E/EP 1.2pre

Stephan Schulz
Technische Universität München, Germany

Architecture

E 1.2pre [Sch02,Sch04] is a purely equational theorem prover. The core proof procedure operates on formulae in clause normal form, using a calculus that combines superposition (with selection of negative literals) and rewriting. No special rules for non-equational literals have been implemented, i.e., resolution is simulated via paramodulation and equality resolution. The basic calculus is extended with rules for AC redundancy elimination, some contextual simplification, and pseudo-splitting with definition caching. The latest versions of E also support simultaneous paramodulation, either for all inferences or for selected inferences.

E is based on the DISCOUNT-loop variant of the given-clause algorithm, i.e., a strict separation of active and passive facts. Proof search in E is primarily controlled by a literal selection strategy, a clause evaluation heuristic, and a simplification ordering. The prover supports a large number of preprogrammed literal selection strategies. Clause evaluation heuristics can be constructed on the fly by combining various parameterized primitive evaluation functions, or can be selected from a set of predefined heuristics. Supported term orderings are several parameterized instances of Knuth-Bendix-Ordering (KBO) and Lexicographic Path Ordering (LPO).

The prover uses a preprocessing step to convert formulae in full first order format to clause normal form. This step may introduce (first-order) definitions to avoid an exponential growth of formulae. Preprocessing also unfolds equational definitions and performs some simplifications on the clause level.

EP 1.2pre is just a combination of E 1.2pre in verbose mode and a proof analysis tool extracting the used inference steps.

Strategies

The automatic mode determines literal selection strategy, term ordering, and search heuristic based on simple problem characteristics of the preprocessed clausal problem. E has been optimized for performance over the TPTP. The automatic mode of E 1.2pre is partially inherited from previous version and is based on about 60 test runs over TPTP 4.0.1. It consists of the selection of one of about 40 different strategies for each problem. All test runs have been performed on Linux/Intel machines with a time limit roughly equivalent to 300 seconds on 300MHz Sun SPARC machines, i.e., around 30 seconds on 2Ghz class machines. All individual strategies are refutationally complete.

E distinguishes problem classes based on a number of features, all of which have between 2 and 4 possible values. The most important ones are:

Is the most general non-negative clause unit, Horn, or Non-Horn?
Is the most general negative clause unit or non-unit?
Are all negative clauses unit clauses?
Are all literals equality literals, are some literals equality literals, or is the problem non-equational?
Are there a few, some, or many clauses in the problem?
Is the maximum arity of any function symbol 0, 1, 2, or greater?
Is the sum of function symbol arities in the signature small, medium, or large?

For classes above a threshold size, we assign the absolute best heuristic to the class. For smaller, non-empty classes, we assign the globally best heuristic that solves the same number of problems on this class as the best heuristic on this class does. Empty classes are assigned the globally best heuristic. Typically, most selected heuristics are assigned to more than one class.

For the LTB part of the competition, E will use a relevancy-based pruning approach and attempt to solve the problems with successively more complete specifications until it succeeds or runs out of time.

Implementation

E is implemented in ANSI C, using the GNU C compiler. At the core is an implementation of aggressively shared first-order terms in a term bank data structure. Based on this, E supports the global sharing of rewrite steps. Rewriting is implemented in the form of rewrite links from rewritten to new terms. In effect, E is caching rewrite operations as long as sufficient memory is available. E uses perfect discrimination trees with age and size constraints for rewriting and unit-subsumption, feature vector indexing [Sch04] for forward- and backward subsumption and contextual literal cutting, and a new technique called fingerprint indexing for backward rewriting and (hopefully) paramodulation. Knuth-Bendix Ordering and Lexicographic Path Ordering are implemented using the linear and polynomial algorithms described by Bernd Löchner [Loe04,Loe06].

The program has been successfully installed under SunOS 4.3.x, Solaris 2.x, HP-UX B 10.20, MacOS-X, and various versions of Linux. Sources of the latest released version are available freely from:

    http://www.eprover.org

EP 1.2pre is a simple Bourne shell script calling E and the postprocessor in a pipeline.

Expected Competition Performance

In the last years, E performed well in most proof categories. We believe that E will again be among the stronger provers in the FOF and CNF categories. We hope that E will at least be a useful complement to dedicated systems in the other categories.

EP 1.2p will be hampered by the fact that it has to analyse the inference step listing, an operation that typically is about as expensive as the proof search itself. Nevertheless, it should be competitive among the FOF systems.

References

Sch2002: Schulz S. (2002), E: A Brainiac Theorem Prover, Journal of AI Communications 15(2/3), 111-126, IOS Press
Sch2004: Schulz S. (2004), System Abstract: E 0.81, Proceedings of the 3rd IJCAR, (Cork, Ireland), Lecture Notes in Artificial Intelligence, Springer-Verlag
Sch2004b: Schulz S. (2004), Simple and Efficient Clause Subsumption with Feature Vector Indexing, Proceedings of the IJCAR-2004 Workshop on Empirically Successful First-Order Theorem Proving, (Cork, Ireland)
Loe2004: Löchner B. (2004), What to know when implementing LPO, Proceedings of the IJCAR-2004 Workshop on Empirically Successful First-Order Theorem Proving, (Cork, Ireland)
Loe06: L¨chner B. (2006), Things to Know When Implementing KBO, Journal of Automated Reasoning 36(4), 289-310.