Entrants' System Descriptions


ATPBoost 1.0

Bartosz Piotrowski
University of Warsaw, Poland

Architecture

ATPBoost 1.0 [
PU18], is a metasystem for ATP in large theories relying on consistent naming of symbols and formulae. It learns premise selection from examples of successful proofs using several machine learning methods (k-nearest neighbors, gradient boosted trees, recurrent neural networks, and graph neural networks). The underlying deductive system is (currently) E prover. The system addresses the fact of a multiplicity of possible proofs. The learning process is reinforced by a feedback loop between the learners and the prover during which new training proofs may be found and wrong classifications of premises may be corrected. The version for CASC J10 will (likely) use non-neural machine learning methods only (due to time and hardware limitations, as the neural methods are time-consuming and often require GPU for efficient training).

Strategies

The main strategy of ATPBoost 1.0 for learning premise selection is to run multiple iterations of the learning-proving feedback loop. During the learning part, machine learning models are trained on the available proofs to select relevant axioms. During the proving part, the prover is run multiple times with axioms advised by the trained models. Newly found proofs give a feedback signal to the learners: highly-ranked premises which were not used in any of the proofs are returned as negative training examples, and new positive examples are extracted from the new proofs. The machine learning models trained in such a way provide premise-selection advice for ATP attempts for new conjectures. For each conjecture, the prover is run with several axiom limits.

Implementation

The metasystem is implemented in Python. It uses several machine learning libraries (XGBoost, OpenNMT, TensorFlow, Scikit-learn). ATPBoost is available from
    https://github.com/BartoszPiotrowski/ATPboost

Expected Competition Performance

ATPBoost is most useful when the number of available axioms is very large but the problems itself are not very difficult provided relevant axioms are selected. ATPBoost has good performance on the MPTP Challenge. The performance on the type of problems in the current LTB division is unknown!


CSE 1.3

Feng Cao
Southwest Jiaotong University, China

Architecture

CSE 1.3 is a developed prover based on the last version of CSE 1.2. It is an automated theorem prover for first-order logic without equality mainly based on a novel inference mechanism, called as Contradiction Separation Based Dynamic Multi-Clause Synergized Automated Deduction (S-CS) [
XL+18], which is able to handle multiple (two or more) clauses dynamically in a synergized way in one deduction step, while binary resolution is its special case. CSE 1.3 also adopts conventional factoring, equality resolution (ER rule), and variable renaming. Some pre-processing techniques, including pure literal deletion and simplification based on the distance to the goal clause, and a number of standard redundancy criteria for pruning the search space: tautology deletion, subsumption (forward and backward) are applied as well.

CSE 1.3 has been improved, compared with CSE 1.2, mainly from the following aspects:

  1. Optimization of contradiction separation algorithm based on full usage of clauses, which is able to increase the sufficiency of clauses participating in deduction.
  2. Optimization of contradiction separation algorithm based on optimized deduction path, which is able to effectively control the unifier complexity in the deduction process.
  3. Dynamic adjustment of clause and literal weight update in the deduction process.
Internally, CSE 1.3 works only with clausal normal form. E prover [Sch13] is adopted with thanks for clausification of full first-order logic problems during preprocessing.

Strategies

CSE 1.3 inherited most of the strategies in CSE 1.2. The main new strategies are:

Implementation

CSE 1.3 is implemented mainly in C++, and JAVA is used for batch problem running implementation. Shared data structure is used for constants and shared variables storage. In addition, special data structure is designed for property description of clause, literal and term, so that it can support the multiple strategy mode. E prover is used for clausification of FOF problems, and then TPTP4X is applied to convert the CNF format into TPTP format.

Expected Competition Performance

CSE 1.3 has made some improvements compared to CSE 1.2, and so we expect a better performance in this year's competition.

Acknowledgement:

Development of CSE 1.3 has been partially supported by the National Natural Science Foundation of China (NSFC) (Grant No.61673320).


CSE_E 1.2

Feng Cao
Southwest Jiaotong University, China

Architecture

CSE_E 1.2 is an automated theorem prover for first-order logic by combining CSE 1.3 and E 2.4, where CSE is based on the Contradiction Separation Based Dynamic Multi-Clause Synergized Automated Deduction (S-CS) [
XL+18] and E is mainly based on superposition. The combination mechanism is like this: E and CSE are applied to the given problem sequentially. If either prover solves the problem, then the proof process completes. If neither CSE nor E can solve the problem, some inferred clauses, especially unit clauses, by CSE will be fed to E as lemmas, along with the original clauses, for further proof search. This kind of combination is expected to take advantage of both CSE and E, and produce a better performance. Concretely, CSE is able to generate a good number of unit clauses, based on the fact that unit clauses are helpful for proof search and equality handling. On the other hand, E has a good ability on equality handling.

Strategies

The strategies of CSE part of CSE_E 1.2 take the same strategies as in CSE 1.3 standalone, e.g., clause/literal selection, strategy selection, and CSC strategy. The main new strategies for the combined systems are:

Implementation

CSE_E 1.2 is implemented mainly in C++, and JAVA is used for batch problem running implementation. The job dispatch between CSE and E is implemented in JAVA.

Expected Competition Performance

We expect CSE_E 1.2 to solve some hard problems and have a satisfying performance.

Acknowledgement:

Development of CSE 1.3 has been partially supported by the National Natural Science Foundation of China (NSFC) (Grant No.61673320). Stephan Schulz for his kind permission on using his E prover that makes CSE_E possible.


CVC4 1.8

Andrew Reynolds
University of Iowa, USA

Architecture

CVC4 [
BC+11] is an SMT solver based on the DPLL(T) architecture [NOT06] that includes built-in support for many theories, including linear arithmetic, arrays, bit vectors, datatypes, finite sets and strings. It incorporates approaches for handling universally quantified formulas. For problems involving free function and predicate symbols, CVC4 primarily uses heuristic approaches based on conflict-based instantiation and E-matching for theorems, and finite model finding approaches for non-theorems. For problems in pure arithmetic, CVC4 uses techniques for counterexample-guided quantifier instantiation [RKK17].

Like other SMT solvers, CVC4 treats quantified formulas using a two-tiered approach. First, quantified formulas are replaced by fresh Boolean predicates and the ground theory solver(s) are used in conjunction with the underlying SAT solver to determine satisfiability. If the problem is unsatisfiable at the ground level, then the solver answers "unsatisfiable". Otherwise, the quantifier instantiation module is invoked, and will either add instances of quantified formulas to the problem, answer "satisfiable", or return unknown. Finite model finding in CVC4 targets problems containing background theories whose quantification is limited to finite and uninterpreted sorts. In finite model finding mode, CVC4 uses a ground theory of finite cardinality constraints that minimizes the number of ground equivalence classes, as described in [RT+13]. When the problem is satisfiable at the ground level, a candidate model is constructed that contains complete interpretations for all predicate and function symbols. It then adds instances of quantified formulas that are in conflict with the candidate model, as described in [RT+13]. If no instances are added, it reports "satisfiable".

CVC4 has native support for problems in higher-order logic, as described in recent work [BR+19]. It uses a pragmatic approach for HOL, where lambdas are eliminated eagerly via lambda lifting. The approach extends the theory solver for quantifier-free uninterpreted functions (UF) and E-matching. For the former, the theory solver for UF in CVC4 now handles equalities between functions using an extensionality inference. Partial applications of functions are handle using a (lazy) applicative encoding where some function applications are equated to the applicative encoding. For the latter, several of the data structures for E-matching have been modified to incorporate matching in the presence of equalities between functions, function variables, and partial function applications.

Strategies

For handling theorems, CVC4 primarily uses conflict-based quantifier instantiation [RTd14,BFR17], enumerative instantiation [RBF18] and E-matching. CVC4 uses a handful of orthogonal trigger selection strategies for E-matching. For handling non-theorems, CVC4 primarily uses finite model finding techniques. Since CVC4 with finite model finding is also capable of establishing unsatisfiability, it is used as a strategy for theorems as well. For problems in pure arithmetic, CVC4 uses variations of counterexample-guided quantifier instantiation [RD+15], which select relevant quantifier instantiations based on models for counterexamples to quantified formulas. At the quantifier-free level, CVC4 uses standard decision procedures for linear arithmetic and uninterpreted functions, as well as heuristic approaches for handling non-linear arithmetic [RT+17].

Implementation

CVC4 is implemented in C++. The code is available from:
    https://github.com/CVC4

Expected Competition Performance

The first-order theorem proving and finite model finding capabilities of CVC4 have not changed much in the past year. Its heuristics for linear and non-linear arithmetic have changed slightly, which should impact TFA. Its higher-order capabilities have undergone some minor improvements, which should lead to slightly better performance in THF.


E 2.4

Stephan Schulz
DHBW Stuttgart, Germany

Architecture

E 2.4pre [
Sch02, Sch13, SCV19] 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 given-clause 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. As of E 2.1, PicoSAT [Bie08] can be used to periodically check the (on-the-fly grounded) proof state for propositional unsatisfiability.

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.

Strategies

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), which can be lifted in different ways to literal orderings.

For CASC-27, 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.

About 230 different strategies have been thoroughly evaluated on all untyped first-order problems from TPTP 7.2.0. In addition, we have explored some parts of the heuristic parameter space with a short time limit of 5 seconds. This allowed us to test about 650 strategies on all TPTP problems, and an extra 7000 strategies on all 1193 UEQ problems from TPTP 7.2.0. About 100 of these strategies are used in the automatic mode, and about 450 are used in at least one schedule.

Implementation

E is build around perfectly shared terms, i.e. each distinct term is represented only 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 [Sch13]. Superposition and backward rewriting use fingerprint indexing [Sch12], 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 [Loe06, Loe06]. The prover and additional information are available at
    https://www.eprover.org

Expected Competition Performance

E 2.4 was the CASC-27 UEQ winner.


E 2.5

Stephan Schulz
DHBW Stuttgart, Germany

Architecture

E 2.5pre [
Sch02, Sch13, SCV19] 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 given-clause 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. As of E 2.1, PicoSAT [Bie08] can be used to periodically check the (on-the-fly grounded) proof state for propositional unsatisfiability.

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.

Strategies

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), which can be lifted in different ways to literal orderings.

For CASC-J10, 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, possibly presence of certain axiom patterns, ...). 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.

About 130 different strategies have been thoroughly evaluated on all untyped first-order problems from TPTP 7.3.0. We have also explored some parts of the heuristic parameter space with a short time limit of 5 seconds. This allowed us to test about 650 strategies on all TPTP problems, and an extra 7000 strategies on UEQ problems from TPTP 7.2.0. About 100 of these strategies are used in the automatic mode, and about 450 are used in at least one schedule.

Implementation

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 [Sch13]. Superposition and backward rewriting use fingerprint indexing [Sch12], 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 [Loe06, Loe06]. The prover and additional information are available at
    https://www.eprover.org

Expected Competition Performance

The inference core of E 2.5pre has been slightly modified since last years pre-release. We have also been able to evaluate some more different search strategies. As a result, we expect performance to be somewhat better than in the last years, especially in UEQ. The system is expected to perform well in most proof classes, but will at best complement top systems in the disproof classes.


Enigma 0.5.1

Jan Jakubuv
Czech Technical University in Prague, Czech Republic

Architecture

ENIGMA (Efficient learNing-based Inference Guiding MAchine) [
JU17,JU18,JU19,GJU19,CJ+19,JC+20] is an efficient implementation of learning-based guidance for given clause selection in saturation-based automated theorem provers. Clauses from many proof searches are classified as positive and negative based on their participation in the proofs. An efficient classification model is trained on this data, using fast feature-based characterization of the clauses. The learned model is then tightly linked with the core prover and used as a basis of a parameterized evaluation heuristic that provides fast ranking of all generated clauses. ENIGMA 0.4 uses E as its underlying prover. The CASC-27 FOF competition version will most likely use a classifier based on gradient-boosted trees (XGBoost or LightGBM) and clause features that abstract from symbol and formula/clause names. We may also include neural classifiers based on clause structure. The system will be pre-trained on the latest public TPTP library and also use a portfolio of strategies pre-trained for TPTP with BliStr(Tune) [Urb13,JU17]. The system may also include strategies for large TPTP problems used previously in the E.T. system.

Strategies

The core system is a modification of E adding learning-based evaluation of the generated clauses. The system is pre-trained on TPTP using abstracted characterization of clauses, and the trained classifier is used during the competition as an additional clause evaluation heuristic that is combined in various ways with the core E strategies.

Implementation

The system modifies E's implementation by adding various features extraction methods and linking E with efficient learning-based classifiers (tree-based, linear and neural). ENIGMA is available at:
    https://github.com/ai4reason/enigmatic

Expected Competition Performance

ENIGMA should be able to improve on E's auto schedule in FOF.


Etableau 0.2

John Hester
University of Florida, USA

Architecture

Etableau 0.2 is an automated theorem prover built alongside E [
Sch13], using it as a library. Etableau at a basic level implements the clausal connection calculus, with enforced regularity, folding up, local unification, and other improvements. The greatest distinguishing feature of Etableau is a novel calculus using E's saturation to close local branches of tableaux. If a branch is local (shares no variables with other branches of the tableau) and cannot be closed by other means, the branch is saturated as an independent problem. If a contradiction is found, this local branch can now be marked as closed. If a local branch is found to be satisfiable, the problem is satisfiable.

Strategies

Etableau uses iterative deepening to control search through the space of tableaux. At a given maximum depth, tableaux to be expanded on are chosen based on the ratio of branches that have been closed. In other words, tableaux that have many closed branches and a small number of remaining open branches are processed first. In situations where attempts to close branches by saturation are made, E is used with its auto heuristic.

Implementation

Etableau is written in C, using many data structures and functions from E and with many added. Etableau is available at this github repository:
    https://github.com/hesterj/E-TAB

Expected Competition Performance

Etableau is expected to perform well on problems with smaller numbers of axioms, and struggle on problems with many axioms.


GKC 0.5.1

Tanel Tammet
Tallinn University of Technology, Estonia

Architecture

GKC [
Tam19] is a resolution prover optimized for search in large knowledge bases. It is used as a foundation (GK Core) for building a common-sense reasoner GK. In particular, GK can handle inconsistencies and perform probabilistic and nonmonotonic reasoning, see [Tam20]. We envision natural language question answering systems as the main potential application for these specialized methods.

These standard inference rules have been implemented in GKC:

GKC does not currently implement any propositional inferences or instance generation.

GKC splits the multiple strategies it decides to try between several forked instances. For the competition the plan is to use eight forks. Each fork runs a subset of strategies sequentially.

Strategies

For CASC GKC uses multiple strategies run sequentially, with the time limit starting at one second for each, increased 5 times once the whole batch has been perforfmed. The strategy selection takes into consideration the basic properties of the problem, in particular its approximate size. There is no interplay between different strategies.

We perform the selection of a given clause by using several queues in order to spread the selection relatively uniformly over these categories of derived clauses and their descendants: axioms, external axioms, assumptions and goals. The queues are organized in two layers. As a first layer we use the common ratio-based algorithm of alternating between selecting n clauses from a weight-ordered queue and one clause from the FIFO queue with the derivation order. As a second layer we use four separate queues based on the derivation history of a clause. Each queue in the second layer contains the two sub-queues of the first layer.

GKC only looks for proofs and does not try to show non-provability.

Implementation

GKC is implemented in C. The data representation machinery is built upon a shared memory graph database whitedb (http://whitedb.org/), enabling it to solve multiple different queries in parallel processeses without a need to repeatedly parse or load the large parsed knowledge base from the disk. An interesting aspect of GKC is the pervasive use of hash indexes, feature vectors and fingerprints, while no tree indexes are used.

GKC can be obtained from

    https://github.com/tammet/gkc/

Expected Competition Performance

Compared to the performance in previous CASC, GKC 0.5.1 should perform somewhat better. In particular, more search strategies have been implemented and the selection of search strategies is wider and more varied. The core algorithms and data structures remain the same. We expect GKC to be in the middle of the final ranking for FOF and below the average in UEQ and LTB. We expect GKC to perform well on very large problems.


iProver 3.3

Konstantin Korovin
University of Manchester, United Kingdom

Architecture

iProver [
Kor08] is an automated theorem prover based on an instantiation calculus Inst-Gen [Kor13, GK03], which is complete for first-order logic. iProver approximates first-order clauses using propositional abstractions which are solved using MiniSAT [ES04] and refined using model-guided instantiations. iProver combines instantiation, resolution and superposition [DK20] and is extended with a general abstraction-refinement framework for under-and over-approximations [HK18,HK19].

Recent features in iProver include:

Strategies

iProver has around 100 options to control the proof search including options for literal selection, passive clause selection, frequency of calling the SAT solver, simplifications and options for combination of instantiation with resolution and superposition. By default iProver will execute a small number of fixed schedules of selected options depending on general syntactic properties such as Horn/non-Horn, equational/non-equational, and maximal term depth. In the competition we will also use a selection of learnt strategies.

Implementation

iProver is implemented in OCaml and for the ground reasoning uses MiniSat [ES04]. iProver accepts FOF, TFF and CNF formats. Vampire [KV13,HK+12] and E prover [Sch13] are used for proof-producing clausification of FOF/TFF problems, Vampire is also used for SInE axiom selection [HV11] in the LTB division. iProver is available at:
    http://www.cs.man.ac.uk/~korovink/iprover/

Expected Competition Performance

We expect improvement in performance compared to the previous year due to improved implementation of superposition, simplifications and heuristic selection.


lazyCoP 0.1

Michael Rawson
University of Manchester, United Kingdom

Architecture

lazyCoP 0.1 is a connection-tableaux system for first-order logic with equality. It implements the lazy paramodulation calculus described in [
Pas08], with some additional inferences such as "shortcut" strict rules and equality lemmas. The system implements well-known refinements of the predicate connection calculus, such as tautology elimination and strong regularity, and these are lifted to equalities where appropriate. The resulting system appears to be complete, but we make no theoretical claim one way or another.

The system was originally conceived to efficiently accommodate a machine-learned heuristic guidance system: this system is not yet guided in this way, but learned heuristics are intended for a future version.

Strategies

The system explores a tableaux-level search space using the classic A* informed-search algorithm. The (admissible) heuristic function is the number of open branches. Typical connection systems explore via some kind of iterative deepening: A* search is a necessity for future learned guidance, and is not as catastrophic in memory consumption as might be expected. No form of strategy scheduling is yet implemented and the system will run for the entire time allowed on all available cores.

Implementation

A finite tree of inference rules forms a search space. To expand a selected leaf node, the system traverses from root to leaf, applying each rule to a new empty tableau. Possible inferences from the resulting tableau are added to the leaf and the resulting nodes are enqueued. The system does not yet include a custom clausification routine: a recent build of Vampire is employed for this purpose. lazyCoP is implemented entirely in the Rust programming language, allowing tight control over memory allocation and layout while avoiding some classes of memory- and thread- safety bugs. The source code (likely to be incomplete and/or buggy up to and including the competition!) is available at:
    https://github.com/MichaelRawson/lazycop

Expected Competition Performance

Performance on problems without equality is hoped to be comparable with other connection systems, if slightly slower. Problems requiring a modest amount of equational reasoning (or problems requiring no equational reasoning but containing extraneous equality axioms) are not expected to perform well, but should not cause catastrophic blowup either. Pure-equality problems (such as UEQ) are not the intended domain and do not perform well, but the first author remains hopeful for At Least One Problem.


leanCoP 2.2

Jens Otten
University of Oslo, Norway

Architecture

leanCoP [
OB03, Ott08] is an automated theorem prover for classical first-order logic with equality. It is a very compact implementation of the connection (tableau) calculus [Bib87, LS01].

Strategies

The reduction rule of the connection calculus is applied before the extension rule. Open branches are selected in a depth-first way. Iterative deepening on the proof depth is performed in order to achieve completeness. Additional inference rules and techniques include regularity, lemmata, and restricted backtracking [Ott10]. leanCoP uses an optimized structure-preserving transformation into clausal form [Ott10] and a fixed strategy schedule that is controlled by a shell script.

Implementation

leanCoP is implemented in Prolog. The source code of the core prover consists only of a few lines of code. Prolog's built-in indexing mechanism is used to quickly find connections when the extension rule is applied.

leanCoP can read formulae in leanCoP syntax and in TPTP first-order syntax. Equality axioms and axioms to support distinct objects are automatically added if required. The leanCoP core prover returns a very compact connection proof, which is then translated into a more comprehensive output format, e.g., into a lean (TPTP-style) connection proof or into a readable text proof.

The source code of leanCoP 2.2 is available under the GNU general public license. It can be downloaded from the leanCoP website at:

    http://www.leancop.de
The website also contains information about ileanCoP [Ott08] and MleanCoP [Ott12, Ott14], two versions of leanCoP for first-order intuitionistic logic and first-order modal logic, respectively.

Expected Competition Performance

As the prover has not changed, the performance of leanCoP 2.2 is expected to be the same as last year.


LEO-II 1.7.0

Alexander Steen
University of Luxembourg, Luxembourg

Architecture

LEO-II [
BP+08], the successor of LEO [BK98], is a higher-order ATP system based on extensional higher-order resolution. More precisely, LEO-II employs a refinement of extensional higher-order RUE resolution [Ben99]. LEO-II is designed to cooperate with specialist systems for fragments of higher-order logic. By default, LEO-II cooperates with the first-order ATP system E [Sch02]. LEO-II is often too weak to find a refutation amongst the steadily growing set of clauses on its own. However, some of the clauses in LEO-II's search space attain a special status: they are first-order clauses modulo the application of an appropriate transformation function. Therefore, LEO-II launches a cooperating first-order ATP system every n iterations of its (standard) resolution proof search loop (e.g., 10). If the first-order ATP system finds a refutation, it communicates its success to LEO-II in the standard SZS format. Communication between LEO-II and the cooperating first-order ATP system uses the TPTP language and standards.

Strategies

LEO-II employs an adapted "Otter loop". Moreover, LEO-II uses some basic strategy scheduling to try different search strategies or flag settings. These search strategies also include some different relevance filters.

Implementation

LEO-II is implemented in OCaml 4, and its problem representation language is the TPTP THF language [BRS08]. In fact, the development of LEO-II has largely paralleled the development of the TPTP THF language and related infrastructure [SB10]. LEO-II's parser supports the TPTP THF0 language and also the TPTP languages FOF and CNF.

Unfortunately the LEO-II system still uses only a very simple sequential collaboration model with first-order ATPs instead of using the more advanced, concurrent and resource-adaptive OANTS architecture [BS+08] as exploited by its predecessor LEO.

The LEO-II system is distributed under a BSD style license, and it is available from

    http://www.leoprover.org

Expected Competition Performance

LEO-II ist not actively being developed anymore, hence there are no expected improvements to last year's CASC results.


Leo-III 1.4

Alexander Steen
University of Luxembourg, Luxembourg

Architecture

Leo-III [
SB18] [SB18], the successor of LEO-II [BP+08] is a higher-order ATP system based on extensional higher-order paramodulation with inference restrictions using a higher-order term ordering. The calculus contains dedicated extensionality rules and is augmented with equational simplification routines that have their intellectual roots in first-order superposition-based theorem proving. The saturation algorithm is a variant of the given clause loop procedure inspired by the first-order ATP system E.

Leo-III cooperates with external first-order ATPs which are called asynchronously during proof search; a focus is on cooperation with systems that support typed first-order (TFF) input. For this year's CASC, CVC4 [BC+11] and E [Sch02, Sch13] are used as external systems. However, cooperation is in general not limited to first-order systems. Further TPTP/TSTP-compliant external systems (such as higher-order ATPs or counter model generators) may be included using simple command-line arguments. If the saturation procedure loop (or one of the external provers) finds a proof, the system stops, generates the proof certificate and returns the result.

For the LTB division, Leo-III is augmented by an external Python3 driver which schedules Leo-III of the batches.

Strategies

Leo-III comes with several configuration parameters that influence its proof search by applying different heuristics and/or restricting inferences. These parameters can be chosen manually by the user on start-up. Leo-III currently does not offer portfolio scheduling (time slicing) nor automatic selection of configuration parameters that seems somehow beneficial for the reasoning problem input at hand. Strategies and strategy scheduling will be addressed in further upcoming versions.

Implementation

Leo-III utilizes and instantiates the associated LeoPARD system platform [WSB15] for higher-order (HO) deduction systems implemented in Scala (currently using Scala 2.12 and running on a JVM with Java 8). The prover makes use of LeoPARD's sophisticated data structures and implements its own reasoning logic on top. A generic parser is provided that supports all TPTP syntax dialects. It is implemented using ANTLR4 and converts its produced concrete syntax tree to an internal TPTP AST data structure which is then transformed into polymorphically typed lambda terms. As of version 1.1, Leo-III supports all common TPTP dialects (CNF, FOF, TFF, THF) as well as its polymorphic variants [BP13, KSR16].

The term data structure of Leo-III uses a polymorphically typed spine term representation augmented with explicit substitutions and De Bruijn-indices. Furthermore, terms are perfectly shared during proof search, permitting constant-time equality checks between alpha-equivalent terms.

Leo-III's saturation procedure may at any point invoke external reasoning tools. To that end, Leo-III includes an encoding module which translates (polymorphic) higher-order clauses to polymorphic and monomorphic typed first-order clauses, whichever is supported by the external system. While LEO-II relied on cooperation with untyped first-order provers, Leo-III exploits the native type support in first-order provers (TFF logic) for removing clutter during translation and, in turn, higher effectivity of external cooperation.

Leo-III is available on GitHub:

    https://github.com/leoprover/Leo-III

Expected Competition Performance

Leo 1.4 was the CASC-27 LTB winner.


Leo-III 1.5

Alexander Steen
University of Luxembourg, Luxembourg

Architecture

Leo-III [
SB18], the successor of LEO-II [BP+08], is a higher-order ATP system based on extensional higher-order paramodulation with inference restrictions using a higher-order term ordering. The calculus contains dedicated extensionality rules and is augmented with equational simplification routines that have their intellectual roots in first-order superposition-based theorem proving. The saturation algorithm is a variant of the given clause loop procedure inspired by the first-order ATP system E. Leo-III cooperates with external first-order ATPs that are called asynchronously during proof search; a focus is on cooperation with systems that support typed first-order (TFF) input. For this year's CASC, CVC4 [BC+11] and E [Sch02,Sch13] are used as external systems. However, cooperation is in general not limited to first-order systems. Further TPTP/TSTP-compliant external systems (such as higher-order ATPs or counter model generators) may be included using simple command-line arguments. If the saturation procedure loop (or one of the external provers) finds a proof, the system stops, generates the proof certificate and returns the result.

For the LTB division, Leo-III is augmented by an external Python3 driver which schedules Leo-III on the batches.

Strategies

Leo-III comes with several configuration parameters that influence its proof search by applying different heuristics and/or restricting inferences. These parameters can be chosen manually by the user on start-up. Leo-III implements a naive time slicing approach of some of these strategies for this CASC.

Implementation

Leo-III utilizes and instantiates the associated LeoPARD system platform [WSB15] for higher-order (HO) deduction systems implemented in Scala (currently using Scala 2.13 and running on a JVM with Java 8). The prover makes use of LeoPARD's sophisticated data structures and implements its own reasoning logic on top. A generic parser is provided that supports all TPTP syntax dialects. It is implemented using ANTLR4 and converts its produced concrete syntax tree to an internal TPTP AST data structure, which is then transformed into polymorphically typed lambda terms. As of version 1.1, Leo-III supports all common TPTP dialects (CNF, FOF, TFF, THF) as well as its polymorphic variants [BP13,KRS16].

The term data structure of Leo-III uses a polymorphically typed spine term representation augmented with explicit substitutions and De Bruijn-indices. Furthermore, terms are perfectly shared during proof search, permitting constant-time equality checks between alpha-equivalent terms.

Leo-III's saturation procedure may at any point invoke external reasoning tools. To that end, Leo-III includes an encoding module which translates (polymorphic) higher-order clauses to polymorphic and monomorphic typed first-order clauses, whichever is supported by the external system. While LEO-II relied on cooperation with untyped first-order provers, Leo-III exploits the native type support in first-order provers (TFF logic) for removing clutter during translation and, in turn, higher effectivity of external cooperation.

Leo-III is available on GitHub:

    https://github.com/leoprover/Leo-III

Expected Competition Performance

Version 1.5 only marginally improves the previous release by fixing some bugs. As CASC is using wall clock (WC) time instead of CPU time usage in all divisions, the Java VM version of Leo-III is used in the competition (as opposed to a native build used last year). We hope that the JRE performs, after a slow start-up, quite well on longer (wrt. WC time) runs. We expect a similar performance as in last year's CASC.

Also in the LTB mode, there are no major novelties: only some timing parameters have been changed compared to last year. Stemming from Leo-III's support for polymorphic HOL reasoning, we expect a reasonable performance compared to the other systems. On the other hand, Leo-III's LTB mode does not do any learning and/or analysis of the learning samples.


MaLARea 0.9

Josef Urban
Czech Technical University in Prague, Czech Republic

Architecture

MaLARea 0.8 [
Urb07,US+08,KUV15] is a metasystem for ATP in large theories where symbol and formula names are used consistently. It uses several deductive systems (now E, SPASS, Vampire, Paradox, Mace), as well as complementary AI techniques like machine learning based on symbol-based similarity, model-based similarity, term-based similarity, and obviously previous successful proofs. The version for CASC-27 will use the E prover with the BliStr(Tune) [Urb13,JU17] large-theory strategies, possibly also Prover9, Mace and Paradox. The premise selection methods will likely also use the distance-weighted k-nearest neighbor [KU13], ATPBoost [PU18], and E's implementation of SInE.

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 time limits 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:
    https://github.com/JUrban/MPTP2/tree/master/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, and on several previous LTB competitions.


Prover9 1109a

Bob Veroff on behalf of William McCune
University of New Mexico, USA

Architecture

Prover9, Version 2009-11A, is a resolution/paramodulation prover for first-order logic with equality. Its overall architecture is very similar to that of Otter-3.3 [
McC03]. It uses the "given clause algorithm", in which not-yet-given clauses are available for rewriting and for other inference operations (sometimes called the "Otter loop").

Prover9 has available positive ordered (and nonordered) resolution and paramodulation, negative ordered (and nonordered) resolution, factoring, positive and negative hyperresolution, UR-resolution, and demodulation (term rewriting). Terms can be ordered with LPO, RPO, or KBO. Selection of the "given clause" is by an age-weight ratio.

Proofs can be given at two levels of detail: (1) standard, in which each line of the proof is a stored clause with detailed justification, and (2) expanded, with a separate line for each operation. When FOF problems are input, proof of transformation to clauses is not given.

Completeness is not guaranteed, so termination does not indicate satisfiability.

Strategies

Prover9 has available many strategies; the following statements apply to CASC.

Given a problem, Prover9 adjusts its inference rules and strategy according to syntactic properties of the input clauses such as the presence of equality and non-Horn clauses. Prover9 also does some preprocessing, for example, to eliminate predicates.

For CASC Prover9 uses KBO to order terms for demodulation and for the inference rules, with a simple rule for determining symbol precedence.

For the FOF problems, a preprocessing step attempts to reduce the problem to independent subproblems by a miniscope transformation; if the problem reduction succeeds, each subproblem is clausified and given to the ordinary search procedure; if the problem reduction fails, the original problem is clausified and given to the search procedure.

Implementation

Prover9 is coded in C, and it uses the LADR libraries. Some of the code descended from EQP [McC97]. (LADR has some AC functions, but Prover9 does not use them). Term data structures are not shared (as they are in Otter). Term indexing is used extensively, with discrimination tree indexing for finding rewrite rules and subsuming units, FPA/Path indexing for finding subsumed units, rewritable terms, and resolvable literals. Feature vector indexing [Sch04] is used for forward and backward nonunit subsumption. Prover9 is available from
    http://www.cs.unm.edu/~mccune/prover9/

Expected Competition Performance

Prover9 is the CASC fixed point, against which progress can be judged. Each year it is expected do worse than the previous year, relative to the other systems.


PyRes 1.3

Stephan Schulz
DHBW Stuttgart, Germany

Architecture

PyRes [
SP20] is a simple resolution-style theorem prover for first-order logic, implemented in very clear and well-commented Python. It has been written as a pedagogical tool to illustrate the architecture and basic algorithms of a saturation-style theorem prover. The prover consists of a parser for (most of) TPTP-3 format, a simple clausifier to convert full first-order format into clause normal form, and a saturation core trying to derive the empty clause from the resulting clause set.

The saturation core is based on the DISCOUNT-loop variant of the given-clause algorithm, i.e., a strict separation of active and passive facts. It implements simple binary resolution and factoring [Rob65], optionally with selection of negative literals [BG+01]. Redundancy elimination is restricted to forward and backward subsumption and tautology deletion. There are no inference rules for equality - if equality is detected, the necessary axioms are added.

Strategies

The prover supports several negative literal selection strategies, as well as selection of the given clause from a set of differently weighted priority queues in the style of E [SCV19]. In the competition, it will always select the syntactically largest literal, and will use weight-age interleaved clause selection with a pick-given ratio of 5 to 1.

Implementation

The prover is implemented in Python 3, with maximal emphasis on clear and well-documented code. Terms are represented as nested lists (equivalent to LISP style s-expressions). Literals, clauses, and formulas are implemented as classes using an object-oriented style.

One of the changes compared to last years version is the introduction of simple indices for unification (which now uses top symbol hashing) and subsumption. Subsumption now uses a new, simple technique we call predicate abstraction indexing, which represents a clause as an ordered sequence of the predicate symbols of its literals. PyRes builds a proof object on the fly, and can print a TPTP-3 style proof or saturation derivation.

The system source is available at:

    https://github.com/eprover/PyRes

Expected Competition Performance

Performance is expected to be better than last year, but still mediocre for non-equational problems, and abysmal for problems with equality. However, per CASC rules, PyRes will still be assumed superior to any non-participating prover.


Satallax 3.4

Michael Färber
ENS Paris-Saclay, France

Architecture

Satallax 3.4 [
Bro12] is an automated theorem prover for higher-order logic. The particular form of higher-order logic supported by Satallax is Church's simple type theory with extensionality and choice operators. The SAT solver MiniSat [ES04] is responsible for much of the proof search. The theoretical basis of search is a complete ground tableau calculus for higher-order logic [BS10] with a choice operator [BB11]. Problems are given in the THF format.

Proof search: A branch is formed from the axioms of the problem and the negation of the conjecture (if any is given). From this point on, Satallax tries to determine unsatisfiability or satisfiability of this branch. Satallax progressively generates higher-order formulae and corresponding propositional clauses [Bro13]. These formulae and propositional clauses correspond to instances of the tableau rules. Satallax uses the SAT solver MiniSat to test the current set of propositional clauses for unsatisfiability. If the clauses are unsatisfiable, then the original branch is unsatisfiable. Optionally, Satallax generates lambda-free higher-order logic (lfHOL) formulae in addition to the propositional clauses [VB+19]. If this option is used, then Satallax periodically calls the theorem prover E [Sch13] to test for lfHOL unsatisfiability. If the set of lfHOL formulae is unsatisfiable, then the original branch is unsatisfiable. Upon request, Satallax attempts to reconstruct a proof which can be output in the TSTP format.

Strategies

There are about 150 flags that control the order in which formulae and instantiation terms are considered and propositional clauses are generated. Other flags activate some optional extensions to the basic proof procedure (such as whether or not to call the theorem prover E). A collection of flag settings is called a mode. Approximately 500 modes have been defined and tested so far. A strategy schedule is an ordered collection of modes with information about how much time the mode should be allotted. Satallax tries each of the modes for a certain amount of time sequentially. Before deciding on the schedule to use, Satallax parses the problem and determines if it is big enough that a SInE-based premise selection algorithm [HV11] should be used. If SInE is not activated, then Satallax uses a strategy schedule consisting of 37 modes. If SInE is activated, than Satallax is run with a SInE-specific schedule consisting of 58 modes with different SInE parameter values selecting different premises. Each mode is tried for time limits ranging from less than a second to just over 1 minute.

Implementation

Satallax is implemented in OCaml, making use of the external tools MiniSat (via a foreign function interface) and E. Satallax is available at:
    http://cl-informatik.uibk.ac.at/~mfaerber/satallax.html

Expected Competition Performance

Satallax 3.4 was the CASC-27 THF winner.


Satallax 3.5

Michael Färber
ENS Paris-Saclay, France

Architecture

Satallax [
Bro12] is an automated theorem prover for higher-order logic. The particular form of higher-order logic supported by Satallax is Church’s simple type theory with extensionality and choice operators. The SAT solver MiniSat [ES04] is responsible for much of the proof search. The theoretical basis of search is a complete ground tableau calculus for higher-order logic [BS10] with a choice operator [BB11]. Problems are given in the THF format. A branch is formed from the axioms of the problem and the negation of the conjecture (if any is given). From this point on, Satallax tries to determine unsatisfiability or satisfiability of this branch. Satallax progressively generates higher-order formulae and corresponding propositional clauses [Bro13]. These formulae and propositional clauses correspond to instances of the tableau rules. Satallax uses the SAT solver MiniSat to test the current set of propositional clauses for unsatisfiability. If the clauses are unsatisfiable, then the original branch is unsatisfiable. Optionally, Satallax generates lambda-free higher-order logic (lfHOL) formulae in addition to the propositional clauses [VB+19]. If this option is used, then Satallax periodically calls the theorem prover E [Sch13] to test for lfHOL unsatisfiability. If the set of lfHOL formulae is unsatisfiable, then the original branch is unsatisfiable. Upon request, Satallax attempts to reconstruct a proof which can be output in the TSTP format.

Strategies

There are about 150 flags that control the order in which formulae and instantiation terms are considered and propositional clauses are generated. Other flags activate some optional extensions to the basic proof procedure (such as whether or not to call the theorem prover E). A collection of flag settings is called a mode. Approximately 500 modes have been defined and tested so far. A strategy schedule is an ordered collection of modes with information about how much time the mode should be allotted. Satallax tries each of the modes for a certain amount of time sequentially. Before deciding on the schedule to use, Satallax parses the problem and determines if it is big enough that a SInE-based premise selection algorithm [HV11] should be used. If SInE is not activated, then Satallax uses a strategy schedule consisting of 37 modes. If SInE is activated, than Satallax is run with a SInE-specific schedule consisting of 58 modes with different SInE parameter values selecting different premises. Each mode is tried for time limits ranging from less than a second to just over 1 minute.

Implementation

Satallax is implemented in OCaml, making use of the external tools MiniSat (via a foreign function interface) and E.

Satallax is available at:

    http://cl-informatik.uibk.ac.at/~mfaerber/satallax.html

Expected Competition Performance

The main change from Satallax 3.4 is the switch to a branch of E that fixes a higher-order unsoundness. The performance of Satallax 3.5 should be comparable to Satallax 3.4.


Twee 2.2.1

Nick Smallbone
Chalmers University of Technology, Sweden

Architecture

Twee 2.2.1 is an equational prover based on unfailing completion [
BDP89]. It features ground joinability testing [MN90] and a connectedness test [BD88], which together eliminate many redundant inferences in the presence of unorientable equations.

Twee's implementation of ground joinability testing performs case splits on the order of variables, in the style of [MN90], and discharges individual cases by rewriting modulo a variable ordering. The case splitting strategy chooses only useful case splits, which prevents the number of cases from blowing up.

Horn clauses are encoded as equations as described in [CS18]. The CASC version of Twee "handles" non-Horn clauses by discarding them.

Strategies

Twee's strategy is simple and it does not tune its heuristics or strategy based on the input problem. The term ordering is always KBO; functions are ordered by number of occurrences and always have weight 1.

The main loop is a DISCOUNT loop. The active set contains rewrite rules and unorientable equations, which are used for rewriting, and the passive set contains unprocessed critical pairs. Twee often interreduces the active set, and occasionally simplifies the passive set with respect to the active set. Each critical pair is scored using a weighted sum of the weight of both of its terms. Terms are treated as DAGs when computing weights, i.e., duplicate subterms are only counted once per term. The weights of critical pairs that correspond to Horn clauses are adjusted by the heuristic described in [CS18], section 5.

For CASC, to take advantage of multiple cores, several versions of Twee run in parallel using different parameters.

Implementation

Twee is written in Haskell. Terms are represented as array-based flatterms for efficient unification and matching. Rewriting uses an imperfect discrimination tree.

The passive set is represented compactly (12 bytes per critical pair) by storing only the information needed to reconstruct the critical pair, not the critical pair itself. Because of this, Twee can run for an hour or more without exhausting memory.

Twee uses an LCF-style kernel: all rules in the active set come with a certified proof object which traces back to the input axioms. When a conjecture is proved, the proof object is transformed into a human-readable proof. Proof construction does not harm efficiency because the proof kernel is invoked only when a new rule is accepted. In particular, reasoning about the passive set does not invoke the kernel. The translation from Horn clauses to equations is not yet certified.

Twee can be downloaded from:

    http://nick8325.github.io/twee

Expected Competition Performance

Twee is quite strong at UEQ, but will be at a disadvantage compared to provers with smart timeslicing (such as E and Vampire) which can make better use of the multiple cores. Prediction: Twee will solve more problems than last year but finish below E and Vampire. It could still solve some problems that no-one else does, though. Twee will do badly in FOF since it throws away all non-Horn clauses. It may get lucky and solve a few hard problems, especially if some mostly-equational problems show up.


Vampire 4.4

Giles Reger
University of Manchester, United Kingdom

There are no major changes to the main part of Vampire since 4.4, beyond some new proof search heuristics and new default values for some options. The biggest addition is support for higher-order reasoning via translation to applicative form and combinators, addition of axioms and extra inference rules, and a new form of combinatory unification.

Architecture

Vampire [
KV13] 4.4 is an automatic theorem prover for first-order logic with extensions to theory-reasoning and higher-order logic. Vampire implements the calculi of ordered binary resolution and superposition for handling equality. It also implements the Inst-gen calculus and a MACE-style finite model builder [RSV16]. Splitting in resolution-based proof search is controlled by the AVATAR architecture which uses a SAT or SMT solver to make splitting decisions [Vor14, RB+16]. Both resolution and instantiation based proof search make use of global subsumption.

A number of standard redundancy criteria and simplification techniques are used for pruning the search space: subsumption, tautology deletion, subsumption resolution and rewriting by ordered unit equalities. The reduction ordering is the Knuth-Bendix Ordering. Substitution tree and code tree indexes are used to implement all major operations on sets of terms, literals and clauses. Internally, Vampire works only with clausal normal form. Problems in the full first-order logic syntax are clausified during preprocessing. Vampire implements many useful preprocessing transformations including the SinE axiom selection algorithm.

When a theorem is proved, the system produces a verifiable proof, which validates both the clausification phase and the refutation of the CNF. Vampire 4.4 provides a very large number of options for strategy selection. The most important ones are:

Implementation

Vampire 4.4 is implemented in C++. It makes use of Minisat and Z3.

Expected Competition Performance

Vampire 4.4 was the CASC-27 TFA, FOF, and FNT winner.


Vampire 4.5

Giles Reger
University of Manchester, United Kingdom

Architecture

Vampire [
KV13] 4.5 is an automatic theorem prover for first-order logic with extensions to theory-reasoning and higher-order logic. Vampire implements the calculi of ordered binary resolution and superposition for handling equality. It also implements the Inst-gen calculus and a MACE-style finite model builder [RSV16]. Splitting in resolution-based proof search is controlled by the AVATAR architecture which uses a SAT or SMT solver to make splitting decisions [Vor14,RB+16]. A number of standard redundancy criteria and simplification techniques are used for pruning the search space: subsumption, tautology deletion, subsumption resolution and rewriting by ordered unit equalities. The reduction ordering is the Knuth-Bendix Ordering. Substitution tree and code tree indexes are used to implement all major operations on sets of terms, literals and clauses. Internally, Vampire works only with clausal normal form. Problems in the full first-order logic syntax are clausified during preprocessing [RSV16]. Vampire implements many useful preprocessing transformations including the SinE axiom selection algorithm. When a theorem is proved, the system produces a verifiable proof, which validates both the clausification phase and the refutation of the CNF.

Strategies

Vampire 4.5 provides a very large number of options for strategy selection. The most important ones are:

Implementation

Vampire 4.5 is implemented in C++. It makes use of minisat and Z3. See the website
    https://vprover.github.io/
for more information and access to the GitHub repository.

Expected Competition Performance

There are four areas of improvement in Vampire 4.5. Firstly, a new layered clause selection approach [GS20] gives Vampire more fine-grained control over clause selection, in particular the way in which clauses involving theory axioms are selected. Secondly, theory evaluation and instantiation methods have been overhauled. Thirdly, a new subsumption demodulation rule [GKR20] improves support for reasoning with conditional equalities. Finally, higher-order reasoning (introduced in Vampire 4.4) has been rewritten based on a new superposition calculus [BG20] utilising a KBO-like ordering [BG20] for orienting combinator equations. Vampire 4.5 should be an improvement on Vampire 4.4.


Zipperposition 2.0

Petar Vukmirović
Vrije Universiteit Amsterdam, The Netherlands

Architecture

Zipperposition is a superposition-based theorem prover for typed first-order logic with equality and higher-order logic. It is a pragmatic implementation of a complete calculus for Boolean-free higher-order logic [
BB+19]. It features a number of extensions that include polymorphic types; user-defined rewriting on terms and formulas ("deduction modulo theories"); a lightweight variant of AVATAR for case splitting; boolean reasoning [VN20]. The core architecture of the prover is based on saturation with an extensible set of rules for inferences and simplifications. Zipperposition uses a recently developed full higher-order unification algorithm that enables efficient integration of procedures for decidable fragments of higher-order unification [VBN20]. The initial calculus and main loop were imitations of an old version of E [Sch02], but there are many more rules nowadays. A summary of the calculus for integer arithmetic and induction can be found in [Cru15].

Strategies

The system uses various strategies in a portfolio. The strategies are run in parallel, making use of all CPU cores available. We designed the portfolio of strategies by manual inspection of different TPTP problems. Heuristics used in Zipperposition are inspired by efficient heuristics used in E. Portfolio mode differentiates higher-order problems from the first-order ones. If the problem is first-order all higher-order prover features are turned off. Other than that, the portfolio is static and does not depend on the syntactic properties of the problem.

Implementation

The prover is implemented in OCaml, and has been around for eight years. Term indexing is done using fingerprints for unification, perfect discrimination trees for rewriting, and feature vectors for subsumption. Some inference rules such as contextual literal cutting make heavy use of subsumption. For higher-order problems some strategies use E prover, running in lambda-free higher-order mode, as an end-game backend prover. The code can be found at
    https://github.com/sneeuwballen/zipperposition
and is entirely free software (BSD-licensed). Zipperposition can also output graphic proofs using graphviz. Some tools to perform type inference and clausification for typed formulas are also provided, as well as a separate library for dealing with terms and formulas [Cru15].

Expected Competition Performance

The prover is expected to have average performance on FOF, similar to Prover9, and a good performance on THF, at the level of last-year's CASC winner.