- CSE 1.5
- CSE_E 1.4
- cvc5 1.0
- Drodi 3.3.3
- E 3.0
- Ehoh 2.7
- Etableau 0.67
- GKC 0.7
- Goéland 1.0.0
- iProver 3.6
- Lash 1.12
- LEO-II 1.7.0
- Leo-III 1.7.0
- Prover9 1109a
- Satallax 3.4
- SnakeForV4.7 1.0
- Toma 0.2
- Twee 2.4
- Twee 2.4.1
- Vampire 4.5
- Vampire 4.6
- Vampire 4.7
- Zipperposition 2.1
- Zipperposition 2.1.999

JiangXi University of Science and Technology, China

CSE 1.5 has been improved compared with CSE 1.4, mainly from the following aspects:

- Optimization of the contradiction separation algorithm based on full usage of clauses, the number of repetitions of the input clause used in the deduction process is calculated by the evaluation function.
- In order to avoid the local optimum caused by multi-clause contradiction separation deduction path search according to the given rules, a restart mechanism is added.

- Deduction control strategy. In the process of proof search, different contradiction separation algorithms can be used interchangeably.
- Clause selection strategy. Add the priority order of long clauses participating in deduction.

**Acknowledgement:**
Development of CSE 1.5 has been partially supported by the General Research
Project of Jiangxi Education Department (Grant No. GJJ200818).

Southwest Jiaotong University, China

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.

- Lemma filtering mainly based on deduction weight of binary clauses.
- Fine-grained dynamic time allocation scheme in different run stages.

**Acknowledgement:**
Development of CSE_E 1.4 has been partially supported by the National Natural
Science Foundation of China (NSFC) (Grant No. 61976130).
Stephan Schulz for his kind permission on using his E prover that makes
CSE_E possible.

University of Iowa, USA

cvc5 has native support for problems in higher-order logic, as described in [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 cvc5 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.

https://github.com/cvc5/cvc5

Amateur Programmer, Spain

- Otter, Discount and Limited Resource Strategy [RV03] saturation algorithms.
- A basic implementation of AVATAR architecture [Vor14].
- Several literal and term reduction orderings.
- Several literal selection options.
- Several clause selection heuristics with adjustable selection ratios, including several types of clause weight queues and one age queue.
- Classical clause relevancy pruning.
- Drodi can generate a learning file from successful proofs and use the file to guide clause selection strategy.

- Predicate is in a selected or unselected literal.
- Predicate or function is in the conjecture.
- Predicate or function is only in input formulas that are not the conjecture nor axioms.
- Predicate or function is only in axiom input formulas.

DHBW Stuttgart, Germany

For CASC-J11, E implements a multi-core 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 140 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.

https://www.eprover.org

Vrije Universiteit Amsterdam, The Netherlands

https://github.com/eprover/eproverwhich includes more details on Ehoh's compilation and installation.

University of Florida, USA

https://github.com/hesterj/Etableau

Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier, France

https://github.com/GoelandProver/Goeland

Tallinn University of Technology, Estonia

GKC 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
[Tam21,
Tam22].
We are working on a natural language question answering system and envision
NLP question answering as the main potential application for these specialized
methods.
The WASM version of the previous GKC 0.6 is used as the prover engine in the
educational `http://logictools.org`
system.
It can read and output proofs in the TPTP, simplified TPTP and JSON format,
the latter compatible with JSON-LD, see
[TS21].

These standard inference rules have been implemented in GKC:

- Binary resolution with optionally the set of support strategy, negative or positive ordered resolution or unit restriction.
- Hyperresolution.
- Factorization.
- Paramodulation and demodulation with the Knuth-Bendix ordering.

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.

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

University of Manchester, United Kingdom

Recent features in iProver include:

- Ground joinability and connectedness in the superposition calculus [DK22].
- Support for quantified linear arithmetic.
- AC joinability and AC normalisation [DK21].
- Superposition calculus with simplifications including: demodulation, light normalisation, subsumption, subsumption resolution and global subsumption. iProver's simplification set up [DK20] is tunable via command line options and generalises common architectures such as Discount or Otter.
- HOS-ML framework for learning heuristics using combination of hyper-parameter optimisation and dynamic clustering together with schedule optimisation using constraint solving [HK21, HK19]

http://www.cs.man.ac.uk/~korovink/iprover/

University of Innsbruck, Austria

University of Greifswald, Germany

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

University of Greifswald, Germany

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.

https://tptp.org/NonClassicalLogic/

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

University of New Mexico, USA

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.

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.

http://www.cs.unm.edu/~mccune/prover9/

Universität Innsbruck, Austria

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.

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

Czech Technical University in Prague, Czech Republic

*Singleton-focused* means that, apriori, each discovered strategy is
optimized for a single (hard) problem.
However, the *sieving* strips down (i.e., makes default) all options
but those necessary for the observed success.
This, together with the fact that Vampire's strategy description language is
relatively crude, helps prevent unwanted overfitting.

The heuristic schedule selection phase uses *(probabilistic) weighted greedy
cover*, where the weight of improving the schedule's expected number of
solved problems by *ΔE* while scheduling a strategy *S* for
*I* additional instructions is *ΔE/I*.

While the schedules (one for each entered division) are constructed to maximize the expected performace on the respective known TPTP problems, the final schedules are monolithic and do not branch based on specific problems' characteristics.

https://github.com/vprover/vampire/tree/shuffling

Japan Advanced Institute of Science and Technology, Japan

https://www.jaist.ac.jp/project/maxcomp/

Chalmers University of Technology, Sweden

Twee features ground joinability testing [MN90] and a connectedness test [BD88], which together eliminate many redundant inferences in the presence of unorientable equations. The ground joinability test performs case splits on the order of variables, in the style of [MN90], and discharges individual cases by rewriting modulo a variable ordering.

Horn clauses are encoded as equations as described in [CS18]. For CASC, Twee accepts non-Horn problems but throws away all the non-Horn clauses.

- Select and normalise the lowest-scored critical pair, and if it is not redundant, add it as a rewrite rule to the active set.
- Normalise the active rules with respect to each other.
- Normalise the goal with respect to the active rules.

For CASC, to take advantage of multiple cores, several versions of Twee run in parallel using different parameters (e.g., with the goal-directed transformation on or off).

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 as open source from:

http://nick8325.github.io/twee

Chalmers University of Technology, Sweden

Twee features ground joinability testing [MN90] and a connectedness test [BD88], which together eliminate many redundant inferences in the presence of unorientable equations. The ground joinability test performs case splits on the order of variables, in the style of [MN90], and discharges individual cases by rewriting modulo a variable ordering.

- Select and normalise the lowest-scored critical pair, and if it is not redundant, add it as a rewrite rule to the active set.
- Normalise the active rules with respect to each other.
- Normalise the goal with respect to the active rules.

For CASC, to take advantage of multiple cores, several versions of Twee run in parallel using different parameters (e.g., with the goal-directed transformation on or off).

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 as open source from:

http://nick8325.github.io/twee

University of Manchester, United Kingdom

- Choices of saturation algorithm:
- Limited Resource Strategy [RV03]
- DISCOUNT loop
- Otter loop
- Instantiation using the Inst-Gen calculus
- MACE-style finite model building with sort inference

- Splitting via AVATAR [Vor14]
- A variety of optional simplifications.
- Parameterized reduction orderings.
- A number of built-in literal selection functions and different modes of comparing literals [HR+16].
- Age-weight ratio that specifies how strongly lighter clauses are preferred for inference selection. This has been extended with a layered clause selection approach [GS20].
- Set-of-support strategy with extensions for theory reasoning.
- For theory-reasoning:
- Ground equational reasoning via congruence closure.
- Addition of theory axioms and evaluation of interpreted functions.
- Use of Z3 with AVATAR to restrict search to ground-theory-consistent splitting branches [RB+16].
- Specialised theory instantiation and unification [RSV18].
- Extensionality resolution with detection of extensionality axioms

- For higher-order problems:
- Translation to polymorphic first-order logic using applicative form and combinators.
- A new superposition calculus [BR20] utilising a KBO-like ordering [BR20] for orienting combinator equations. The calculus introduces an inference, narrow, for rewriting with combinator equations.
- Proof search heuristics targeting the growth of clauses resulting from narrowing.
- An extension of unification with abstraction to deal with functional and boolean extensionality.

- Various inferences to deal with booleans

https://vprover.github.io/for more information and access to the GitHub repository.

University of Manchester, United Kingdom

There are only small changes between Vampire 4.5 and Vampire 4.6 in the tracks relevant to CASC. Most of our efforts have been spent on theory reasoning (which are not relevant as TFA is not running) and efforts to parallelise Vampire which are too immature for CASC this year. One significant engineering effort has been to incorporate higher-order and polymorphic reasoning into the "main branch" such that a single executable is used for all divisions.

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.

- Choices of saturation algorithm:
- Limited Resource Strategy [RV03]
- DISCOUNT loop
- Otter loop
- Instantiation using the Inst-Gen calculus
- MACE-style finite model building with sort inference

- Splitting via AVATAR [Vor14]
- A variety of optional simplifications.
- Parameterized reduction orderings.
- A number of built-in literal selection functions and different modes of comparing literals [HR+16].
- Age-weight ratio that specifies how strongly lighter clauses are preferred for inference selection. This has been extended with a layered clause selection approach [GS20].
- Set-of-support strategy with extensions for theory reasoning.
- For theory-reasoning:
- Ground equational reasoning via congruence closure.
- Addition of theory axioms and evaluation of interpreted functions [RSV21].
- Use of Z3 with AVATAR to restrict search to ground-theory-consistent splitting branches [RB+16].
- Specialised theory instantiation and unification [RSV18].
- Extensionality resolution with detection of extensionality axioms

- For higher-order problems:
- Translation to polymorphic first-order logic using applicative form and combinators
- A superposition calculus [BR20] utilising a KBO-like ordering [BR20] for orienting combinator equations. The calculus introduces an inference, narrow, for rewriting with combinator equations.
- Proof search heuristics targeting the growth of clauses resulting from narrowing.
- An extension of unification with abstraction to deal with functional and boolean extensionality.
- Various inferences to deal with booleans

https://vprover.github.io/

University of Manchester, United Kingdom

There are only small changes between Vampire 4.7 and Vampire 4.6 in the tracks relevant to CASC. As TFA did not run in 2021, the updates related to the paper "Making Theory Reasoning Simpler" [RSV21] that were present last year should have an impact this year. This work introduces a new set of rules for the evaluation and simplification of theory literals. We have also added some optional preprocessing steps inspired by Twee (see "Twee: An Equational Theorem Prover" [Sma21]) but these have not been fully incorporated into our strategy portfolio so are unlikely to make a significant impact.

- Choices of saturation algorithm:
- Limited Resource Strategy [RV03]
- DISCOUNT loop
- Otter loop
- Instantiation using the Inst-Gen calculus
- MACE-style finite model building with sort inference

- Splitting via AVATAR [Vor14]
- A variety of optional simplifications.
- Parameterized reduction orderings.
- A number of built-in literal selection functions and different modes of comparing literals [HR+16].
- Age-weight ratio that specifies how strongly lighter clauses are preferred for inference selection. This has been extended with a layered clause selection approach [GS20].
- Set-of-support strategy with extensions for theory reasoning.
- For theory-reasoning:
- Ground equational reasoning via congruence closure.
- Addition of theory axioms and evaluation of interpreted functions [RSV21].
- Use of Z3 with AVATAR to restrict search to ground-theory-consistent splitting branches [RB+16].
- Specialised theory instantiation and unification [RSV18].
- Extensionality resolution with detection of extensionality axioms

- For higher-order problems:
- Translation to polymorphic first-order logic using applicative form and combinators
- A superposition calculus [BR20] utilising a KBO-like ordering [BR20] for orienting combinator equations. The calculus introduces an inference, narrow, for rewriting with combinator equations.
- Proof search heuristics targeting the growth of clauses resulting from narrowing.
- An extension of unification with abstraction to deal with functional and boolean extensionality.
- Various inferences to deal with booleans

https://vprover.github.io/

Vrije Universiteit Amsterdam, The Netherlands

https://github.com/sneeuwballen/zipperpositionand 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]. The code can be found at

https://github.com/sneeuwballen/zipperpositionand 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].

Vrije Universiteit Amsterdam, The Netherlands

Zipperposition's code can be found at

https://github.com/sneeuwballen/zipperpositionand 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].