Papers › Schematic Unification
Schematic Unification
David M. Cerna
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We present a generalization of first-order unification to a term algebra where variable indexing is part of the object language. We exploit variable indexing by associating some sequences of variables (X₀, X₁, X₂,…) with a mapping σ whose domain is the variable sequence and whose range consist of terms that may contain variables from the sequence. From a given term t, an infinite sequence of terms may be produced by iterative application of σ. Given a unification problem U and mapping σ, the \textit{schematic unification problem} asks whether all unification problems U, σ(U), σ(σ(U)), … are unifiable. We provide a terminating and sound algorithm. Our algorithm is \textit{complete} if we further restrict ourselves to so-called ∞-stable problems. We conjecture that this additional requirement is unnecessary for completeness. Schematic unification is related to methods of inductive proof transformation by resolution and inductive reasoning.
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