Sublogarithmic ambiguity

Klaus Wich
<span title="">2005</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Context-free grammars and languages with infinite ambiguity can be distinguished by the growth rate of their ambiguity with respect to the length of the words. So far the least growth rate known for a divergent inherent ambiguity function was logarithmic. Roughly speaking we show that it is possible to stay below any computable function. More precisely let f : N → N be an arbitrary computable divergent total non-decreasing function. Then there is a context-free language L with a divergent
more &raquo; ... nt ambiguity function g below f , i.e., g(n) f (n) for each n ∈ N. This result is an immediate consequence of two other results which are of independent interest. The first result says that there is a linear context-free grammar G with so called unambiguous turn position whose ambiguity function is below f . The second one states that any ambiguity function of a cycle-free context-free grammar is an inherent ambiguity function of some context-free language. → . In this case X is called the root of . The root of is denoted by ↑( ). It can be easily seen that each tree has a unique root. (Note that a tree either consist of the root only or of a string which begins with a production. In the latter case this production is the first production applied which uniquely determines the root.) The frontier of a string ∈ T G is ↓( ) := N∪ ( ). The arrows for the root and the frontier of trees point into the direction where they are usually displayed in a diagram. A node of a tree over T is an element of [1, is a leaf and it is the left-hand side of the production [i] if i is an internal node, i.e., it is X ∈ N if [i] ∈ {X} × (N ∪ )
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="">doi:10.1016/j.tcs.2005.07.024</a> <a target="_blank" rel="external noopener" href="">fatcat:lvbrv4ak45dfboxmpji4tt6fye</a> </span>
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