CHewTPTP 1.0
Eric McGregor
Clarkson University, USA
Architecture
ChewTPTP version 1.0 implements the method described in
[BK+08,DH+07].
ChewTPTP transforms a set of first order clauses into a propositional encoding
(modulo recursive type theory) of the existence of a rigid first order
connection tableau and the satisfiability of unification constraints, which
is then fed to Yices.
For the unification constraints, terms are represented as recursive datatypes,
and unification constraints are equations on terms.
Additional instances of the first order clauses may be added for the
non-rigid case.
Darwin 1.3
Peter Baumgartner1,
Alexander Fuchs2,
Cesare Tinelli2
1National ICT Australia, Australia,
2The University of Iowa, USA
Architecture
Darwin [BFT04,BFT06a]
is an automated theorem prover for first order clausal logic.
It is an implementation of the Model Evolution Calculus
[BT03].
The Model Evolution Calculus lifts the propositional DPLL procedure to
first-order logic. One of the main motivations for this approach
was the possibility of migrating to the first-order level some of
those very effective search techniques developed by the SAT community
for the DPLL procedure.
The current version of Darwin implements first-order versions of unit propagation inference rules analogously to a restricted form of unit resolution and subsumption by unit clauses. To retain completeness, it includes a first-order version of the (binary) propositional splitting inference rule.
Proof search in Darwin starts with a default interpretation for a given clause set, which is evolved towards a model or until a refutation is found.
Improvements to the previous version include additional preprocessing steps, less memory requirements, and lemma learning [BFT06b].
Darwin is implemented in OCaml and has been tested under various Linux distributions. It is available from:
http://goedel.cs.uiowa.edu/Darwin/
E 1.0pre and EP 1.0pre
Stephan Schulz
Institut für Informatik, Technische Universität, Germany
Architecture
E 1.0pre [Sch02,Sch04a]
is a purely equational theorem prover. The core proof procedure operates on
formulas 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 supports 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, many of which are only experimental. 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 formulas in full first order format to clause normal form. This step may introduce (first-order) definitions to avoid an exponential growth of the formula. Preprocessing also unfolds equational definitions and performs some simplifications on the clause level.
The automatic mode determines literal selection strategy, term ordering, and search heuristic based on simple problem characteristics of the preprocessed clausal problem.
EP 1.0pre is just a combination of E 1.0pre in verbose mode and a proof analysis tool extracting the used inference steps.
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:
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.
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.orgEP 1.0pre is a simple Bourne shell script calling E and the postprocessor in a pipeline.
EP 1.0p 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 in the FOF proof class.
E-Darwin 1.0
Björn Pelzer
University Koblenz-Landau, Germany
Architecture
E-Darwin 1.0 is an automated theorem prover for first order clausal logic
with equality.
It is a modified version of the Darwin prover [BFT04],
and it implements the Model Evolution Calculus with Equality
[BT05].
The general operation of the original Darwin has been extended by inferences
for handling equality, specifically a paramodulation and a reflexivity
inference.
Whereas the original Darwin derived units only, the new inferences in
E-Darwin allow the derivation of multi-literal clauses.
A more extensive detection and handling of redundancy has been added as well.
http://www.uni-koblenz.de/~bpelzer/edarwin
Darwin's method of storing partial unifiers has been adapted to equations and subterm positions for the paramodulation inference in E-Darwin. A combination of perfect and non-perfect discrimination tree indexes is used to store the context and the clauses.
E-KRHyper 1.1
Björn Pelzer
University Koblenz-Landau, Germany
Architecture
E-KRHyper 1.1 [PW07] is a theorem proving and
model generation system for first-order logic with equality.
It is an implementation of the E-hyper tableau calculus
[BFP07], which integrates a superposition-based
handling of equality [BG98] into the hyper
tableau calculus [BFN96].
The system is an extension of the KRHyper theorem prover
[Wer03], which implements the original hyper tableau
calculus.
An E-hyper tableau is a tree whose nodes are labeled with clauses and which is built up by the application of the inference rules of the E-hyper tableau calculus. The calculus rules are designed such that most of the reasoning is performed using positive unit clauses. A branch can be extended with new clauses that have been derived from the clauses of that branch.
A positive disjunction can be used to split a branch, creating a new branch for each disjunct. No variables may be shared between branches, and if a case-split creates branches with shared variables, then these are immediately substituted by ground terms. The grounding substitution is arbitrary as long as the terms in its range are irreducible: the branch being split may not contain a positive equational unit which can simplify a substituting term, i.e., rewrite it with one that is smaller according to a reduction ordering. When multiple irreducible substitutions are possible, each of them must be applied in consecutive splittings in order to preserve completeness.
Redundancy rules allow the detection and removal of clauses that are redundant with respect to a branch.
The hyper extension inference from the original hyper tableau calculus is equivalent to a series of E-hyper tableau calculus inference applications. Therefore the implementation of the hyper extension in KRHyper by a variant of semi-naive evaluation [Ull89] is retained in E-KRHyper, where it serves as a shortcut inference for the resolution of non-equational literals.
E-KRHyper is available under the GNU Public License from:
http://www.uni-koblenz.de/~bpelzer/ekrhyper
For the LTB division a small script partitions
the time available to the given problem set and runs
iProver with the allocated time limit, and
time is repartitioned after each solved problem.
iProver is available at:
To compute an effective schedule, the solvers entered in last year's CASC
competition were run on all the TPTP-v3.4.1 benchmark instances in the SAT
and FNT divisions, subject to a five minute time limit per instance (the SAT
and FNT divisions were chosen because preliminary experiments indicated that
a scheduling approach would work well in those divisions).
Using the timing data obtained from these runs, a schedule was then computed
that, if run on each benchmark instance, would solve the instances in the
minimum average time.
Two schedules were computed, one for the FNT instances and one for the SAT
instances.
The schedules were computed using the greedy approximation algorithm described
in [SGS07].
When executing each step of a schedule, we run the specified solver from
scratch and terminate it once it has run for the specified amount of time.
Alternatively, we could suspend the solver (rather than terminate it) at
the end of the step and resume the solver if and when it is used by a later
step.
We took the former approach for simplicity, and because we were concerned
about having multiple solvers in memory at the same time.
Because the time required to solve a particular benchmark instance varies
across machines, a schedule must be calibrated for use on a particular machine.
This calibration is performed as part of MetaProver's installation script.
The example schedule shown above is the schedule used by MetaProver for
instances in the FNT division, calibrated for execution on an Intel
Xeon 3.6 GHz machine with 4 GB of memory.
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 2.1 implements.
Since equality often appears in interactive theorem prover goals,
Metis 2.1 also implements the ordered paramodulation
calculus.
In addition to standard size and distance measures, Metis 2.1 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 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 2.1 will back-off to interpreting all functions and
relations randomly (except equality), using a finite model with 4 elements.
Metis 2.1 reads problems in TPTP format and outputs detailed proofs in
TSTP format, where each proof step is one of 6 simple inference rules.
Metis 2.1 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 2.1 is free software, released under the GPL. It can be
downloaded from:
The system is also able to work with second order statements. It
may also receive knowledge and know-how for a specific domain from a
human user; see [Pas89] and
[Pas93]. These two possibilities
are not used while working with the TPTP Library.
Proofs are given in natural style
(for each step, that is for each action or rule application,
the system gives the new fact, the precedent facts its comes from
and an explanation).
Muscadet 3.0 is available from:
Equinox 3.0
Koen Claessen
Chalmers University of Technology, Sweden
Architecture
Equinox is an experimental theorem prover for pure first-order logic
with equality. It finds ground proofs of the input theory, by solving
successive ground instantiations of the theory using an incremental
SAT-solver. Equality is dealt with using a Nelson-Oppen framework.
Implementation
The main part of Equinox is implemented in Haskell using the GHC
compiler. Equinox also has a built-in incremental SAT solver (MiniSat)
which is written in C++. The two parts are linked together on the
object level using Haskell's Foreign Function Interface.
Strategies
There is only one strategy in Equinox:
Expected Competition Performance
Equinox should perform reasonably well. There should be problems that
it can solve that few other provers can handle.
iProver 0.5
Konstantin Korovin
The University of Manchester, England
Architecture
iProver is an automated theorem prover which is based on an instantiation
calculus Inst-Gen [GK03,
Kor08a], complete for first-order logic.
One of the distinctive features of iProver is a modular combination of
first-order reasoning with ground reasoning.
In particular, iProver currently integrates MiniSat for reasoning with
ground abstractions of first-order clauses.
In addition to instantiation, iProver implements ordered resolution calculus
and a combination of instantiation and ordered resolution, see
[Kor08b] for the implementation details.
The saturation process is implemented as a modification of a given
clause algorithm.
We use non-perfect discrimination trees for the unification indexes,
priority queues for passive clauses and
a compressed vector index for subsumption and subsumption resolution
(both forward and backward).
The following redundancy eliminations are implemented:
blocking non-proper instantiations; dismatching constraints
[GK04,Kor08b];
global subsumption [Kor08b]; resolution-based
simplifications and propositional-based simplifications.
Equality is dealt with (internally) by adding the necessary axioms of equality.
iProver has a satisfiablity mode which includes a finite model finder,
based on an adaptation of ideas from DarwinFM and Paradox to the iProver
setting.
Strategies
iProver v0.5 has around 40 options to control the proof
search including options for literal selections,
passive clause selections, frequency of calling the SAT solver,
simplifications and options for combination of instantiation with resolution.
At the CASC competition iProver
will execute a fixed schedule of selected options.
Implementation
iProver is implemented in OCaml and for the ground reasoning uses MiniSat.
iProver accepts cnf and fof formats.
In the case of fof format, either Vampire or E prover is used for
clausification.
http://www.cs.man.ac.uk/~korovink/iprover/
Expected Competition Performance
The instantiation method behind iProver is a decision procedure for the
EPR class, and we expect a good performance in this class.
We also expect a reasonable performance in FOF and CNF classes.
MaLARea 0.3
Josef Urban
Charles University in Prague, Czech Republic
Architecture
MaLARea 0.3 [Urb07,
US+08] is a metasystem for ATP in large theories
where symbol and formula names are used consistently.
It uses several deductive systems (now E,SPASS,Paradox,Mace), as well
as complementary AI techniques like machine learning (the SNoW system)
based on symbol-based similarity, model-based similarity, term-based
similarity, and obviously previous successful proofs.
Strategies
The basic strategy is to run ATPs on problems, then use the machine learner
to learn axiom relevance for conjectures from solutions, and use the most
relevant axioms for next ATP attempts.
This is iterated, using different timelimits and axiom limits.
Various features are used for learning, and the learning is complemented
by other criteria like model-based reasoning, symbol and term-based
similarity, etc.
Implementation
The metasystem is implemented in ca. 2500 lines of Perl.
It uses many external programs - the above mentioned ATPs and machine
learner, TPTP utilities, LADR utilities for work with models, and some
standard Unix tools.
MaLARea is available at
http://kti.ms.mff.cuni.cz/cgi-bin/viewcvs.cgi/MPTP2/MaLARea/
The metasystem's Perl code is released under GPL2.
Expected Competition Performance
Thanks to machine learning, MaLARea is strongest on batches of many
related problems with many redundant axioms where some of the problems
are easy to solve and can be used for learning the axiom relevance.
MaLARea is not very good when all problems are too difficult (nothing to
learn from), or the problems (are few and) have nothing in common.
Some of its techniques (selection by symbol and term-based similarity,
model-based reasoning) could however make it even there slightly stronger
than standard ATPs.
MaLARea has a very good performance on the MPTP Challenge, which is a
predecessor of the LTB division.
MetaProver 1.0
Matthew Streeter
Carnegie Mellon University, USA
Architecture
MetaProver is a hybridization of solvers entered in last year's CASC
competition.
When given a problem instance, MetaProver runs one or more solvers subject to
time limits, according to a fixed schedule.
The following is an example of such a schedule.
Note that the average time such a schedule requires to solve a problem instance
can be much lower than that of any of the algorithms used in the schedule.
Strategies
The strategies used by MetaProver include those used by its component
algorithms: DarwinFM 1.4.1, E 0.999, Geo 2007f, iProver 0.2, Metis 2.0,
Paradox 1.3, and Paradox 2.2.
At a higher level, MetaProver adopts the strategy of running its component
algorithms according to a schedule derived from the algorithms' performance
on the relevant TPTP benchmark instances, as described in the previous
section.
This schedule is computed using performance data for a large number of
benchmark instances, and thus it is reasonable to expect the schedule's
performance to generalize well to new, previously unseen instances.
Implementation
MetaProver is implemented as a set of bash scripts, and runs on Linux.
It is available online at:
http://www.cs.cmu.edu/~matts/MetaProver
Expected Competition Performance
MetaProver outperforms last year's SAT division winner (Paradox 1.3) and
last year's FNT division winner (Paradox 2.2) on the relevant TPTP benchmarks.
Metis 2.1
Joe Hurd
Galois, Inc., USA
Architecture
Metis 2.1 [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.
Strategies
Metis 2.1 uses a fixed strategy for every input problem.
Negative literals are always chosen in favour of positive literals, and
terms are ordered using the Knuth-Bendix ordering with uniform symbol
weight and precedence favouring reduced arity.
Implementation
Metis 2.1 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.
http://www.gilith.com/software/metis
Expected Competition Performance
The major change between Metis 2.0, which was entered into CASC-21, and
Metis 2.1 is the TSTP proof format. There were only minor changes to the
core proof engine, so Metis 2.1 is expected to perform at approximately
the same level and end up in the lower half of the table.
Muscadet 3.0
Dominique Pastre
Université René Descartes Paris‑5, France
Architecture
The MUSCADET theorem prover is a knowledge-based system.
It is based on Natural Deduction, following the terminology of
[Ble71] and
[Pas78],
and uses methods which resembles those used by humans.
It is composed of an inference engine, which interprets and executes rules,
and of one or several bases of facts,
which are the internal representation of "theorems to be proved".
Rules are either universal and put into the system, or built by the system
itself by metarules from data (definitions and lemmas).
Rules may add new hypotheses, modify the conclusion, create objects,
split theorems into two or more subtheorems
or build new rules which are local for a (sub-)theorem.
Strategies
There are specific strategies for existential, universal, conjonctive or
disjunctive hypotheses and conclusions.
Functional symbols may be used, but an automatic creation of intermediate
objects allows deep subformulae to be flattened and treated as if the
concepts were defined by predicate symbols.
The successive steps of a proof may be forward deduction (deduce new hypotheses
from old ones), backward deduction (replace the conclusion by a new one) or
refutation (only if the conclusion is a negation).
Implementation
Muscadet 3.0 [Pas01] is implemented in SWI-Prolog.
Rules are written as declarative Prolog clauses.
Metarules are written as sets of Prolog clauses, more or less declarative.
The inference engine includes the Prolog interpreter and some procedural Prolog
clauses.
http://www.math-info.univ-paris5.fr/~pastre/muscadet/muscadet.html
Expected Competition Performance
The best performances of Muscadet will be for problems
manipulating many concepts in which all statements (conjectures,
definitions, axioms) are expressed in a manner similar to the
practice of humans, especially of mathematicians
[Pas02,Pas07].
It will have poor performances for problems using few concepts but large
and deep formulas leading to many splittings.
Muscadet 3.0 will probably have the same performances as Muscadet 2.7a (2007 CASC-21 version + bugfixed), but will give an out proof for most solved problems.
OSHL-S 0.1
Hao Xu, David Plaisted
University of North Carolina at Chapel Hill, USA
Architecture
OSHL-S is a theorem prover based on the architecture and strategy
introduced in [PZ00] with a few improvements.
A preliminary form of type inference is employed to reduce the number of
instances that are generated before a contradicting instance is found.
Otter 3.3
William McCune
Argonne National Laboratory, USA
Architecture
Otter 3.3 [McC03a] is an ATP system for
statements in first-order (unsorted) logic with equality.
Otter is based on resolution and paramodulation applied to clauses.
An Otter search uses the "given clause algorithm", and typically involves
a large database of clauses; subsumption and demodulation play an important
role.
http://www-unix.mcs.anl.gov/AR/otter/
Acknowledgments: Ross Overbeek, Larry Wos, Bob Veroff, and Rusty Lusk contributed to the development of Otter.
Paradox 1.3
Koen Claessen, Niklas Sörensson
Chalmers University of Technology and
Gothenburg University, Sweden
Architecture
Paradox [CS03] is a
finite-domain model generator. It is based on a MACE-style
[McC94] flattening and instantiating of the
first-order clauses into propositional clauses, and then the use of
a SAT solver to solve the resulting problem.
Paradox incorporates the following features: Polynomial-time clause splitting heuristics, the use of incremental SAT, static symmetry reduction techniques, and the use of sort inference.
The main differences with Paradox 1.0 are: a better SAT-solver, better memory behaviour, and a faster clause instantiation algorithm.
Paradox 2.2 and 3.0
Koen Claessen, Niklas Sörensson
Chalmers University of Technology, Sweden
Architecture
Paradox 2.2 is a rewrite of Paradox 1.3.
Paradox 2.2 does not have all
the features yet that Paradox 1.3 has. Some experimental features,
such as type-based model finding, have been added.
Paradox 3.0 has the same description as Paradox 2.2.
See the description of Paradox 1.3 for general information.
randoCoP 1.1
Jens Otten, Thomas Raths
University of Potsdam, Germany
Architecture
randoCoP [RO08] is an automated theorem prover
for classical first-order logic. It is an extension of
the leanCoP [Ott08a,OB03]
prover, which is a very compact implementation of the connection calculus.
It integrates a technique that randomly alters the proof
search order by reordering the axioms of the given problem and
the literals within its clausal form.
http://www.leancop.de
There are two underlying provers supported: E and Vampire9.
E will be used in the competition division and Vampire9 in the demonstration
division (as the usage of Vampire in the competition division was not
allowed by its developers).
SInE is available from:
No system description supplied.
However, see the
description of Vampire 8.0 for general information.
Minor changes have been made, including a bugfix in the FOF to
CNF conversion.
None supplied.
No system description supplied.
However, see the
description of Waldmeister 704 for general information.
Zenon outputs totally formal proofs that can be checked by Coq.
SInE 0.3 and SInE-VD 0.3
Krystof Hoder
Charles University in Prague, Czech Republic
Architecture
SiNE 0.3 is an axiom selection system for first order theories.
It uses a syntactic approach based on symbols presence in axioms and
conjecture.
(When we say symbols, we mean functional, predicate and constant symbols
taken together.)
A relation D (as in "Defines") is created between symbols and axioms which
represents the fact that for a symbol there are some axioms that "give it
its meaning".
When the relation is constructed, the actual axiom selection starts.
At the beginning only the conjecture is selected, in each iteration the
selection is extended by all axioms that are D-related to any symbol used
in already selected axioms.
The iteration goes until no more axioms are selected.
Then the selected axioms are handed to an underlying inference engine.
Strategies
The construction of the D relation is inspired by the idea that general
symbols are more likely to define the meaning of more specific axioms than
vice-versa.
So given a generality measure on symbols, SiNE puts each axiom into the
relation with the least general of its symbols.
(When there are more of them, all are put in the relation.)
The generality measure used is the number of axioms in which the symbol
occurs.
(General symbols as s_Entity are likely to be used more often
than specific symbols like s_Monday.)
One slight optimisation for SUMO problems is also used: When we run into
a symbol ending with "_M" we remove the suffix for the selection process.
The "_M" suffix is used when a predicate symbol should be used as functional.
This strategy selects about 2% of axioms on problems CSR(075-109).
Implementation
The axiom selection is implemented in Python.
At first all problem files are read and include directives are extracted,
then problems which include the same sets of axioms are grouped together.
After that an iteration over phases starts.
Each phase defines one proving attempt on each problem.
It specifies the amount of time (relative to remaining time and unsolved
problem count), whether axioms from other successful proofs (in the same
group of problems) are included, and benevolence.
Benevolence greater than 1 means that not only the least general symbols can
be D-related to their formulas.
In each phase groups are iterated through, in each group the axioms which
would be included by problems are loaded and preprocessed, constructing the
D relation.
(This takes most of time of the whole axiom selection.)
Then for each problem a set of axioms is selected based on all symbols that
occur in the problem file, and an underlying prover is called.
http://www.ms.mff.cuni.cz/~hodek4am/sine.html
Expected Competition Performance
The batch mode was run on the problems from CSR[075-109]+[1-3].p whose status
was Theorem.
Using Vampire SInE proved 67 of 69 problems with an average time 130 seconds
per problem.
Using E, 64 problems were proved which took 20 seconds per problem.
Vampire 8.1
Andrei Voronkov
University of Manchester, England
Expected Competition Performance
Vampire 8.1 is the CASC-21 CNF division winner.
Vampire 9.0
Andrei Voronkov
University of Manchester, England
Expected Competition Performance
Vampire 9.0 is the CASC-21 FOF division winner.
Waldmeister 806
Thomas Hillenbrand1, Bernd Löchner2
1Max-Planck-Institut für Informatik Saarbrücken, Germany
2Technische Universität Kaiserslautern, Germany,
Expected Competition Performance
Waldmeister 806 is the CASC-21 UEQ division winner.
Zenon 0.5.0
Damien Doligez
INRIA, France
Architecture
Zenon 0.5.0 [BDD07] is based on the tableau method
with free variables.
It uses a nondestructive way of handling free variables, which enables
a purely local search procedure: each branch is closed before the
next one is explored.
Strategies
Implementation
Zenon is written in Objective Caml.
It can be downloaded from:
http://focal.inria.fr/zenon
Expected Competition Performance
Zenon is still in the prototype stage and we don't really expect
brilliant results at this point.