## Modular Proof Systems for Partial Functions with Evans Equality

**Harald Ganzinger, Viorica Sofronie-Stokkermans, and Uwe Waldmann**
*Information and Computation*,
204:1453-1492, 2006.

*Earlier version:*
Modular Proof Systems for
Partial Functions with Weak Equality,
in *IJCAR 2004*, pp. 168-182, 2004.

Draft version:
[Postscript file, 257 kB]

**Abstract:**
The paper presents a modular superposition calculus for the
combination of first-order theories involving both total
and partial functions.
The modularity of the calculus is a consequence of the fact that
all the inferences are pure
- only involving clauses over the alphabet of either
one, but not both, of the theories - when refuting goals represented
by sets of pure formulae.
The calculus is shown to be complete provided that functions
that are not in the intersection of the component signatures
are declared as partial. This result also means that if the
unsatisfiability of a goal modulo the combined theory does
not depend on the totality of the functions in the
extensions, the inconsistency will be effectively found.
Moreover, we consider a constraint superposition calculus
for the case of hierarchical theories and show that it has
a related modularity property. Finally we identify cases
where the partial models can always be made total so that
modular superposition is also complete with respect to the
standard (total function) semantics of the theories.

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Uwe Waldmann
<

uwe@mpi-inf.mpg.de>,
2005-11-10.