Entrants' System Descriptions

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.

Strategies

The strategy is as follows:
  1. Encode the existence of a rigid connection tableau with unification constraints.
  2. Run SMT solver on encoding.
  3. If the solver returns satisfiable, we verify that the model represents a rigid tableau. If it does we return unsatisfiable, otherwise we add additional clauses and goto 2.
  4. If the solver returns unsatisfiable we add additional instances of the initial clauses to the problem and goto 1.

Implementation

ChewTPTP is written in C++ and uses Yices SMT solver. Yices requires GMP, the GNU Multiprecision Library.

Expected Competition Performance

The system is a demonstration of the method described above and is not expected to win its division.

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.

Strategies

Darwin traverses the search space by iterative deepening over the term depth of candidate literals. Darwin employs a uniform search strategy for all problem classes.

Implementation

The central data structure is the context. A context represents an interpretation as a set of first-order literals. The context is grown by using instances of literals from the input clauses. The implementation of Darwin is intended to support basic operations on contexts in an efficient way. This involves the handling of large sets of candidate literals for modifying the current context. The candidate literals are computed via simultaneous unification between given clauses and context literals. This process is sped up by storing partial unifiers for each given clause and merging them for different combinations of context literals, instead of redoing the whole unifier computations. For efficient filtering of unneeded candidates against context literals, discrimination tree or substitution tree indexing is employed. The splitting rule generates choice points in the derivation which are backtracked using a form of backjumping similar to the one used in DPLL-based SAT solvers.

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/

Expected Competition Performance

Darwin 1.3 is the CASC-21 EPR division winner.

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.

Strategies

E has been optimized for performance over the TPTP. The automatic mode of E 1.0pre is inherited from previous version and is based on about 90 test runs over TPTP 3.1.1. It consists of the selection of one of about 40 different strategies for each problem. All test runs have been performed on SUN Ultra 60/300 machines with a time limit of 300 seconds (or roughly equivalent configurations). All individual strategies are refutationally complete. The worst one solves about 49% of TPTP 3.0.1, the best one about 60%.

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:

Wherever there is a three-way split on a numerical feature value, the limits are selected automatically with the aim of splitting the set of problems into approximately equal sized parts based on this one feature.

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.

Implementation

E is implemented in ANSI C, using the GNU C compiler. At the core is a 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. Other important features are the use of perfect discrimination trees with age and size constraints for rewriting and unit-subsumption, feature vector indexing [Sch04b] for forward- and backward subsumption and contextual literal cutting, and a new polynomial implementation of LPO [Loe04].

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 1.0pre 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 1.0p 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.

Strategies

The uniform search strategy is identical to the one employed in the original Darwin.

Implementation

E-Darwin is implemented in the functional/imperative language OCaml. The system has been tested on Unix and is available under the GNU Public License from the E-Darwin website at: 
    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.

Expected Competition Performance

The original Darwin has performed very well in the previous years, but since E-Darwin differs significantly from the original, it should be regarded as a new system. Very little testing has been done so far, making it difficult to estimate the performance.

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.

Strategies

E-KRHyper uses a uniform search strategy for all problems. The E-hyper tableau is generated depth-first, with E-KRHyper always working on a single branch. Refutational completeness and a fair search control are ensured by an iterative deepening strategy with a limit on the maximum term weight of generated clauses.

Implementation

E-KRHyper is implemented in the functional/imperative language OCaml, and Runs on Unix and MS-Windows platforms. The system accepts input in the TPTP format and in the TPTP-supported Protein format. The calculus implemented by E-KRHyper works on clauses, so first order formula input is converted into CNF by an algorithm which follows the transformation steps as used in the clausification of Otter [McC94]. E-KRHyper operates on an E-hyper tableau which is represented by linked node records. Several layered discrimination-tree based indexes (both perfect and non-perfect) provide access to the clauses in the tableau and support backtracking.

E-KRHyper is available under the GNU Public License from: 

    http://www.uni-koblenz.de/~bpelzer/ekrhyper

Expected Competition Performance

Most work on E-KRHyper since version 1.0 has been concerning its interfaces and embedding in natural language processing applications, so only a minor improvement in comparison to last year's performance can be expected.

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:
  1. Give all ground clauses in the problem to a SAT solver.
  2. Run the SAT-solver.
  3. If a contradiction is found, we have a proof and we terminate.
  4. If a model is found, we use the model to indicate which new ground instantiations should be added to the SAT-solver.
  5. Goto 2.

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.

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.

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.

iProver is available at: 

    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.
  1. Run E 0.999 for 0.01 seconds.
  2. Run Geo 2007f for 0.02 seconds.
  3. Run Metis 2.0 for 0.01 seconds.
  4. Run E 0.999 for 0.03 seconds.
  5. Run Geo 2007f for 0.06 seconds.
  6. Run Paradox 2.2 for 0.3 seconds.
  7. Run E 0.999 for 0.07 seconds.
  8. Run Paradox 2.2 for 1.48 seconds.
  9. Run Metis 2.0 for 0.43 seconds.
  10. Run Paradox 2.2 for 3.34 seconds.
  11. Run iProver 0.2 for 32.75 seconds.
  12. Run E 0.999 for 164.54 seconds.
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.

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.

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.

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.

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.

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: 

    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).

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.

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.

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: 

    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.

Strategies

OSHL-S employs a uniform strategy for all problems, which is similar to OSHL: It starts with a all negative or all positive model. In each iteration, it finds a contradicting instance according to a relaxed ordering and type information, and then modifies the model to make the instance satisfiable, if possible.

Implementation

OSHL-S is implemented in Java using Java SE 6 SDK and the Netbeans IDE. The profiler and the GUI provided in Java SE 6 and the Netbeans IDE are employed to profile the system, which provide performance data that are used for optimization. The implementation of OSHL-S combines a few strategies for improving the performance of the system, including caching of terms, clauses, and solutions to integer programming problems, lazy-evaluation, and backtracking.

Expected Competition Performance

The performance should be good for near propositional problems; otherwise, the competition performance is unknown.

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.

Strategies

Otter's original automatic mode, which reflects no tuning to the TPTP problems, will be used.

Implementation

Otter is written in C. Otter uses shared data structures for clauses and terms, and it uses indexing for resolution, paramodulation, forward and backward subsumption, forward and backward demodulation, and unit conflict. Otter is available from: 
    http://www-unix.mcs.anl.gov/AR/otter/

Expected Competition Performance

Otter has been entered into CASC as a stable benchmark against which progress can be judged (there have been only minor changes to Otter since 1996 [MW97], nothing that really affects its performace in CASC). This is not an ordinary entry, and we do not hope for Otter to do well in the competition.

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.

Strategies

There is only one strategy in Paradox:
  1. Analyze the problem, finding an upper bound N on the domain size of models, where N is possibly infinite. A finite such upper bound can be found, for example, for EPR problems.
  2. Flatten the problem, and split clauses and simplify as much as possible.
  3. Instantiate the problem for domain sizes 1 up to N, applying the SAT solver incrementally for each size. Report "SATISFIABLE" when a model is found.
  4. When no model of sizes smaller or equal to N is found, report "CONTRADICTION".
In this way, Paradox can be used both as a model finder and as an EPR solver.

Implementation

The main part of Paradox is implemented in Haskell using the GHC compiler. Paradox also has a built-in incremental SAT solver which is written in C++. The two parts are linked together on the object level using Haskell's Foreign Function Interface.

Expected Competition Performance

Paradox 1.3 is the CASC-21 SAT division winner.

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.

Expected Competition Performance

Paradox 2.2 is the CASC-21 FNT division winner.

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.

Strategies

The shell script of randoCoP repeatedly reorders the axioms and the literals within the clausal form, before invoking the two most successful variants of the leanCoP 2.0 core prover. These variants use restricted backtracking [Ott08b] and benefit significantly from the reordering techniques used by randoCoP.

Implementation

randoCoP is implemented in Prolog. The essential extensions consist of a modified shell script, which is used to invoke the leanCoP 2.0 core prover, and a few predicates that realize the reordering of axioms and literals. A preprocessing component translates first-order formulas into a (definitional) clausal form. Equality can be handled by adding the equality axioms. Version 1.1 of randoCoP returns a compact connection proof, which is then translated into a readable proof. The source code of randoCoP is available at 
    http://www.leancop.de

Expected Competition Performance

The performance of randoCoP is in particular good on problems involving large theories, i.e., problems that contain a large number of axioms.

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.

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: 

    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

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.

Expected Competition Performance

Vampire 8.1 is the CASC-21 CNF division winner.

Vampire 9.0

Andrei Voronkov
University of Manchester, England

None supplied.

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,

No system description supplied. However, see the description of Waldmeister 704 for general information.

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.

Zenon outputs totally formal proofs that can be checked by Coq.

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.

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