productions A ?aBb and A ?aCb appear in
  the sets of productions, then
• ? x ?L(B), ? y ?L(C) Ter(x)?Ter(y)

13
Proposed Method to Infer TDELG

• Union-Find approach
• Simple to understand and easy to implement
• Incorporates all properties of TDELG
• Minimizing computing overheads
• Suitable for Incremental inferencing

14
Definitions

• S A set of given sample strings.
• ND(S) A set of nodes found in all skeletons of
  S.
• Parent(p) Immediate predecessor in a skeleton
  having p.
• H(S) A set of SSNF of S
• F(S) A set of frontier strings of S

15
Method

• Initial partition ?0 is obtained using SSNFs
• A list of pair of nodes is maintained to denote
  possible merging of blocks of partitions.
• A pair of nodes is removed from the list
  repeatedly until list becomes empty and the
  blocks containing the nodes are tested for
  merging to get new partition ?n . New pairs are
  added to list if PTF of nodes of other blocks are
  changed.

16
Example

• Let the sample set be ab, aabb, aaabbb.
• Computation of head, tail, SSNF, FS, and PTF of
  the nodes in the even linear skeletons are shown
  in the table.
• Initial partition is ?0 N11 , N21 ,N31 ,
• , N22 , ,N32 , N33

17
Computation of Functions
18
Example

• LIST (N11, N22), (N11, N33), (N22, N33),
  (N21, N32) .
• Consider (N11, N22) ? LIST
• This pair forces the merging of blocks
• B1 N11, N21, N31 and B1 N22, N32
• New partition ?1 N11,N21,N31, N22, N32, N33
  .
• Final partition N11, N21, N31, N22, N32 ,
  N33
• TDELG is (S,S1, a,b, S? S1, S1? aS1b, S1?
  ab , S )

19
Complexity Analysis

• Let S w1 , w2 wn be the input sample.
• Let k be the size of the input i.e sum of lengths
  of all input strings belonging to S .
• Construction of initial partition of ND(S )
  requires a computation time O(k2) .
• The partition may be queried and updated using
  collapsing FIND and weighted UNION operations
  which lead to a running time of O(k?(k)) where ?
  is a very slowly growing function.

20
Comparison with the related works

• Methods in the literature
• Takada proposed a method of inferring a subclass
  of even linear languages using control sets.
• Fernau also proposed similar method.
• Basic idea in their methods extend regular
  language inference algorithm.
• However, their methods require extra
  pre-processing and post processing steps.
• Their approaches are difficult to understand.

• Our method
• We propose a characterizable subclass of even
  linear languages.
• Our method is not an extension of regular
  inference mechanism.
• We need not to preprocess the input or post
  process the output.
• An algorithm is proposed which takes only
  positive samples as the input.
• The tabular approach which is simple to
  understand can be used to implement.

21
References

• D.Angluin, Inference of Reversible Languages, J.
  ACM 29, pp. 741-765, 1982.
• D. Angluin, C. H. Smith. Inductive inference
  theory and methods, Computing Surveys, 15, pp.
  237-269, 1983.
• E. M. Gold, Language identification in the limit,
  Information and Control, 10, pp. 447-474, 1967.
• V. Radhakrishnan. Grammatical inference from
  positive data An effective integrated approach.
  PhD thesis, Department of Computer Science and
  Engineering, Indian Institute of Technology,
  Bombay (India), 1987.