Galois, Inc., USA

Metis 2.2 [Hur03] is a proof tactic used in the HOL4 interactive theorem prover. It works by converting a higher order logic goal to a set of clauses in first order logic, with the property that a refutation of the clause set can be translated to a higher order logic proof of the original goal.

Experiments with various first order calculi [Hur03] have shown
a *given clause algorithm* and ordered resolution to best suit
this application, and that is what Metis 2.2 implements. Since
equality often appears in interactive theorem prover goals, Metis 2.2
also implements the ordered paramodulation calculus.

Metis 2.2 uses a fixed strategy for every input problem. Negative literals are always chosen over positive literals, and terms are ordered using the Knuth-Bendix ordering with uniform symbol weight and precedence favouring reduced arity.

Metis 2.2 is written in Standard ML, for ease of integration with
HOL4. It uses indexes for resolution, paramodulation, (forward)
subsumption and demodulation. It keeps the *Active* clause set
reduced with respect to all the unit equalities so far derived.

In addition to standard age and size measures, Metis 2.2 uses
finite models to weight clauses in the *Passive* set. When
integrated with higher order logic, a finite model is manually
constructed to interpret standard functions and relations in such a
way as to make many standard axioms true and negated goals false.
Non-standard functions and relations are interpreted randomly, but
with a bias towards making negated goals false. Since it is part of
the CASC competition rules that standard functions and relations are
obfuscated, Metis 2.2 will back off to the biased random
interpretation of all functions and relations (except equality), using
a finite model with 4 elements.

Metis 2.2 reads problems in TPTP format and outputs detailed proofs in TSTP format, where each refutation step is one of 6 simple inference rules. Metis 2.2 implements a complete calculus, so when the set of clauses is saturated it can soundly declare the input problem to be unprovable (and outputs the saturation set).

Metis 2.2 is free software, released under the GPL. It can be downloaded from

http://www.gilith.com/software/metis

The major change between Metis 2.1, which was entered into CASC J4, and Metis 2.2 is the representation of normalization steps in TSTP proof format. There were only minor changes to the core proof engine, so Metis 2.2 is expected to perform at approximately the same level and end up in the lower half of the table.

- Hur03
- Hurd J. (September 2003),
**First-Order Proof Tactics in Higher-Order Logic Theorem Provers**, Archer M., Di Vito B., Muñoz C.,*Design and Application of Strategies/Tactics in Higher Order Logics (STRATA 2003)*(Rome, Italy), pp.56-68, NASA Technical Report CP-2003-212448, NASA.