@article{KraanBasin96,
AUTHOR = "Ina Kraan and David Basin and Alan Bundy",
TITLE = "Middle-Out Reasoning for Synthesis and Induction",
journal = "Journal of Automated Reasoning",
volume = 16,
number = "1--2",		  
pages = "113--145",
year = 1996,
ABSTRACT      = { 
We develop two applications of middle-out reasoning in inductive proofs: Logic
program synthesis and the selection of induction schemes.  Middle-out reasoning as
part of proof planning was first suggest by Bundy et. al.  Middle-out reasoning
uses variables to represent unknown terms and formulae.  Unification instantiates
the variables in the subsequent planning, while proof planning provides the
necessary search control.

Middle-out reasoning is used for synthesis by planning the verification of an
unknown logic program:  The program body is represented with a meta-variable.
The planning results both in an instantiation of the program body and a plan for
the verification of that program.  If the plan executes successfully, the
synthesized program is partially correct and complete.

Middle-out reasoning is also used to select induction schemes.  Finding an
appropriate induction scheme during synthesis is difficult, because the
recursion of the program, which is unknown at the outset, determines
the induction in the proof.  In middle-out induction, we set up a 
schematic step case by representing the constructors that are applied to induction
variables with meta-variables.  Once the step case is complete, the instantiated
variables correspond to an induction appropriate to the recursion of the program.

We have implemented these techniques as an extension of the proof planning
system CLAM called Periwinkle, and synthesized a variety of 
programs fully automatically.  },
}