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<html><head><meta http-equiv="Content-Type" content="text/html; charset=UTF-8"></head><body><hr><!------------------------------------------------------------------------>
<P>
<A NAME="E---2.2pre">
<HR><!------------------------------------------------------------------------>
<H2>E 2.2pre</H2>
Stephan Schulz<BR>
DHBW Stuttgart, Germany
<H3>Architecture</H3>
E 2.2pre [Sch2002, Sch2013] is a purely equational theorem prover for
many-sorted first-order logic with equality. It consists of an
(optional) clausifier for pre-processing full first-order formulae
into clausal form, and a saturation algorithm implementing an instance
of the superposition calculus with negative literal selection and a
number of redundancy elimination techniques. E is based on the
DISCOUNT-loop variant of the <EM>given-clause</EM> algorithm, i.e., a
strict separation of active and passive facts. No special rules for
non-equational literals have been implemented. Resolution is
effectively simulated by paramodulation and equality
resolution. However, as of E 2.1, PicoSAT [Bie2008] can be used to
periodically check the (on-the-fly grounded) proof state for
propositional unsatisfiability.
<P>
For the LTB divisions, a control program uses a SInE-like analysis to
extract reduced axiomatizations that are handed to several instances
of E. E will not use on-the-fly learning this year.
<H3>Strategies</H3>
Proof search in E is primarily controlled by a literal selection
strategy, a clause selection heuristic, and a simplification
ordering. The prover supports a large number of pre-programmed literal
selection strategies. Clause selection heuristics can be constructed
on the fly by combining various parameterized primitive evaluation
functions, or can be selected from a set of predefined
heuristics. Clause evaluation heuristics are based on symbol-counting,
but also take other clause properties into account. In particular, the
search can prefer clauses from the set of support, or containing many
symbols also present in the goal. Supported term orderings are several
parameterized instances of Knuth-Bendix-Ordering (KBO) and
Lexicographic Path Ordering (LPO).
<P>
For CASC-J9, E implements a strategy-scheduling automatic mode. The
total CPU time available is broken into several (unequal) time
slices. For each time slice, the problem is classified into one of
several classes, based on a number of simple features (number of
clauses, maximal symbol arity, presence of equality, presence of
non-unit and non-Horn clauses,...). For each class, a schedule of
strategies is greedily constructed from experimental data as follows:
The first strategy assigned to a schedule is the the one that solves
the most problems from this class in the first time slice. Each
subsequent strategy is selected based on the number of solutions on
problems not already solved by a preceding strategy.
<p>
About 220 different strategies have been evaluated on all untyped
first-order problems from TPTP 6.4.0. About 90 of these
strategies are used in the automatic mode, and about 210 are used in
at least one schedule.
<H3>Implementation</H3>
E is build around perfectly shared terms, i.e. each distinct term is
only represented once in a term bank. The whole set of terms thus
consists of a number of interconnected directed acyclic graphs. Term
memory is managed by a simple mark-and-sweep garbage collector.
Unconditional (forward) rewriting using unit clauses is implemented
using perfect discrimination trees with size and age constraints.
Whenever a possible simplification is detected, it is added as a
rewrite link in the term bank. As a result, not only terms, but also
rewrite steps are shared. Subsumption and contextual literal cutting
(also known as subsumption resolution) is supported using feature
vector indexing [Sch2013a]. Superposition and backward rewriting use
fingerprint indexing [Sch2012], a new technique combining ideas from
feature vector indexing and path indexing. Finally, LPO and KBO are
implemented using the elegant and efficient algorithms developed by
Bernd Löchner in [Loe2006, Loe2006a]. The prover and additional
information are available at
<PRE>
<A HREF="https://www.eprover.org">https://www.eprover.org</A></PRE>
<H3>Expected Competition Performance</H3>
E 2.2pre has only minor changes compared to last years
pre-releases. The major change is the integration of PicoSAT, however,
very few PicoSAT strategies have been evaluated. As a result, we
expect performance to be similar to last years, with maybe most
improvements in the EPR division. The system is expected to perform
well in most proof classes, but will at best complement top systems in
the disproof classes.
<P>
<a NAME="References">
<h3>References</h3>
<dl>
<dt> Sch2013
<dd> Schulz S. (2013),
<strong>System Description: E 1.8</strong>,
<em>Proc. of the 19th LPAR, Stellenbosch</em>,
LNCS 8312, pp.735-743, Springer
</dd>
<dt> Sch2002
<dd> Schulz S. (2002),
<strong>E: A Brainiac Theorem Prover</strong>,
<em>Journal of AI Communications</em> 15(2/3), pp.111-126, IOS Press
</dd>
<dt> Sch2013a
<dd> Schulz S. (2013),
<strong>Simple and Efficient Clause Subsumption with Feature
Vector Indexing</strong>,
<em>Automated Reasoning and Mathematics: Essays in
Memory of William W. McCune</em>, LNAI 7788, pp. 45-67,
Springer
</dd>
<dt> Sch2012
<dd> Schulz S. (2012),
<strong>Fingerprint Indexing for Paramodulation and
Rewriting</strong>,
<em>Proceedings of the 6th IJCAR (Manchester, UK)</em>,
LNAI 7364, pp.477-483, Springer
</dd>
<dt> Loe2006
<dd> Löchner B. (2004),
<strong>Things to Know when Implementing LPO</strong>,
<em>International Journal on Artificial Intelligence Tools</em>,
15(1), pp.53–80, 2006.
</dd>
<DT> Loe2006a
<DD> Löchner B. (2006),
<strong>Things to Know when Implementing KBO</strong>,
<em>Journal of Automated Reasoning</em> 36(4),
pp.289-310.
</dd>
<dt> Bie2008
<dd> Biere A. (2008),
<strong>PicoSAT essentials</strong>,
<em>Journal on Satisfiability, Boolean Modeling and Computation</em>
36, pp.75-97, 2008
</dd>
</dl>
<p>
</p><hr><!------------------------------------------------------------------------>
</body></html>
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