Finite thickness




In formal language theory, in particular in algorithmic learning theory, a class C of languages has finite thickness if every string is contained in at most finitely many languages in C. This condition was introduced by Dana Angluin as a sufficient condition for C being identifiable in the limit.
[1]



The related notion of M-finite thickness


Given a language L and an indexed class C = { L1, L2, L3, ... } of languages, a member language Lj ∈ C is called a minimal concept of L within C if L ⊆ Lj, but not L ⊊ Li ⊆ Lj for any Li ∈ C.[2]
The class C is said to satisfy the MEF-condition if every finite subset D of a member language Li ∈ C has a minimal concept Lj ⊆ Li. Symmetrically, C is said to satisfy the MFF-condition if every nonempty finite set D has at most finitely many minimal concepts in C. Finally, C is said to have M-finite thickness if it satisfies both the MEF- and the MFF-condition.
[3]


Finite thickness implies M-finite thickness.[4] However, there are classes that are of M-finite thickness but not of finite thickness (for example, any class of languages C = { L1, L2, L3, ... } such that L1 ⊆ L2 ⊆ L3 ⊆ ...).



References





  1. ^ Dana Angluin (1980). "Inductive Inference of Formal Languages from Positive Data" (PDF). Information and Control. 45 (2): 117–135. doi:10.1016/s0019-9958(80)90285-5..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em} (citeseer.ist.psu.edu); here: Condition 3, p.123 mid. Angluin's original requirement (every non-empty string set be contained in at most finitely many languages) is equivalent.


  2. ^ Andris Ambainis; Sanjay Jain; Arun Sharma (1997). "Ordinal mind change complexity of language identification". Computational Learning Theory (PDF). LNCS. 1208. Springer. pp. 301–315.; here: Definition 25


  3. ^ Ambainis et al. 1997, Definition 26


  4. ^ Ambainis et al. 1997, Corollary 29











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