Beagle first converts the input formulas into clause normal form. Pure arithmetic (sub-)formulas are treated by eager application of quantifier elimination. The core reasoning component implements the Hierarchic Superposition Calculus with Weak Abstraction (HSPWA) [BW13]. Extensions are a splitting rule for clauses that can be divided into variable disjoint parts, and a partial instantiation rule for variables with finite domain, and two kinds of background-sorted variables trading off completeness vs. search space.
The HSPWA calculus generalizes the superposition calculus by integrating theory reasoning in a black-box style. For the theories mentioned above, Beagle combines quantifier elimination procedures and other solvers to dispatch proof obligations over these theories. The default solvers are an improved version of Cooper's algorithm for linear integer arithmetic, and the CVC4 SMT solver for linear real/rational arithmetic. Non-linear integer arithmetic is treated by partial instantiation and additional lemmas.
Beagle uses strategy scheduling by trying (at most) three flag settings sequentially.
https://bitbucket.org/peba123/beagle
CSE 1.6
Feng Cao
JiangXi University of Science and Technology, China
Architecture
CSE 1.6 is a developed prover based on the last version - CSE 1.5.
It is an automated theorem prover for first-order logic without equality, based mainly on a novel
inference mechanism called Contradiction Separation Based Dynamic Multi-Clause Synergized
Automated Deduction (S-CS)
[XL+18].
S-CS is able to handle multiple (two or more) clauses dynamically in a synergized way in one
deduction step, while binary resolution is a special case.
CSE 1.6 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 1.6 has been improved compared with CSE 1.5, mainly from the following aspects:
Acknowledgement: Development of CSE 1.6 has been partially supported by the General Research Project of Jiangxi Education Department (Grant No. GJJ200818).
CSE_E 1.5
Peiyao Liu
Southwest Jiaotong University, China
Architecture
CSE_E 1.5 is an automated theorem prover for first-order logic by combining CSE 1.6 and E 3.0,
where CSE 1.6 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 with no more than two literals,
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.
Acknowledgement: Development of CSE_E 1.5 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.
cvc5 1.0
Andrew Reynolds
University of Iowa, USA
Architecture
cvc5 is the successor of CVC4
[BC+11].
It is an SMT solver based on the CDCL(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,
cvc5 primarily uses heuristic approaches based on conflict-based instantiation
and E-matching for theorems, and finite model finding approaches for
non-theorems.
Like other SMT solvers, cvc5 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 cvc5 targets problems containing background theories
whose quantification is limited to finite and uninterpreted sorts.
In finite model finding mode, cvc5 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".
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
cvc5 1.0.5
Andrew Reynolds
University of Iowa, USA
Architecture
cvc5
[BB+22]
is the successor of CVC4
[BC+11].
It is an SMT solver based on the CDCL(T) architecture~\cite{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, cvc5 primarily uses heuristic
approaches based on conflict-based instantiation and E-matching for theorems, and finite model
finding approaches for non-theorems.
Like other SMT solvers, cvc5 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 cvc5 targets problems containing background theories whose quantification is limited to finite and uninterpreted sorts. In finite model finding mode, cvc5 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".
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
Drodi 3.5.1
Oscar Contreras
Amateur Programmer, Spain
Architecture
Drodi 3.5.1 is a very basic and lightweight automathed theorem prover.
It implements ordered resolution and equality paramodulation inferences as well
as demodulation and some other standard simplifications.
It also includes its own basic implementations of clausal normal form
conversion
[NW01],
AVATAR architecture with a SAT solver
[Vor14],
Limited Resource Strategy
[RV03],
discrimination trees as well as KBO, non recursive and lexicographic reduction
orderings.
Drodi produces a (hopefully) verifiable proof in TPTP format.
The following features have been added since last CASC competition:
Duper's core architecture is based on saturation with a set of inference and
simplification rules that
operate on dependently-typed terms but primarily perform first-order and some
higher-order reasoning.
The calculus is closely based on Zipperposition's
[BB+21]
though it has been adapted to be sound in Lean's type theory.
The unification procedure extends
[VBN20]
to dependent type theory, and remains complete
for the higher order fragment of Lean. Duper uses streams to represent
conclusions of inference rules and interleaves between unification
and inference, as in
[BB+21].
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.
Duper 1.0
Joshua Clune
Carnegie Mellon University, USA
Architecture
Duper is a superposition-based theorem prover for dependent type theory,
designed to prove theorems in the proof assistant Lean 4.
To solve problems in TPTP format, Duper first translates them into Lean goals,
using a shallow embedding
of first-order and higher-order logic into dependent type theory.
Translation is currently possible for FOF, TFF, and THF problems without arithmetic.
When run in Lean, Duper produces proof terms which get verified by Lean's kernel.
When run as a standalone executable for the purposes of this competition, Duper still produces
Lean proof terms but they are not checked by Lean's kernel.
Strategies
Currently, Duper has only a single strategy without time slicing and does not
take advantage of multiple cores.
The strategy is simple and does not tune its heuristics based on the input
problem.
Duper uses a KBO term ordering with a basic precedence and weight heuristic.
Implementation
The prover is implemented in Lean 4 as a Lean tactic.
The script provided to solve TPTP problems parses the given file, converts it to a Lean 4 goal,
then attempts to solve said goal with the Duper tactic.
Duper's source code can be found at
https://github.com/leanprover-community/duper
Expected Competition Performance
Duper is still in an early stage of development and is not expected to compete with matured tools.
E 3.0
Stephan Schulz
DHBW Stuttgart, Germany
Architecture
E
[Sch02,
Sch13,
SCV19]
is a purely equational theorem prover for many-sorted first-order logic with
equality, and for monomorphic higher-order logic.
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, optionally with higher-order extensions
[VB+21].
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.
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
For CASC-29, E implements a two-stage 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. The strategies are then scheduled onto the available cores and run in parallel.
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
GKC 0.8
Tanel Tammet
Tallinn University of Technology, Estonia
Architecture
GKC
[Tam19]
is a resolution prover optimized for search in large knowledge bases.
The GKC version 0.8 running at CASC-29 is a marginally improved version of the GKC 0.7 running
in two previous CASCs.
Almost all of the GKC development effort this year has gone to the commonsense superstructure GK
(https://logictools.org/gk/) and the natural
language reasoning pipeline
(https://github.com/tammet/nlpsolver)
[TJ+23].
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 [Tam21, Tam22].
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 [TS21].
GKC only looks for proofs and does not try to show non-provability. These standard inference rules have been implemented in GKC:
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/
None provided
iProver 3.8
Konstantin Korovin
University of Manchester, United Kingdom
Lash 1.13
Cezary Kaliszyk
University of Innsbruck, Austria
Architecture
Lash
[BK22]
is a higher-order automated theorem prover created as a fork of the theorem
prover Satallax.
The basic underlying calculus of Satallax is a ground tableau calculus whose
rules only use shallow information about the terms and formulas taking part
in the rule.
Strategies
There are about 113 flags that control Lash's behavior, most of them inherited
from Satallax.
A mode is a collection of flag values.
Starting from 10 Satallax modes, Grackle was used to derive 61 modes
automatically, and grouped into two schedules of 15 modes each.
Implementation
Lash uses new, efficient C representations of vital structures and operations.
Most importantly, Lash uses a C representation of (normal) terms with perfect
sharing along with a C implementation of normalizing substitutions.
Lash's version 1.12 additionally includes a new term enumeration scheme, and
Grackle-based strategy schedule.
Expected Competition Performance
Comparable to Satallax.
LEO-II 1.7.0
Alexander Steen
University of Greifswald, Germany
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
Leo-III 1.7.8
Alexander Steen
University of Greifswald, Germany
Architecture
Leo-III
[SB21],
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.
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
Princess 230619
Philipp Rümmer
University of Regensburg, Germany
Architecture
Princess
[Rue08]
is a theorem prover for first-order logic modulo linear integer arithmetic.
The prover uses a combination of techniques from the areas of first-order reasoning and SMT
solving.
The main underlying calculus is a free-variable tableau calculus, which is extended with
constraints to enable backtracking-free proof expansion, and positive unit hyper-resolution
for lightweight instantiation of quantified formulae.
Linear integer arithmetic is handled using a set of built-in proof rules resembling the Omega
test, which altogether yields a calculus that is complete for full Presburger arithmetic, for
first-order logic, and for a number of further fragments.
Princess also contains theory modules for, among others, non-linear arithmetic, rationals,
bit-vectors, arrays, heaps, algebraic data-types, strings.
The list of contributors is available on https://github.com/uuverifiers/princess/blob/master/AUTHORS.
The following options are used in the competition:
-portfolio=casc +threads +printProof -inputFormat=tptp
The version 230619 submitted to CASC-29 is the standard version of Princess. This is in contrast to CASC-26 (2017), when Princess was participating the last time, when a version tailor-made for CASC was used.
Sources and binaries are available from
https://github.com/uuverifiers/princess
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.
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/
Satallax 3.4
Cezary Kaliszyk
Universität Innsbruck, Austria
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.
http://cl-informatik.uibk.ac.at/~mfaerber/satallax.html
Toma 0.4
Teppei Saito
Japan Advanced Institute of Science and Technology, Japan
Architecture
Toma 0.4 is an automatic equational theorem prover.
It proves unsatisfiability of a UEQ problem as follows:
A given problem is transformed into a word problem whose validity entails unsatisfiability of the
original problem.
The word problem is solved by a new variant of maximal (ordered) completion
[WM18,
Hir21].
https://www.jaist.ac.jp/project/maxcomp/
Twee 2.4.1
Nick Smallbone
Chalmers University of Technology, Sweden
Architecture
Twee
[Sma21]
is a theorem prover for unit equality problems based on unfailing completion
[BDP89].
It implements a DISCOUNT loop, where the active set contains rewrite rules
(and unorientable equations) and the passive set contains critical pairs.
The basic calculus is not goal-directed, but Twee implements a transformation
which improves goal direction for many problems.
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.
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
Twee 2.4.2
Nick Smallbone
Chalmers University of Technology, Sweden
Architecture
Twee 2.4.2
[Sma21]
is a theorem prover for unit equality problems based on unfailing completion
[BDP89].
It implements a DISCOUNT loop, where the active set contains rewrite rules (and unorientable
equations) and the passive set contains critical pairs.
The basic calculus is not goal-directed, but Twee implements a transformation which improves goal
direction for many problems.
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.
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 counted only once per term.
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).
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.
Twee can be downloaded as open source from:
https://nick8325.github.io/twee
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.
Vampire 4.7
Giles Reger
University of Manchester, United Kingdom
Architecture
Vampire
[KV13]
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.
https://vprover.github.io/
There have been a number of changes and improvements since Vampire 4.7, although it is still the
same beast.
Most significant from a competition point of view are long-awaited refreshed strategy schedules.
As a result, several features present in previous competitions will now come into full force,
including new rules for the evaluation and simplification of theory literals.
A large number of completely new features and improvements also landed this year: highlights
include a significant refactoring of the substitution tree implementation, the arrival of
encompassment demodulation to Vampire, and support for parametric datatypes.
Vampire's higher-order support has also been re-implemented from the ground up.
The new implementation is still at an early stage and its theoretical underpinnings are being
developed.
There is currently no documentation of either.
Vampire 4.8
Michael Rawson
TU Wien, Austria
Architecture
Vampire
[KV13]
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.
Zipperposition 2.1.999
Jasmin Blanchette
Vrije Universiteit Amsterdam, The Netherlands
Architecture
Zipperposition is a superposition-based theorem prover for typed first-order
logic with equality and for higher-order logic.
It is a pragmatic implementation of a complete calculus for full higher-order
logic
[BB+21].
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
[EBT21],
and 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 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 earlier version of E
[Sch02].
With the implementation of higher-order superposition, the main loop had to be
adapted to deal with possibly infinite sets of unifiers
[VB+21].
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].
Zipperposition 2.1.9999
Jasmin Blanchette
Ludwig-Maximilians-Universität München, Germany
Architecture
Zipperposition is a superposition-based theorem prover for typed first-order
logic with equality and for higher-order logic.
It is a pragmatic implementation of a complete calculus for full higher-order
logic
[BB+21].
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
[EBT21],
and 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 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 earlier version of E
[Sch02].
With the implementation of higher-order superposition, the main loop had to be
adapted to deal with possibly infinite sets of unifiers
[VB+21].
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].