CSE 1.0
Feng Cao (Yang Xu, Jun Liu, Shuwei Chen, Xiaomei Zhong, Peng Xu, Qinghua Liu,
Huimin Fu, Xiaodong Guan, Zhenming Song)
Southwest Jiaotong University, China
Ulster University, United Kingdom (Jun Liu)
Architecture
CSE 1.0 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].
This S-CS inference rule 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.
In each step, multiple clauses are selected and separated into two parts,
sub-clauses, while one part of each clause is used to form a contradiction,
and the disjunction of the remaining literals forms the logical consequence
of the selected clauses, called contradiction separation clause (CSC).
CSE 1.0 adopts conventional factoring, equality resolution, 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.
Internally, CSE 1.0 works only with clausal normal form. E prover [Sch13] is adopted with thanks for clausification of full first-order logic problems during preprocessing.
CSE 1.1
Feng Cao (Yang Xu, Jun Liu, Shuwei Chen, Xingxing He, Xiaomei Zhong, Peng Xu,
Qinghua Liu, Huimin Fu, Jian Zhong, Guanfeng Wu, Xiaodong Guan,
Zhenming Song)
Southwest Jiaotong University, China
Ulster University, United Kingdom (Jun Liu)
Architecture
The basic inference mechanism of CSE 1.1 is similar to CSE 1.0, i.e., 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.
The difference between CSE 1.0 and CSE 1.1 is that there are two S-CS deduction
mechanisms in CSE 1.1, where one is called from left to right, which refers
the clauses that are not in the contradiction under construction, and another
is named from right to left, which considers the clauses that are already in
the contradiction under construction.
In addition, it supports the repeat usage of the same clause in one deduction
step.
These characteristics make the S-CS deduction be able to produce more unit
clauses.
CSE 1.1 adopts conventional factoring, equality resolution, 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.
Internally, CSE 1.1 works only with clausal normal form. E prover [Sch13] is adopted with thanks for clausification of full first-order logic problems during preprocessing.
CSE_E 1.0
Feng Cao (Yang Xu, Stephan Schulz, Jun Liu, Shuwei Chen, Xingxing He,
Xiaomei Zhong, Peng Xu, Qinghua Liu, Huimin Fu, Jian Zhong, Guanfeng Wu,
Xiaodong Guan, Zhenming Song)
Southwest Jiaotong University, China
DHBW Stuttgart, Germany (Stephan Schulz)
Ulster University, United Kingdom (Jun Liu)
Architecture
CSE_E 1.0 is an automated theorem prover for first-order logic by combining
CSE 1.1 and E 2.1, where CSE is based on the Contradiction Separation Based
Dynamic Multi-Clause Synergized Automated Deduction (S-CS)
[XL+18] and E is 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.
CVC4 1.6pre
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 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
[RD+15].
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".
https://github.com/CVC4
E 2.2pre
Stephan Schulz
DHBW Stuttgart, Germany
Architecture
E 2.1
[Sch02,Sch13] 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.
However, as of E 2.1, PicoSAT
[Bie08]
can be used to periodically check the (on-the-fly grounded) proof state for
propositional unsatisfiability.
For LTB division, a control program uses a SInE-like analysis to extract reduced axiomatizations that are handed to several instances of E. E will probably not use on-the-fly learning this year.
For CASC-J9, 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 220 different strategies have been evaluated on all untyped first-order problems from TPTP 6.4.0. About 90 of these strategies are used in the automatic mode, and about 210 are used in at least one schedule.
http://www.eprover.org
Geo-III 2018C
Hans de Nivelle
Nazarbayev University, Kazakhstan
Architecture
Geo III is a theorem prover for Partial Classical Logic
[deN11],
based on reduction to Kleene Logic
[deN17].
Currently, only Chapters 4 and 5 are implemented.
Since Kleene logic generalizes 2-valued logic, Geo III can take part in CASC.
Apart from being 3-valued, the main differences with earlier
versions of Geo are the following:
https://cs-sst.github.io/faculty/nivelle/implementation/index
Grackle 0.1
Jan Jakubuv
Czech Technical University in Prague, Czech Republic
Architecture
Grackle is a bird species found in large numbers through much of North America.
Different subspecies of the grackle family evolved a different bill length.
This has the effect that different subspecies feed on different nutriment and
do not compete with each other.
This motivates the Grackle system.
Grackle 0.1 is a generalization of BliStrTune
[Urb13,JU17,Ju18], a system for invention of complementary E prover strategies, based
on ParamILS system
[HH+09].
BliStrTune was previously extended to invent Vampire strategies
[JSU17] but this is not used here.
Grackle is a next step in this direction of generalization, and it is able to
develop complementary strategies of an arbitrary parametrized algorithm, not
only E or Vampire.
In CASC-J9, however, Grackle is used only to develop E strategies and the main
difference from BliStrTune is a separate invention of SinE parameters for E
prover.
https://github.com/ai4reason/atpyThe Grackle system is a part of this library. Grackle itself uses ParamILS to improve an existing strategy. Grackle requires a set of reasonably performing E prover strategies to start with. These are extracted from E's auto mode and previous Grackle/BliStrTune runs on a subset of TPTP library. The code of ATPy and Grackle is released under GPL2.
iProver 2.6
Konstantin Korovin
University of Manchester, United Kingdom
Architecture
iProver is an automated theorem prover based on an instantiation calculus
Inst-Gen
[GK03,Kor13] which is complete for first-order logic.
iProver combines first-order reasoning with ground reasoning for which it uses
MiniSat
[ES04] and optionally PicoSAT
[Bie08] (only MiniSat will be used at this CASC).
iProver also combines instantiation with ordered resolution; see
[Kor08,Kor13] for the implementation details.
The proof search is implemented using a saturation process based on the given
clause algorithm.
iProver uses non-perfect discrimination trees for the unification indexes,
priority queues for passive clauses, and a compressed vector index for
subsumption and subsumption resolution (both forward and backward).
The following redundancy eliminations are implemented: blocking non-proper
instantiations; dismatching constraints
[GK04,Kor08]; global subsumption
[Kor08]; resolution-based simplifications and propositional-based
simplifications.
A compressed feature vector index is used for efficient forward/backward
subsumption and subsumption resolution.
Equality is dealt with (internally) by adding the necessary axioms of equality.
Recent changes in iProver include improved preprocessing and incremental
finite model finding; support for the TFF format restricted to clauses; the
AIG format for hardware verification and QBF reasoning.
In the LTB and SLH divisions, iProver combines an abstraction-refinement framework [HK17] with axiom selection based on the SinE algorithm [HV11] as implemented in Vampire [KV13], i.e., axiom selection is done by Vampire and proof attempts are done by iProver.
Some of iProver features are summarised below.
http://www.cs.man.ac.uk/~korovink/iprover/
iProver 2.8
Konstantin Korovin
University of Manchester, United Kingdom
Description Missing
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.
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.deThe 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.
LEO-II 1.7.0
Alexander Steen
Freie Universität Berlin, 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.
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 1.3
Alexander Steen
Freie Universität Berlin, Germany
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 is augmented with dedicated extensionality rules and equational
simplification routines that have their intellectual roots in first-order
superposition-based theorem proving.
Although Leo-III is originally designed as an agent-based reasoning system,
its current version utilizes one sequential saturation algorithm only.
The saturation algorithm itself is a variant of the given clause loop
procedure.
Leo-III heavily relies on cooperation with external (first-order) ATPs that are called asynchronously during proof search. At the moment, first-order cooperation focuses on typed first-order (TFF) systems, where CVC4 [BC+11] and E [Sch02,Sch13] are used as default external systems. Nevertheless, cooperation is 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.
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
MaLARea 0.6
Josef Urban
Czech Technical University in Prague, Czech Republic
Architecture
MaLARea 0.6
[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 (the SNoW system)
based on symbol-based similarity, model-based similarity, term-based
similarity, and obviously previous successful proofs.
The version for CASC-J9 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]
and E's implementation of SInE.
https://github.com/JUrban/MPTP2/tree/master/MaLAReaThe metasystem's Perl code is released under GPL2.
nanoCoP---1.1
Jens Otten
University of Oslo, Norway
Architecture
nanoCoP
[Ott16,Ott17]
is an automated theorem prover for classical first-order logic with equality.
It is a very compact implementation of the non-clausal connection calculus
[Ott11].
nanoCoP can read formulae in leanCoP/nanoCoP syntax and in TPTP first-order syntax. Equality axioms are automatically added if required. The nanoCoP core prover returns a compact non-clausal connection proof.
The source code of nanoCoP 1.1 is available under the GNU general public license. It can be downloaded from the nanoCoP website at:
http://www.leancop.de/nanocopThe provers nanoCoP-i and nanoCoP-M are version of nanoCoP for first-order intuitionistic logic and first-order modal logic, respectively. They are based on an adapted non-clausal connection calculus for non-classical logics [Ott17].
Princess 170717
Philipp Rümmer
Uppsala University, Sweden
Architecture
Princess
[Rue08,Rue12] 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.
In addition, some built-in procedures for nonlinear integer arithmetic are
available.
The internal calculus of Princess only supports uninterpreted predicates; uninterpreted functions are encoded as predicates, together with the usual axioms. Through appropriate translation of quantified formulae with functions, the e-matching technique common in SMT solvers can be simulated; triggers in quantified formulae are chosen based on heuristics similar to those in the Simplify prover.
Princess is available from:
http://www.philipp.ruemmer.org/princess.shtml
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 3.2
Michael Färber
Universität Innsbruck, Austria
Architecture
Satallax 3.2
[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 first-order formulae in addition to the propositional clauses. If this option is used, then Satallax periodically calls the first-order theorem prover E [Sch13] to test for first-order unsatisfiability. If the set of first-order 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. The proof reconstruction has been significantly changed since Satallax 3.0 in order to make proof reconstruction more efficient and thus less likely to fail within the time constraints.
http://satallaxprover.com
Satallax 3.3
Michael Färber
Universität Innsbruck, Austria
Architecture
Satallax 3.3
[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 first-order formulae in addition to the propositional clauses. If this option is used, then Satallax periodically calls the first-order theorem prover E [Sch13] to test for first-order unsatisfiability. If the set of first-order 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://satallaxprover.com
Twee 2.2
Nick Smallbone
Chalmers University of Technology, Sweden
Architecture
Twee 2.2 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. It is able to pick only useful case splits and to case split on a subset of the variables, which makes it efficient enough to be switched on unconditionally.
Horn clauses are encoded as equations as described in [CS18]. The CASC version of Twee "handles" non-Horn clauses by discarding them.
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.
The passive set is represented as a heap. It achieves high space efficiency (12 bytes per critical pair) by storing the parent rule numbers and overlap position instead of the full critical pair and by grouping all critical pairs of each rule into one heap entry.
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
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.
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.
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 4.0
Giles Reger
University of Manchester, United Kingdom
Architecture
Vampire 4.0 is an automatic theorem prover for first-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.
Splitting in resolution-based proof search is controlled by the AVATAR
architecture, which uses a SAT solver to make splitting decisions.
Both resolution and instantiation based proof search make use of global
subsumption.
Strategies
Vampire 4.0 provides a very large number of options for strategy selection. The most important ones are:
Implementation
Vampire 4.0 is implemented in C++.
Expected Competition Performance
Vampire 4.0 is the CASC-26 LTB division winner.
Vampire 4.1
Giles Reger
University of Manchester, United Kingdom
Architecture
Vampire
[KV13]
4.1 is an automatic theorem prover for first-order logic.
Vampire implements the calculi of ordered binary resolution and superposition
for handling equality.
It also implements the Inst-gen calculus
[Kor13]
and a MACE-style finite model builder
[RSV16].
Splitting in resolution-based proof search is controlled by the AVATAR
architecture
[Vor14]
which uses a SAT or SMT solver to make splitting decisions.
Both resolution and instantiation based proof search make use of global
subsumption
[Kor13].
Strategies
Vampire 4.1 provides a very large number of options for strategy selection.
The most important ones are:
Implementation
Vampire 4.1 is implemented in C++.
Expected Competition Performance
Vampire 4.1 is the CASC-26 TFA and FNT division winner.
Vampire 4.2
Giles Reger
University of Manchester, United Kingdom
Architecture
Vampire
[KV13] 4.2 is an automatic theorem prover for first-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.
Strategies
Vampire 4.2 provides a very large number of options for strategy selection.
The most important ones are:
Implementation
Vampire 4.2 is implemented in C++. It makes use of minisat and z3.
Expected Competition Performance
Vampire 4.2 is the CASC-26 FOF division winner.
Vampire 4.3
Giles Reger
University of Manchester, United Kingdom
This description is very similar to that of Vampire 4.2.
The main difference is the use of theory instantiation and unification with
abstraction
[RSV18]
for theory reasoning (this was experimental in 4.2). The set-of-support
strategy for theory reasoning has also been extended.
Little has changed in other areas of Vampire.
As always there have been some small improvements to heuristics, data
structures and schedules but nothing fundamentally new.
Architecture
Vampire
[KV13] 4.3 is an automatic theorem prover for first-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.