We give an algorithm for the class of second order unification problems in which second order variables have at most one occurrence.

Higher-order unification and matching

pdf/a-unification-algorithm-for-second-order-linear-terms-1zri31rt7x.pdf

A Unification Algorithm for Second-Order Linear Terms

:tb.~tract. Theorem-proving on the computer, using procedures based on the fund~mental theorem of Herbrand concerning the first-order predicate etdeulus, is examined with ~ view towards improving the efticieney and widening the range of practical applicability of these procedures. A elose analysis of the process of substitution (of terms for variables), and the process of t ruth-funct ional analysis of the results of such substitutions, reveals that both processes can be combined into a single new process (called resolution), i terating which is vastty more ef[ieient than the older cyclic procedures consisting of substitution stages alternating with truth-functional analysis stages. The theory of the resolution process is presented in the form of a system of first<~rder logic with .just one inference principle (the resolution principle). The completeness of the system is proved; the simplest proof-procedure based oil the system is then the direct implementation of the proof of completeness. Howew~r, this procedure is quite inefficient, ~nd the paper concludes with a discussion of several principles (called search principles) which are applicable to the design of efficient proof-procedures employing resolution as the basle logical process.

A Machine-Oriented Logic Based on the Resolution Principle

Second-order unification is undecidable in general. Miller showed that unification of so-called higher-order patterns is decidable and unitary. We show that the unification of a linear higher-order pattern s with an arbitrary second-order term that shares no variables with s is decidable and finitary. A few extensions of this unification problem are still decidable: unifying two second-order terms, where one term is linear, is undecidable if the terms contain bound variables but decidable if they don't.

Decidable Higher-Order Unification Problems

We prove that the second order predicate unification problem is undecidable by reducing the second order term unification problem to it.

The undecidability of the second order predicate unification problem

We propose a program transformation method based on rewriting-rules composed of second-order schemas. A complete second-order matching algorithm is presented that allows effective use of these rules. We show how to formally prove the correctness of the rules using a denotational semantics for the programming language. We establish the correctness of the transformation method itself, and give techniques pertaining to its actual implementation. The paper is illustrated with recursion removal examples.

Proving and applying program transformations expressed with second-order patterns

https://repository.upenn.edu/cgi/viewcontent.cgi?article=1470&context=cis_reports

Unification under a mixed prefix

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A Unification Algorithm for Second-Order Linear Terms

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