Metatheory and Reflection in Theorem Proving: A Survey and Critique

John Harrison.

Technical Report CRC-053, SRI International Cambridge Computer Science Research Centre, 1995.


One way to ensure correctness of the inference performed by computer theorem provers is to force all proofs to be done step by step in a simple, more or less traditional, deductive system. Using techniques pioneered in Edinburgh LCF, this can be made palatable. However, some believe such an approach will never be efficient enough for large, complex proofs. One alternative, commonly called reflection, is to analyze proofs using a second layer of logic, a metalogic, and so justify abbreviating or simplifying proofs, making the kinds of shortcuts humans often do or appealing to specialized decision algorithms. In this paper we contrast the fully-expansive LCF approach with the use of reflection. We put forward arguments to suggest that the inadequacy of the LCF approach has not been adequately demonstrated, and neither has the practical utility of reflection (notwithstanding its undoubted intellectual interest). The LCF system with which we are most concerned is the HOL proof assistant.

The plan of the paper is as follows. We examine ways of providing user extensibility for theorem provers, which naturally places the LCF and reflective approaches in opposition. A detailed introduction to LCF is provided, emphasizing ways in which it can be made efficient. Next, we present a short introduction to metatheory and its usefulness, and, starting from Gödel's proofs and Feferman's transfinite progressions of theories, look at logical `reflection principles'. We show how to introduce computational `reflection principles' which do not extend the power of the logic, but may make deductions in it more efficient, and speculate about their practical usefulness. Applications or proposed applications of computational reflection in theorem proving are surveyed, following which we draw some conclusions. In an appendix, we attempt to clarify a couple of other notions of `reflection' often encountered in the literature.

The paper questions the too-easy acceptance of reflection principles as a practical necessity. However I hope it also serves as an adequate introduction to the concepts involved in reflection and a survey of relevant work. To this end, a rather extensive bibliography is provided.

PDF or DVI or PostScript

Bibtex entry:

        author          = "John Harrison",
        title           = "Metatheory and Reflection in Theorem Proving:
                           A Survey and Critique",
        institution     = "SRI Cambridge",
        address         = "Millers Yard, Cambridge, UK",
        year            = 1995,
        type            = "Technical Report",
        number          = "CRC-053",
        note            = "Available on the Web as