Extensions of iterative congruences on free iterative algebras

Francesco Parisi-Presicce
1983 Journal of computer and system sciences (Print)  
This paper investigates special congruences on R, (the L-algebra of partial regular trees) which extend congruences on Ez (the C-algebra of total regular trees). It is proved that if a congruence on Ez induces an iterative factor algebra, then so does its extension on R,. This result is used to show that if an iterative algebra admits a faithful regular extension, then that extension is again iterative. The least fixed point approach [l-3] and the unique fixed point approach [8] to the
more » ... of programming languages have been compared by Tiuryn [ 121 using the framework of universal algebras. In [ 121, Tiuryn proves the existence of extensions of algebras with the unique fixed point property (iterative algebras) to ordered algebras with the least fixed point property (regular algebras), extensions preserving the fixed-point solutions. These regular extensions need not in general be iterative [ 131. We will prove that if the extension is faithful (i.e., if no two elements of the original carrier are identified), then the extension is both iterative and regular. Then we will carry the idea of the proof one step further to prove the following: let K be a congruence on RX (the Z-algebra of total regular trees) inducing an iterative factor algebra and let K' be an "extension" (in terms made precise later) of K in R, (the Z-algebra of partial regular trees). Then Rx/K' is iterative. The notion of extension of a congruence K on R, is a natural one. Two (possibly partial) regular trees t, and t, are equivalent if they can be decomposed into two components, one containing the "bottom" element I (representing the undefined parts), identical for both t, and t" and the other one consisting of an n-tuple of total regular trees, the n-tuple for t, K-congruent to the n-tuple for t,. Roughly speaking, these extensions reflect the idea that where the undefined parts occur is also a valuable piece of information.
doi:10.1016/0022-0000(83)90039-9 fatcat:ezrlqiupkvgwjhuuapsrq2j3ce