A Type System Describing Unboundedness

Abstract

We consider nondeterministic higher-order recursion schemes as recognizers of languages of finite words or finite trees. We propose a type system that allows to solve the simultaneous-unboundedness problem (SUP) for schemes, which asks, given a set of letters A and a scheme G, whether it is the case that for every number n the scheme accepts a word (a tree) in which every letter from A appears at least n times. Using this type system we prove that SUP is (m-1)-EXPTIME-complete for word-recognizing schemes of order m, and m-EXPTIME-complete for tree-recognizing schemes of order m. Moreover, we establish the reflection property for SUP: out of an input scheme G one can create its enhanced version that recognizes the same language but is aware of the answer to SUP.

Keywords

• simultaneous-unboundedness problem
• higher-order recursion schemes
• intersection types
• reflection
• [info.info-fl]computer science [cs]/formal languages and automata theory [cs.fl]
• [info.info-lo]computer science [cs]/logic in computer science [cs.lo]
• [info.info-cc]computer science [cs]/computational complexity [cs.cc]