| Department of Computer Science University of Miami geoff@cs.miami.edu |
The TPTP (Thousands of Problems for Theorem Provers) is a library of problems, in first-order logic with an interpreted equality symbol, for Automated Theorem Proving (ATP) systems. The principal motivation for the TPTP is to move the testing and evaluation of ATP systems from the previously ad hoc situation onto a firm footing. This became necessary, as results being published do not always accurately reflect the capabilities of the ATP system being considered. A common library of problems is necessary for meaningful system evaluations, meaningful system comparisons, repeatability of testing, and the production of statistically significant results. The TPTP is such a library. The TPTP provides a simple, unambiguous, source and reference mechanism for ATP problems. It is comprehensive and up-to-date, and thus provides an overview of the current application of ATP. The TPTP problems are stored in a specifically designed, easy to understand format. Utilities for manipulating the problems, for converting the problems to other known ATP formats, and for finding problems with certain characteristics, are provided. Since its first release in 1993, many researchers have used the TPTP as an appropriate and convenient basis for ATP system evaluation.
This technical report explains the motivations and reasoning behind the development of the TPTP, and thus implicitly explains the design decisions made. It also serves as a manual explaining the structure and use of the TPTP: it provides a full description of the TPTP contents and organization, details of the utility programs, and guidelines for obtaining and using the TPTP.
| A Quick Installation Guide for the TPTP is given in Section Getting and Using the TPTP. Please be sure to read the guidelines for using TPTP problems and presenting results, given in Section Using the TPTP. |
What's New in TPTP v3.4.0 :
The sparsness of research into ATP systems for FOF problems meant that no electronic collections of FOF test problems had previously been commonly available. A CNF problem collection in electronic form was made publicly available by Argonne National Laboratories (in Otter format [McC94]) in 1988 [ANL]. This collection was a major source of problems for ATP researchers. Other electronic collections of CNF problems have been available, but not announced officially (e.g., that distributed with the SPRFN ATP system [SPRFN]). Although some of these collections provided significant support to researchers, and formed the early core of the TPTP library, none (with the possible exception of the ANL collection) was specifically designed to serve as a common basis for ATP research. Rather, these collections typically were built in the course of research into a particular ATP system. As a result there are several factors that limited their usefulness as a common basis for research. In particular, previously existing problem collections:
The problem of meaningfully interpreting results can be even worse than indicated. A few problems may be selected and hand-tuned (formulae arranged in a special way, irrelevant formulae omitted, lemmas added in, etc) specifically for the ATP system being tested. The presentation of a problem can significantly affect the nature of the problem, and changing the formulae clearly makes a different problem altogether. Nevertheless the problem may be referenced under the same name as it was presented elsewhere. As a consequence the experimental results reveal little. Some researchers avoid this ambiguity by listing the formulae explicitly, but obviously this usually cannot be done for a large number of problems or for large individual problems. The only satisfactory solution to these issues is a common and stable library of problems. The TPTP is such a library.
The development of the TPTP problem library is an ongoing project, with the aim to provide all of the desired properties.
Current Limitations of the TPTP.
The current release of the TPTP library is limited to problems expressed in 1st
order logic.
There are no problems for induction, or for non-classical theorem proving.
However, see Sections Current Activities
and Further Plans for upcoming and planned extensions.
The problems in the TPTP are syntactically diverse, as is indicated by the ranges of the values in the tables. The problems in the TPTP are also semantically diverse, as is indicated by the range of domains that are covered. The problems are grouped into domains, covering topics in the fields of logic, mathematics, computer science, engineering, social sciences, and others. The domains are presented and discussed in Section The TPTP Domain Structure.
Sources
The problems have been collected from various sources.
The two principal sources have been existing electronic problem collections
and the ATP literature.
Other sources include logic programming, mathematics, puzzles, and
correspondence with ATP researchers.
Many people and organizations have contributed towards the TPTP:
the foundations of the TPTP were laid with David Plaisted's SPRFN collection;
many problems were taken from Argonne National Laboratory's ATP problem library
(special thanks to Bill McCune here); Art Quaife provided several hundred
problems in set theory and algebra; the Journal of Automated Reasoning, CADE
Proceedings, and Association for Automated Reasoning Newsletters have provided
a wealth of material; smaller numbers of problems have been provided by a
number of further contributors (see the
Acknowledgements).
Releases
An attempt has been made to classify the totality of the TPTP problems in a
systematic and natural way.
The resulting domain scheme reflects the natural hierarchy of scientific
domains, as presented in standard subject classification literature.
The current classification is based mainly on the Dewey Decimal
Classification (DDC)
[CB+89]
and the Mathematics Subject Classification (MSC)
[MSC92]
used for the Mathematical Reviews by the American Mathematical Society.
Five main fields are defined:
logic, mathematics, computer science, engineering, and other.
Each field contains further subdivisions, called domains.
Each domain is identified by a three-letter mnemonic.
These mnemonics are also part of the TPTP problem naming scheme
(see Section
Problem and Axiomatization Naming).
The TPTP domains constitute the basic units of the classification.
The full classification scheme is shown in
Figure
The Domain Structure of the TPTP,
and the numbers of abstract problems, problems, and generic problems
in each domain are shown in the TPTP document
OverallSynopsis
A brief description of the domains, with a non-ATP reference for a
general introduction and a generic ATP reference, is given below.
For each domain, appropriate DDC and MSC numbers are also given:
Different Axiomatizations
In the TPTP an axiomatization is standard if it is a
complete axiomatization of
an established theory,
no lemmas have been added, and
it is not designed specifically to be suited or
ill-suited to any ATP system, calculus, or control strategy.
A problem version is standard if it uses a standard axiomatization.
(Note: A standard axiomatization may be redundant, because some axioms are
dependent on others.
In general, it is not known whether or not an axiomatization is minimal, and
thus the possibility of redundancy in standard axiomatizations must be
tolerated.)
In the TPTP, standard axiomatizations are kept in separate axiom files,
and are included in problems as appropriate.
Sets of specialization axioms may be used to extend or constrain an
axiomatization.
Specialization axioms are also kept in separate axiom files.
Within the ATP community, some problems have been created with
non-standard axiomatizations.
A non-standard axiomatization may be formed by modifying a standard
axiomatization:
the standard axiomatization may be reduced (i.e., axioms are removed) and
the result is an incomplete axiomatization, or it may be
augmented
(i.e., lemmas are added) and the result is a redundant axiomatization.
Incomplete and redundant axiomatizations are typically used to find a proof
of a conjecture (based on the axiomatization) using a particular ATP system.
An axiomatization may also be non-standard because it is biased,
i.e., designed specifically to be suited or ill-suited to some ATP system,
calculus, or control strategy.
A problem version is incomplete, redundant, or biased if its axiomatization is.
In the TPTP, for each incomplete, redundant, and biased problem, a new version
of the problem with a standard axiomatization is usually created.
Finally, an axiomatization may be non-standard because it does not capture
any established theory. i.e., a standard axiomatization does not exist, but
the axioms are not biased.
A problem version with such an axiomatization is especial.
Typically, the axioms in an especial problem are specific to that problem,
and are not used in any other problem.
In any 'real' application of an ATP system, a standard or especial
problem version would typically be used, at least initially.
The use of standard axiomatizations is particularly desirable.
Equality Axiomatization
Up to TPTP v1.1.3, the TPTP contained problem files for particular sizes of
generic problems.
This, however, was undesirable.
Firstly, only a finite number of different problem sizes could be
included, and therefore a desired size may have been missing.
Secondly, even a small number of different problem sizes for each
generic problem could consume a considerable amount of disk space.
To overcome these problems, the TPTP contains generator files.
Generator files are used to generate instances of generic problems, according
to user supplied size parameters.
The generators are used in conjunction with the tptp2X utility, and a full
description of their use is given in
Section The tptp2X Utility.
For convenience, the TPTP still contains, where they exist, a theorem and
a non-theorem size instance of each generic problem.
The TPTP sizes chosen are non-trivial,
unless the problem remains easy up to sizes that have very large files.
In the latter situation the TPTP size is that used in TPTP v1.2.0.
The statistics in the TPTP documents
Overall synopsis,
FOF synopsis, and
CNF synopsis
take into account these instances of generic problems.
Problem file naming scheme.
The abstract problem numbers within each domain are not always successive.
This is because some numbers have already been allocated to problems that
will be part of a future TPTP release (see
Section The Present).
The version numbers used for each abstract problem do not always start
at 1, and are not always successive.
This is because the same version number is assigned (wherever possible) to
all problems that come from the same source, within each domain.
Axiom file naming scheme.
If a file is ever removed from or renamed in the TPTP, then the extension
is changed to .rm.
The file is not physically removed, and a comment is added to the file to
explain what has happened.
This mechanism maintains the unique identification of problems and
axiomatizations.
The syntax of all files is that of Prolog (with some operators defined).
This conformance makes it trivial to manipulate the files using Prolog.
All information in the files that is not for use by
ATP systems is formatted as Prolog comments, with a leading %.
The (annotated) formulae are formatted as Prolog facts.
A
full BNF specification of the problem and axiom file formats is provided
in the Documents directory of the TPTP (see
Section Physical Organization).
The tptp2X utility can be used to convert TPTP files to other known ATP system
formats (see
Section The tptp2X Utility).
A description of the information contained in TPTP files is given below.
The % File field.
The % Domain field.
The % Problem field.
The % Version field.
The second possible differentiation is the status of the axiomatization,
as discussed in Section Problem Versions.
There are 12 possiblities:
The third possible differentiation between problem versions is in the
theorem formulation. Variations in the theorem formulation are noted in
a Theorem formulation entry, e.g.,
The % English field.
The % Refs field.
The % Source field.
The % Names field.
The % Status field.
The % Rating field.
The % Syntax field.
The % Comments field.
The % Bugfixes field.
Each of the include instructions indicates that the formulae in the
named file should be included at that point.
Include files with relative path names are expected to be found either under
the directory of the current file, or if not found there then under the
directory specified in the $TPTP environment variable.
If the include instruction has a []ed list of formulae names
as a second argument, the results of parsing the named file are filtered
to retain only those formulae (thus this filter applies to formulae that
may have been recursively included into the named file).
If any member of the list cannot be found, or there are multiple formulae
with a given name, that is an error condition.
If there is no second argument, or the second argument is all,
then all the formulae in the file are included.
In Figure Example include section only
connect_defn and symmetry_of_at_the_same_time are included
from Axioms/GEO004+3.ax.
Full versions of TPTP problems (without include instructions) can be
created by using the tptp2X utility (see
Section The tptp2X Utility).
A side effect of separating out the axioms into axiom files is that the
formula order in the TPTP presentation of problems is typically different
from that of any original presentation.
This reordering is acceptable because the performance of a decent ATP
system should not be very dependent on a particular formula ordering.
Each annotated formula has a name, in the form of a Prolog atom.
Each annotated formula has a role, one of
axiom, hypothesis, definition,
lemma, theorem, conjecture,
lemma_conjecture, negated_conjecture, plain,
and unknown.
axioms are accepted, without proof, as a basis for proving
conjectures and lemma_conjectures in FOF problems.
In CNF problems axioms are accepted as part of the set whose
satisfiability has to be established.
There is no guarantee that the axioms of a problem are consistent.
hypothesiss are assumed to be true for a particular problem,
and are used like axioms.
definitions are used to define symbols, and are used like
axioms.
lemmas and theorems have been proven from the
axioms, can be used like axioms, but are redundant
wrt the axioms.
lemma is used as the role of proven lemma_conjectures,
and theorem is used as the role of proven conjectures,
in output.
A problem containing a lemma or theorem that is not
redundant wrt the axioms is ill-formed.
theorems are more important than lemmas from the
user perspective.
conjectures occur in only FOF problems, and are to all be proven
from the axiom(-like) formulae.
A problem is solved only when all conjectures are proven.
lemma_conjectures are expected to be provable, and may be useful
to prove, while proving conjectures.
negated_conjectures ocuur in only CNF problems, and are formed from
negation of a conjecture in a FOF to CNF conversion.
plains have no special user semantics, and can be used like
axioms.
unknowns have unknown role, and this is an error situation.
The atoms that appear in logic formulae are presented in the form of Prolog
terms, except if the predicate is equality in which case infix = and
!= are used.
The connectives used to build non-atomic formulae are
prefix ~ for negation,
infix | for disjunction,
infix & for conjunction,
infix <=> for equivalence,
infix => for implication,
infix <= for reverse implication,
infix <~> for non-equivalence (XOR),
infix ~| for negated disjunction (NOR), and
infix ~& for negated conjunction (NAND).
Negation has higher precedence than the binary connectives, but no
precedence between binary connectives is assumed; brackets are used to
ensure the correct association.
The binary connectives are right associative.
The universal quantifier is ! and the existential quantifier is
?.
Quantified formulae are written in the form
<Quantifier> [<Variables>] : <Formula>.
Logic formulae, including equality and inequality literals, can be read as
Prolog terms using appropriate operator definitions.
An example of a FOF formula section, extracted from the problem file
GRP194+1.p, is shown in Figure
Example FOF formulae.
An example of a clause section, extracted from the problem file
GRP039-7.p, is shown in Figure
Example CNF clauses.
Hint: If your command shell complains about too many arguments as a
result of expanding the <TPTPFiles> argument to a too large number
of files, e.g., ~/TPTP/Problems/S*/*.p, place the
<TPTPFiles> argument in 'single quotes', and
tptp2X will do the expansion internally.
Finally an extension to indicate the output format is appended to the file
name. The suffixes for the output formats are:
To record how a tptp2X output file has been formed, the tptp2X parameters
are given in a % Comments field entry of the output file.
Example
Example
Example
?-tptp2X(<TPTPFile>,<Size>,<Transform>,<Format>,<Directory>).
The parameters are similar to the tptp2X script command line parameters.
A summary, highlighting differences with "(!)", is given here.
See Section Using tptp2X for parameter
options.
The TPTP is managed in the manner of a software product, in the sense that
fixed releases are made.
Each release of the TPTP is identified by a release number,
in the form
v<Version>.<Edition>.<Patch level>.
The Version number enumerates major new releases of the TPTP, in which
important new features have been added.
The Edition number is incremented each time new problems are added
to the current version.
The Patch level is incremented each time errors, found
in the current edition, are corrected.
All non-trivial changes are recorded in a history file, as well as
in the file for an affected problem.
The TPTP Domain Structure
This section provides the structure according to which
the problems are grouped into domains.
Some information about the domains is also given.
The Domain Structure of the TPTP.
Logic
Combinatory logic
COL
Logic calculi
LCL
Henkin models
HEN
Mathematics
Set theory
SET and SEU
Graph theory
GRA
Algebra
Boolean algebra
BOO
Robbins algebra
ROB
Left distributive
LDA
Lattices
LAT
Groups
GRP
Rings
RNG
Fields
FLD
Homological algebra
HAL
General algebra
ALG
Number theory
NUM
Topology
TOP
Analysis
ANA
Geometry
GEO
Category theory
CAT
Computer science
Computing theory
COM
Knowledge representation
KRS
Natural Language Processing
NLP
Planning
PLA
Agents
AGT
Commonsense Reasoning
CSR
Software creation
SWC
Software verification
SWV
Science and Engineering
Hardware creation
HWC
Hardware verification
HWV
Medicine
MED
Social sciences
Management
MGT
Other
Syntactic
SYN
Puzzles
PUZ
Miscellaneous
MSC
An agent is an autonomous software component of a computer program,
typicallly designed to act intelligently and communicate with other
agents.
Indices : DDC ???; MSC 68T35.
References : General
[RN95];
An algebra is a set with a system of operations defined on it.
Indices : DDC 512; MSC 06XX, 20XX.
References : General
[Bou89,
BM65,
BB70]; ATP --.
Analysis is a branch of mathematics concerned with functions and
limits.
The main parts of analysis are differential calculus, integral
calculus, and the theory of functions.
Indices : DDC 515; MSC 26XX.
References : General
[Ros90]; ATP
[Ble90].
A Boolean algebra is a set of elements with two binary operations
which are idempotent, commutative, and associative.
These operations are mutually distributive, there exist universal
bounds 0, 1, and there is a unary operation of
complementation.
Indices : DDC 511.324, 512.89; MSC 06EXX.
References : General
[Whi61,
BM65,
BB70]; ATP --.
A category is a mathematical structure together with the morphisms
that preserve this structure.
Indices : DDC 512.55; MSC 18XX.
References : General
[Mac71]; ATP
[MOW76].
Combinatory logic is about applying one function to another.
It can be viewed as an alternative foundation of mathematics
(or, due to its Turing-completeness, as a programming language).
More formally, it is a system satisfying two combinators and
satisfying reflexivity, symmetry, transitivity, and two equality
substitution axioms for the function that exists implicitly for
applying one combinator to another.
Indices : DDC 510.101; MSC 03B40.
References : General
[CF58,
CHS72,
Bar81]; ATP
[WM88].
Computing theory is a subfield of computer science dealing with
theoretical issues such as decidability (whether or not a given
problem admits an algorithmic solution), completeness (does
an algorithm always find a solution if one exists?),
correctness (are only solutions produced?),
and computational complexity (the resource requirements of
algorithms).
Indices : DDC 004-006; MSC 68XX.
References : General
[HU79]; ATP --.
Commonsense reasoning is the branch of artificial intelligence concerned
with replicating human thinking.
There are several components to this problem, including:
developing adequately broad and deep commonsense knowledge bases;
developing reasoning methods that exhibit the features of human thinking,
including
the ability to reason with knowledge that is true by default,
the ability to reason rapidly across a broad range of domains, and
the ability to tolerate uncertainty in your knowledge;
developing new kinds of cognitive architectures that support multiple
reasoning methods and representations.
Indices : DDC 121.3; MSC 68TXX.
References : General
[Sha97]; ATP
[Lif95,
McC59,
SME04]
A field is ring (see below) in which the * operation is
commutative, and for which there is an identity element in G,
and for which each non-identity element of G has an inverse in G.
Indices : DDC 512.32; MSC 12XX.
References : General
[Ada82]; ATP
[Dra93].
Geometry is a branch of mathematics that deals with the measurement,
properties, and relationships of points, lines, angles, surfaces,
and solids.
Indices : DDC 516; MSC 51.
References : General
[Tar51,
Tar59]; ATP
[Qua92].
A graph consists of a finite non-empty set of vertices together
with a prescribed set of edges, each edge connecting a pair of
vertices.
Indices : DDC 510.09; MSC 05CXX, 68R10.
References : General
[Har69,
BB70]; ATP --.
A group is a set G and a binary operation +:GxG -> G which
is associative, and for which there is an identity element in G,
and for which each element of G has an inverse in G.
Indices : DDC 512.2; MSC 20
References : General
[Bou89,
BM65]; ATP
[MOW76].
Homological algebra is an abstract algebra concerned with results
valid for many different kinds of spaces.
Modules are the basic tools used in homological algebra.
Indices : DDC 512.55; MSC 18XX.
References : General
[Wei94]; ATP --.
Henkin models provide a generalized semantics for higher order
logics. This leads to a larger class of models and, as a
consequence, fewer true sentences. However, in contrast to
standard semantics, complete and correct calculi can be found.
Indices : DDC 160; MSC 03CXX.
References : General
[Hen50,
Leb83]; ATP --.
Computer hardware is created by inter-connecting logic gates. Hardware
creation is used to form a circuit that will transform given input
patterns to required output patterns.
Indices : DDC 621.395; MSC 94CXX.
References : General
[Hay93]; ATP
[WW83].
Hardware verification is used to ensure that a previously designed
circuit performs the desired transformation of input patterns to
required output patterns.
One approach is to check the performance of the circuit for every
possible combination of given inputs. Other techniques are also used.
Indices : DDC 621.395; MSC 94CXX.
References : General
[Hay93]; ATP
[Woj83].
Knowledge Representation is concerned with writing down descriptions
that can be manipulated by a machine.
Indices : DDC 006.3; MSC 68T30.
References : General
[BL85,
CM87]; ATP --.
A lattice is a set of elements, with two binary operations
which are idempotent, commutative, and associative, and which
satisfy the absorption law.
Indices : DDC 512.865; MSC 06BXX.
References : General
[BM65]; ATP
[McC88].
A logic calculus defines axioms and rules of inference that
can be used to prove theorems.
This domain currently contains the following logical calculi:
Indices : DDC 511.3; MSC 03XX.
References : General
[Luk63]; ATP
[MW92].
LD-algebras are related to large cardinals. Under a very strong
large cardinal assumption, the free-monogenic LD-algebra can be
represented by an algebra of elementary embeddings. Theorems about
this algebra can be proven from a small number of properties,
suggesting the definition of an embedding algebra.
Indices : DDC 512; MSC 20N02, 03E55, 08B20
References : General --; ATP
[Jec93].
The science of diagnosing and treating illness, disease, and injury.
Indices : DDC 610; MSC --.
References : General
[LH+01];
ATP
[HLB05].
Management is the study of systems, and their use and production
of resources.
Indices : DDC 302-303; MSC 90XX.
References : General --; ATP
[PM94,
PB+94].
A collection of problems which do not fit well into any
other domain.
Natural language processing considers the automated generation and
comprehension of languages used by humans.
Indices : DDC 006.3; MSC 68T50
References : General
[Obe89i,
GWS86]; ATP
[Bos00].
Number theory is the study of integers and their properties.
Indices : DDC 512.7; MSC 11YXX.
References : General
[HW92]; ATP
[Qua92].
Planning is the process of determining the sequence of actions
to be performed by an agent, to reach a specified desired state
from a specified initial state.
Indices : DDC 006.3; MSC 68T99.
References : General
[AK+91]; ATP
[Pla81,
Pla82].
Puzzles are designed to test the ingenuity of humans.
Indices : DDC 510, 793.73; MSC --.
References : General
[Car86,Smu78,
Smu85]; ATP --.
A ring is a group (see above) in which the binary operation is
commutative, with an associative and distributive operation
*:GxG -> G for which there is an identity element in G.
Indices : DDC 512.4; MSC 13XX, 16XX.
References : General
[Bou89,
BB70]; ATP
[MOW76].
The Robbins Algebra domain revolves around the question "Is every
Robbins algebra Boolean?".
Most of the problems in this domain identify conditions that make a
near-Boolean algebra Boolean.
Indices : DDC 512; MSC 03G15.
References : General
[HMT71]; ATP
[Win90].
Classically, a set is a totality of certain definite, distinguishable
objects of our intuition or thought - called the elements of the
set. Due to paradoxes that arise from such a naive definition,
mathematicians now regard the notion of a set as an undefined,
primitive concept
[How72].
Indices : DDC 511.322, 512.817; MSC 03EXX, 04XX.
References : General
[Neu25,
Qui69]; ATP
[Qua92].
Software creation is used to form a computer program that meets given
specifications.
Indices : DDC ???.?; MSC 68N30.
References : General --; ATP --.
Software verification formally establishes that a computer program
does the task it is designed for.
Indices : DDC 005.14; MSC 68Q60.
References : General --; ATP
[WO+92,
MOW76].
Syntactic problems have no obvious semantic interpretation.
Indices : DDC --; MSC --.
References : General
[Chu56]; ATP
[Pel86].
Topology is the study of properties of geometric configurations
(e.g., point sets) which are unaltered by elastic deformations
(homeomorphisms, i.e., functions that are 1-1 mappings between sets,
such that both the function and its inverse are continuous).
Indices : DDC 514; MSC 46AXX.
References : General
[Kel55,
Mun75]; ATP
[WM89].
Problem Versions and Standard Axiomatizations
There are often many ways to formulate a problem for presentation
to an ATP system.
Thus, in the TPTP, there are often alternative presentations of a problem.
The alternative presentations are called versions
of the underlying abstract problem.
As the problem versions are the objects that ATP systems must deal
with, they are referred to simply as problems, and the
abstract problems are referred to explicitly as such.
Each problem is stored in a separate physical file.
In the TPTP the most coarse grain differentiation between problem versions
is whether the problem is presented in FOF or CNF.
Within a given presentation, the primary reason for different versions of
an abstract problem is the use of different axiomatizations.
This issue is discussed below.
A secondary reason is different formulations of the theorem to be proven,
e.g., different clausal forms of a FOF problem.
Commonly, many different axiomatizations of a theory exist.
By using different axiomatizations of a particular theory, different
versions of a problem can be formed.
In the TPTP equality is represented in infix using = and !=
for equality and inequality.
An inequality has the same meaning as a negated equality.
If equality is present in a problem, axioms of equality are not provided;
it is assumed that equality reasoning is builtin to every ATP system.
If equality axioms are required by an ATP system they can be added using
the tptp2X utility
(see Section The tptp2X Utility).
If any axioms are added when the problems are submitted to an ATP
system, then the addition must be explicitly noted in reported results (see
Section Using the TPTP).
Problem Generators
Some abstract problems have a generic nature, and particular instances of
the abstract problem are formed according to some size parameter(s).
An example of a generic problem is the N-queens problem: place N queens
on a N by N chess board such that no queen attacks another.
The formulae for any size of this problem can be generated automatically,
for any size of N >= 2.
Note that satisfiability may depend on the problem size.
Problem, Generator, and Axiomatization Naming
Providing unambiguous names for all problems is necessary in a problem library.
A naming scheme has been developed for the TPTP, to provide unique,
stable names for abstract problems, problems, axiomatizations, and generators.
Files are assigned names according to the schemes depicted in
Sections Problem Naming and
Axiom Naming.
The DDDNNN combination provides an unambiguous name for an abstract
problem or axiomatization.
The DDDNNNFV[.SSS] combination provides an unambiguous name for a
problem or generator, and the DDDNNNFE combination provides an
unambiguous name for a set of axioms.
The complete file names are unique within the TPTP.
For example, the file GRP135-1.002.p contains the group theory
problem number 135, version number 1, generated size
2.
Similarly, the file CAT001-0.ax contains the basic category theory
axiomatization number 1.
DDD
NNN
F
V
.SSS
.T
DDD - Domain name abbreviation.
The domain names and their abbreviations are listed in
Section
The Domain Structure of the TPTP.
NNN - Abstract problem number.
It is unique within the domain.
F - Form.
- for CNF and + for FOF.
V - Version number.
It differentiates between different versions of the abstract
problem, as discussed in
Section Problem Versions.
SSS - Size parameter(s).
These only occur for generated problems, and give the size
parameter(s).
T - File name extension.
p for problem files, g for generator files.
DDD
NNN
F
V
.TT
DDD - Domain name abbreviation.
The domain names and their abbreviations
are listed in
Section
The Domain Structure of the TPTP.
NNN - Axiomatization number.
It is unique within the domain.
F - Form.
- for CNF and + for FOF.
V - Specialization number.
It identifies sets of axioms that are used to specialize an
axiomatization.
Axiomatizations of basic theories are allocated the number
0, and specialization axiom sets are numbered from
1 onwards.
T - File name extension.
An extension ax denotes a file containing axioms
specific to a theory.
An extension eq denotes a file containing equality
substitution axioms for the corresponding theory specific axioms.
Problem Presentation
The physical presentation of the TPTP problem library is such that ATP
researchers can easily use the problems.
The TPTP file format, for problem files and axiom files, has three main
sections.
The first section is a header section that contains information for the user.
This information is not for use by ATP systems (see
Section Using the TPTP).
The second section contains include instructions for axiom files.
The last section contains the formulae that are specific to the problem or
axiomatization.
TPTP generator files have three sections, different from the problem and
axiom files.
The header section of generator files is similar to that of problem and axiom
files, but with place-holders for information that is filled in based on
the size of problem and the formulae generated.
Following that comes Prolog source code to generate the formulae, and finally
data describing the permissible sizes and the chosen TPTP size for the problem.
More details are given in
Section
Writing your own Problem Generators.
The Header Section
This section contains information about the problem, for the user.
It is divided into four parts separated by blank lines.
The first part identifies and describes the problem.
The second part provides information about occurrences of the problem
in the literature and elsewhere.
The third part gives the problem's ATP status and a table of syntactic
measurements made on the problem.
The last part contains general information about the problem.
Examples of TPTP headers, extracted from the problem files GRP194+1.p
and GRP039-7.p, are shown in
Figures Example FOF header and
Example CNF header.
%--------------------------------------------------------------------------
% File : GRP194+1 : TPTP v2.2.0. Released v2.0.0.
% Domain : Group Theory (Semigroups)
% Problem : In semigroups, a surjective homomorphism maps the zero
% Version : [Gol93] axioms.
% English : If (F,*) and (H,+) are two semigroups, phi is a surjective
% homomorphism from F to H, and id is a left zero for F,
% then phi(id) is a left zero for H.
% Refs : [Gol93] Goller (1993), Anwendung des Theorembeweisers SETHEO a
% Source : [Gol93]
% Names :
% Status : Theorem
% Rating : 0.11 v2.7.0, 0.17 v2.6.0, 0.14 v2.5.0, 0.12 v2.4.0, 0.25 v2.3.0, 0.33 v2.2.1, 0.00 v2.1.0
% Syntax : Number of formulae : 8 ( 2 unit)
% Number of atoms : 21 ( 4 equality)
% Maximal formula depth : 8 ( 4 average)
% Number of connectives : 13 ( 0 ~ ; 0 |; 6 &)
% ( 1 <=>; 6 =>; 0 <=)
% ( 0 <~>; 0 ~|; 0 ~&)
% Number of predicates : 3 ( 0 propositional; 2-2 arity)
% Number of functors : 5 ( 3 constant; 0-3 arity)
% Number of variables : 15 ( 0 singleton; 14 !; 1 ?)
% Maximal term depth : 3 ( 1 average)
% Comments :
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
% File : GRP039-7 : TPTP v2.7.0. Bugfixed v1.0.1.
% Domain : Group Theory (Subgroups)
% Problem : Subgroups of index 2 are normal
% Version : [MOW76] (equality) axioms : Augmented.
% English : If O is a subgroup of G and there are exactly 2 cosets
% in G/O, then O is normal [that is, for all x in G and
% y in O, x*y*inverse(x) is back in O].
% Refs : [MOW76] McCharen et al. (1976), Problems and Experiments for a
% Source : [MOW76]
% Names : GP2 [MOW76]
% Status : Unsatisfiable
% Rating : 0.45 v2.7.0, 0.50 v2.6.0, 0.40 v2.5.0, 0.42 v2.4.0, 0.33 v2.3.0, 0.44 v2.2.1, 0.44 v2.2.0, 0.67 v2.1.0, 0.89 v2.0.0
% Syntax : Number of clauses : 16 ( 2 non-Horn; 12 unit; 8 RR)
% Number of literals : 24 ( 11 equality)
% Maximal clause size : 4 ( 2 average)
% Number of predicates : 2 ( 0 propositional; 1-2 arity)
% Number of functors : 8 ( 5 constant; 0-2 arity)
% Number of variables : 16 ( 0 singleton)
% Maximal term depth : 3 ( 2 average)
% Comments : Used to define a subgroup of index two is a theorem which
% says that {for all x, for all y, there exists a z such that
... some lines removed here for brevity
% Bugfixes : v1.0.1 - Duplicate axioms multiply_inverse_left and
% multiply_inverse_right removed.
%--------------------------------------------------------------------------
This field contains three items of information.
The first item is the problem's name.
The current TPTP release number is given next, followed by the TPTP release
in which the problem was released or last bugfixed (i.e., the formulae were
changed).
The problem name in
Figure Example CNF header is GRP039-7.
The TPTP release is v2.2.0, and the problem clauses were last
bugfixed in release v1.0.1.
The domain field identifies the domain, and possibly a subdomain, from which
the problem is drawn (see Section
The TPTP Domain Structure).
The domain corresponds to the first three letters of the problem name.
The domain of the problem of
Figure Example CNF header is
Group Theory, and the subdomain is Subgroups.
This field provides a one-line, high-level description of the abstract problem.
In axiom files, this field is called % Axioms, and provides a
one-line, high-level description of the axiomatization.
Thus, the problem of Figure Example CNF header
proves that Subgroups of index 2 are normal.
This field gives information that differentiates this version of the problem
from other versions of the problem.
The first possible differentiation is the axiomatization that is used.
If a specific axiomatization is used, a citation is provided.
In the problem of Figure Example CNF header
the axiomatization used is that of
[MOW76].
If the axiomatization is a pure equality axiomatization (uses only the
equal/2 predicate) then this is noted too, as is the case in
Figure Example CNF header.
In the problem of Figure Example CNF header
an existing standard axiomatization has been Augmented, and has
become non-standard due to redundancy.
% Version : [McCharen, et al., 1976] (equality) axioms.
% Theorem formulation : Explicit formulation of the commutator.
This field provides a full description of the problem, if the one-line
description in the % Problem field is too terse.
This field provides a list of abbreviated references for material in which
the problem has been presented.
Other relevant references are also listed.
The reference keys identify BibTeX entries in the Bibliography.bib
file supplied with the TPTP.
The problems in the TPTP have been
(physically) obtained from a variety of sources, both hardcopy and
electronic. In this field the source of the problem is acknowledged,
usually as a citation. If the problem was sourced from an
existing problem collection then the collection name is given in
[ ] brackets.
The problem collections used thus far are:
The problem of Figure Example CNF header was
taken from
[MOW76].
Problems which have appeared in other problem collections or the literature,
often have names which are known in the ATP community.
This field lists all such names known to us for the problem, along with the
source of the name.
If the source is an existing problem collection then the collection name
is cited, as in the % Source field.
If the source of a name is a paper then a citation is given.
If a problem is not given a name in a paper then ``-'' is used as the known
name and a citation is given.
Problems which first appeared in the TPTP have source TPTP, and no other name.
In generator files all known names for instances of the abstract
problem are listed, with the instance size given before the source.
A reverse index, from known names to TPTP names, is
distributed with the TPTP (see
Section Physical Organization).
The problem of Figure Example CNF header is
called GP2 in
[MOW76].
This field gives the ATP status of the problem, according to the
SZS problem status ontology.
For FOF problems it is one of:
For CNF problems it is one of:
In Figure Example CNF header the status is
unsatisfiable.
This field gives the difficulty of the problem, measured relative to
state-of-the-art ATP systems.
The rating is a real number in the range 0.0 to 1.0, where 0.0 means that all
state-of-the-art ATP systems can solve the problem (i.e., the problem is easy),
and 1.0 means no state-of-the-art ATP system can solve the problem (i.e., the
problem is hard).
The rating is followed by a TPTP release number, indicating in which release
the rating was assigned.
If no rating has been assigned, a ? is given.
In Figure Example CNF header the rating is
0.45, assigned in release v2.7.0.
This field lists various syntactic measures of the problem's clauses.
The measures for CNF problems are:
the number of clauses,
the number of non-Horn clauses,
the number of unit clauses,
the number of range restricted clauses,
the number of literals,
the number of equality literals,
the maximal and average clause size (measured by number of literals),
the number of distinct predicate symbols,
the number of distinct propositional symbols,
the minimal and maximal predicate symbol arities,
the number of distinct function symbols,
the number of distinct constants,
the minimal and maximal functor arities,
the number of distinct variables,
the number of singleton variables
(variables that appear only once in a clause),
and the maximal and average term depth (with constants and variables having
depth 1).
The measures for FOF problems are:
the number of formulae,
the number if unit formulae,
the number of atoms,
the number of equality atoms,
the maximal and average formula depth,
the number of connectives,
the numbers of each type of connective,
the number of distinct propositional symbols,
the minimal and maximal predicate symbol arities,
the number of distinct function symbols,
the number of distinct constants,
the minimal and maximal functor arities,
the number of distinct variables,
the number of singleton variables,
the numbers of universally and existentially quantified variables,
and the maximal and average term depth.
See Section Inside the TPTP
for a summary of this information over the entire TPTP.
This field contains free format comments about the problem, e.g.,
the significance of the problem, or the reason for creating the problem.
If the problem was created using the tptp2X utility (see
Section The tptp2X Utility) then the
tptp2X parameters are given.
This field describes any bugfixes that have been done to the formulae of
the problem.
Each entry gives the release number in which the bugfix was done, and
attempts to pinpoint the bugfix accurately.
In the problem of Figure Example CNF header
a bugfix was made in release v1.0.1, by removing duplicate
multiply_inverse_left and multiply_inverse_right clauses.
The Include Section
The include section contains include instructions for TPTP axiom
files.
An example is shown in
Figure Example include section,
extracted from the problem file GRP194+1.p.
%--------------------------------------------------------------------------
%----Include simple curve axioms
include('Axioms/GEO004+0.ax').
%----Include axioms of betweenness for simple curves
include('Axioms/GEO004+1.ax').
%----Include oriented curve axioms
include('Axioms/GEO004+2.ax').
%----Include trajectory axioms
include('Axioms/GEO004+3.ax',[connect_defn,symmetry_of_at_the_same_time]).
%--------------------------------------------------------------------------
The Formulae Section
Each logical formula is wrapped in an annotated formula structure,
whose principle symbol is fof or cnf to indicate the
form of the enclosed logic formula.
The annotated formulae are in the formulae section of the problem file.
%--------------------------------------------------------------------------
%----Definition of a homomorphism
fof(homomorphism1,axiom,
( ! [X] :
( group_member(X,f)
=> group_member(phi(X),h) ) )).
fof(homomorphism2,axiom,
( ! [X,Y] :
( ( group_member(X,f)
& group_member(Y,f) )
=> multiply(h,phi(X),phi(Y)) = phi(multiply(f,X,Y)) ) )).
fof(surjective,axiom,
( ! [X] :
( group_member(X,h)
=> ? [Y] :
( group_member(Y,f)
& phi(Y) = X ) ) )).
%----Definition of left zero
fof(left_zero,axiom,
( ! [G,X] :
( left_zero(G,X)
<=> ( group_member(X,G)
& ! [Y] :
( group_member(Y,G)
=> multiply(G,X,Y) = X ) ) ) )).
%----The conjecture
fof(left_zero_for_f,hypothesis,
( left_zero(f,f_left_zero) )).
fof(prove_left_zero_h,conjecture,
( left_zero(h,phi(f_left_zero)) )).
%--------------------------------------------------------------------------
%--------------------------------------------------------------------------
%----Redundant two axioms
cnf(right_identity,axiom,
( multiply(X,identity) = X )).
cnf(right_inverse,axiom,
( multiply(X,inverse(X)) = identity )).
... some clauses omitted here for brevity
cnf(property_of_O2,axiom,
( subgroup_member(X)
| subgroup_member(Y)
| multiply(X,element_in_O2(X,Y)) = Y )).
%----Denial of theorem
cnf(b_in_O2,negated_conjecture,
( subgroup_member(b) )).
cnf(b_times_a_inverse_is_c,negated_conjecture,
( multiply(b,inverse(a)) = c )).
cnf(a_times_c_is_d,negated_conjecture,
( multiply(a,c) = d )).
cnf(prove_d_in_O2,negated_conjecture,
( ~ subgroup_member(d) )).
%--------------------------------------------------------------------------
Physical Organization
The TPTP is physically organized into six subdirectories, as follows:
The files in the Documents directory contain comprehensive online
information about the TPTP.
They summarize much of the information contained in this report, in
specific files.
Their format provides quick access to the data, using standard system
tools (e.g., grep, awk).
Utilities
The tptp2X Utility
The tptp2X utility is a multi-functional utility for reformatting,
transforming, and generating TPTP problem files. In particular, it
tptp2X is written in Prolog, and should run on most Prolog systems.
It is known to run on current versions of
Eclipse,
SICStus,
SWI,
and
YAP
Prolog.
Before using tptp2X, it is necessary to install the code for the dialect of
Prolog that is to be used.
This and other installation issues are described next.
Installation
The tptp2X utility consists of the following files:
tptp2X is installed by running tptp2X_install, which will prompt
for required information.
To install tptp2X by hand, the following must be attended to:
Using tptp2X
The most convenient way of using the tptp2X utility is through the
tptp2X script.
The use of this script is:
tptp2X [-h][-q<Level>][-i][-s<Size>][-t<Transform>][-f<Format>] [-d<Directory>] -l<NamesFile>|<TPTPFiles>
The -h flag specifies that usage information should be output.
The -q
-s<Size> is ignored for problem (.p) files.
The transformations are:
The default <Transform> is none.
Details of these algorithms can found in
[SM96].
Note that these transformations do not simplify the resultant
clause set; see the next two transformations for this.
For example, -t rm_equality:stfp would remove symmetry,
transitivity, function substitution, and predicate substitution.
This transformation works only if the equality axiomatization is
complete (i.e., including the axioms of reflexivity, symmetry,
transitivity, function substitution for all functors, and predicate
substitution for all predicate symbols).
The default <Format> is tptp.
'<Otter options>' is a quoted (to avoid
UNIX shell errors), comma separated list of Otter options, which
will be output before the formula lists.
See the Otter Reference Manual and Guide
[McC94]
for details of the available options.
For example,
-f otter:none:'set(auto),assign(max_seconds,5)'
would configure Otter to use its auto mode with a
time limit of 5 seconds.
As the auto mode is commonly used with Otter, the tptp2X
script allows the abbreviation -f otter to specify
-f otter:none:'set(auto),set(tptp_eq),clear(print_given)'.
If no -t parameter is specified then -f otter also
sets -t equality:stfp.
<Cardinality> is the required cardinailty of the model.
<Style> specifies the style of SETHEO clauses to write. It
can be one of:
The default style is sign, i.e., the abbreviation
-f setheo means -f setheo:sign.
A common first use of tptp2X is to convert TPTP problems to another format.
To convert all TPTP problems to another format, the use is
tptp2X -f<Format>, e.g., tptp2X -f otter.
To limit the conversion to one or more domains, the domain names are
specified after the <Format>, e.g.,
tptp2X -f leantap ALG GRP LDA.
If you are a new TPTP user, these are probably the uses that you want
to start with.
The tptp2X Output Files
The output files produced by tptp2X are named according to the input file
name and the options used.
The TPTP problem name (the input file name without the .p or
.g) forms the basis of the output file names.
For files created from TPTP generators, the size parameters are appended to
the base name, separated from the base name by a ".".
Then, for each transformation applied a suffix is appended.
The suffixes for the transformations are:
Transformation
Suffix
stdfof
+stdfof
clausify:<Algorithm>
+cls_<Algorithm>
simplify
+simp
cnf:<Algorithm>
+cnf_<Algorithm>
propify
+prop
lr
+lr
cr
+cr
clr
+clr
fr
+fr
random
+ran
er
+er
ran_er
+ran_er
add_equality:<Add>
+eq_<Add>
rm_equality:<Remove>
+rm_eq_<Remove>
set_equality:<Set>
+seq_<Set>
to_equality
+2eq
magic
+nhms
shorten
+short
none
Format
Extension
Format
Extension
bliksem
.blk
carine
.car
dedam
.dedam
dfg
.dfg
dimacs
.dimacs
eqp
.eqp
finder
.fin
geo
.geo
kif
.kif
leancop
.leancop
lf
.lf
oscar
.oscar
otter
.in
protein
.tme
prover9
.in
sem
.sem
setheo
.lop
smt
.smt
thinker
.thinker
oldtptp
.oldtptp
tptp
.tptp
waldmeister
.pr
~/TPTP/TPTP2X> tptp2X -tlr,cr,clr -fkif ../Problems/ALG/ALG001-1.p
---------------------------------------------------------------------
TPTP2X directory = /home/graph/tptp/TPTP/TPTP2X
TPTP directory = /home/graph/tptp/TPTP
Prolog interpreter = /usr/local/bin/eclipse
Files to convert = ../Problems/ALG/ALG001-1.p
Size =
Transformation = lr,cr,clr
Format to convert to = kif
Output directory = /home/graph/tptp/TPTP/TPTP2X/kif
---------------------------------------------------------------------
Made the directory /home/graph/tptp/TPTP/TPTP2X/kif/ALG
ALG001-1 -> /home/graph/tptp/TPTP/TPTP2X/kif/ALG/ALG001-1+lr.kif
ALG001-1 -> /home/graph/tptp/TPTP/TPTP2X/kif/ALG/ALG001-1+cr.kif
ALG001-1 -> /home/graph/tptp/TPTP/TPTP2X/kif/ALG/ALG001-1+clr.kif
~/TPTP/TPTP2X>
This run applies three separate transformations to the problem
ALG001-1.
The transformations are literal order reversal, clause order reversal, and
reversal of both literal and clause order.
The transformed problems are output in kif format, in the directory
kif/ALG below the TPTP2X directory.
The file names are ALG001-1+lr.kif, ALG001-1+cr.kif, and
ALG001-1+clr.kif.
~/TPTP/TPTP2X> tptp2X -q2 -s3..5 -fotter -d~tptp/tmp SYN001-1.g
SYN001-1 -> /home/tptp/tmp/SYN/SYN001-1.003+rm_eq_stfp.in
SYN001-1 -> /home/tptp/tmp/SYN/SYN001-1.004+rm_eq_stfp.in
SYN001-1 -> /home/tptp/tmp/SYN/SYN001-1.005+rm_eq_stfp.in
~/TPTP/TPTP2X>
This run generates three sizes of the generic problem SYN001-1.
The sizes are 3, 4, and 5.
The output files are formatted for Otter, to use its auto mode.
The files are placed in \verb/~/tptp/tmp/SYN, and are named
SYN001-1.003.lop, SYN001-1.004.lop, and
SYN001-1.005.lop.
The quietness level is set to 2, thus suppressing all informational output
except the lines showing what files are produced.
Note that the file SYN001-1.g is automatically found in the generators
directory.
~/TPTP/TPTP2X> tptp2X -tmagic+shorten - < ~tptp/TPTP/Problems/GRP/GRP001-1.p
---------------------------------------------------------------------
TPTP2X directory = /home/tptp/TPTP/TPTP2X
TPTP directory = /home/tptp/TPTP
Prolog interpreter = /usr/local/bin/sicstus
Files to convert = -
Size =
Transformation = magic+shorten
Format to convert to = tptp
Output directory = -
---------------------------------------------------------------------
%--------------------------------------------------------------------------
% File : Shortened file, so there is no header
%--------------------------------------------------------------------------
input_clause(clause_1,axiom,
[++equal(X1,X1)]).
... Lots of output omitted here for brevity
input_clause(clause_41,theorem,
[--p2(c8,c7,c9)]).
%--------------------------------------------------------------------------
~/TPTP/TPTP2X>
This run uses the tptp2X script as a filter, to apply the non-Horn magic set
transformation and then the symbol shortening transformation to
GRP001-1.p.
GRP001-1.p is fed in from standard input, and the output is written to
standard output (thus all informational output is suppressed).
The tptp2X Interactive Mode
If the -i flag is set, the tptp2X script enters interactive mode after
processing all other command line parameters. In interactive mode the user can
change the value of any of the command line parameters.
The user is prompted for the <TPTPFiles>, the <Size>,
the <Transform>, the <Format>, and the <Directory>.
In each prompt the current value is given.
The user may respond by specifying a new value or by entering <cr>
to accept the current value.
The prompt-respond loop continues for each parameter until the user
accepts the value for the parameter.
Example
~/TPTP/TPTP2X> tptp2X -q0 -d~tptp/tmp -i
---- Interactive mode -----------------------------------------------
Files to convert [Problems/*/*.p] : ../Problems/GRP/GRP001-1.p
Files to convert [../Problems/GRP/GRP001-1.p] :
Size [] :
Transformation [none] : lr+rm_equality:stfp
Transformation [lr+rm_equality:stfp] :
Format to convert to [tptp] : setheo
Format to convert to [setheo] :
Output directory [/home/tptp/tmp] :
---- End of Interactive mode ----------------------------------------
---------------------------------------------------------------------
TPTP2X directory = /home/2/tptp/TPTP/TPTP2X
TPTP directory = /home/2/tptp/TPTP
Prolog interpreter = /usr/local/bin/sicstus
Files to convert = ../Problems/GRP/GRP001-1.p
Size =
Transformation = lr+rm_equality:stfp
Format to convert to = setheo:sign
Output directory = /home/tptp/tmp
---------------------------------------------------------------------
Made the directory /home/tptp/tmp/GRP
SICStus 3 #6: Mon Nov 3 18:32:08 MET 1997
... Lots of informational output omitted here for brevity
{/home/2/tptp/TPTP/TPTP2X/tptp2X.main compiled, 12080 msec 785600 bytes}
yes
yes
GRP001-1 -> /home/tptp/tmp/GRP/GRP001-1+lr+rm_eq_stfp.lop
~/TPTP/TPTP2X>
This run converts the problem GRP001-1 to a SETHEO format file.
The literals are reversed and all equality clauses except reflexivity
are removed.
The top level output directory is specified as \verb/~/tptp/tmp, below
which the subdirectory GRP is made.
The output file is named GRP001-1+lr+eq_stfp.lop.
Verbose mode is used, so all informational output is given.
The following subsections are only of interest to real Prolog users.
If you do not want to use Prolog directly, skip to the next section.
Running tptp2X from within Prolog
The tptp2X utility may also be run from within a Prolog interpreter.
The tptp2X.main file has to be loaded, and the entry point is then
tptp2X/5, in the form:
Exampl