Automated reasoning in extensions of theories of constructors with recursively defined functions and homomorphisms

Abstract

We study possibilities of reasoning about extensions of base theories 
with functions which satisfy certain recursion and homomorphism 
properties.  Our focus is on emphasizing possibilities of hierarchical 
and modular reasoning in such extensions and combinations thereof. 

\begin{itemize}
item[(1)] We show that the theory of absolutely free constructors is local, 
and locality is preserved also in the presence of selectors. These results are 
consistent with existing decision procedures for this theory (e.g. by Oppen). 

item[(2)] We show that, under certain assumptions,  extensions of the theory of 
absolutely free constructors with functions satisfying a certain type of recursion axioms 
satisfy  locality properties, and show that for functions with values in an ordered domain  
we can combine recursive definitions with boundedness axioms  without sacrificing locality. 
We also address the problem of only considering models whose data part is 
the {em initial} term algebra of such theories. 

item[(3)] We analyze conditions which ensure that similar results can be obtained 
if we relax some assumptions about the absolute freeness of the underlying 
theory of data types, and illustrate the ideas on an example from cryptography.
end{itemize}
The locality results we establish allow us to reduce the task of reasoning about the 
class of recursive functions we consider to reasoning in the underlying theory of 
data structures (possibly combined with the theories associated with 
the co-domains of the recursive functions). 

As a by-product, the methods we use provide a possibility of presenting in a different light 
(and in a different form) locality phenomena studied in cryp-to-gra-phy; we believe that 
they will allow to better separate rewriting from proving, and thus to give simpler proofs.