AUT-QE without type inclusion

Publisher Summary This chapter discusses the language AUT–QE without Type Inclusion. A language AUT–QE–NTI is also considered. Its definition is identical to the one of AUT–QE but the type inclusion rule is omitted. The letters NTI stand for ‘‘no type inclusion”. In AUT–QE–NTI, a set of axioms on universal quantifications can lead to a set of theorems which can take over the role of type inclusion. It is to be expected that AUT–QE–NTI enriched with automatic devices has about the same expressive power as AUT–QE and AUT–П. The ways by which the power of this language can be increased are also discussed. The various sets of axioms for universal quantification are also considered. The generalization of “All” to multiple quantification and generalization of “Ax1” and “Ax2” are considered as inverse operations. AUT–QE is very convenient for writing and checking. In order to facilitate the relation between the various languages, the term “abstraction index” of an expression in an AUT–QE–NTI is coined. A comparison of AUT–QE–NTI with AUT– П is also provided. To get a quick survey of the various operations, schematic presentation is also given.