This document contains information about the:

The CASC rules, specifications, and deadlines are absolute. Only the competition panel has the right to make exceptions. It is assumed that all entrants have read the web pages related to the competition, and have complied with the competition rules. Non-compliance with the rules can lead to disqualification. A "catch-all" rule is used to deal with any unforeseen circumstances:

- The TFA division has returned from hiatus.
- The LTB division will not have FOF versions of the problems, but will have TXF versions.

- The
**THF**division: Typed (monomorphic) Higher-order Form theorems (axioms with a provable conjecture). The THF division has two problem categories:- The
**TNE**category: THF with No Equality - The
**TEQ**category: THF with EQuality

- The
- The
**TFA**division: Typed (monomorphic) First-order with Arithmetic theorems (axioms with a provable conjecture). The TFA division has two problem categories:- The
**TFI**category: TFA with only Integer arithmetic - The
**TFE**category: TFA with only rEal arithmetic

- The
- The
**FOF**division: First-Order Form theorems (axioms with a provable conjecture). The FOF division has two problem categories:- The
**FNE**category: FOF with No Equality - The
**FEQ**category: FOF with EQuality

- The
- The
**FNT**division: First-order form Non-Theorems (axioms with a countersatisfiable conjecture, and satisfiable axiom sets). The FNT division has two problem categories:- The
**FNN**category: FNT with No equality - The
**FNQ**category: FNT with eQuality

- The
- The
**UEQ**division: Unit EQuality clause normal form theorems (unsatisfiable clause sets). - The
**SLH**division: Typed (monomorphic) higher-order theorems without arithmetic (axioms with a provable conjecture), generated by Isabelle's Sledgehammer system. - The
**LTB**division: Theorems (axioms with a provable conjecture) from Large Theories, presented in Batches. A large theory has many functions and predicates, and many axioms of which typically only a few are required for the proof of a theorem. The problems in a batch are given to an ATP system all at once, and typically have a common core set of axioms. The batch presentation allows the ATP systems to load and preprocess the common core set of axioms just once, and to share logical and control results between proof searches. Each problem category might be accompanied by a set of training problems and their solutions, taken from the same source as the competition problems. The training data can be used for ATP system tuning during (typically at the start of) the competition. In this CASC the LTB division has one problem category:- The
**VBT**category: Problems generated by Isabelle's Sledgehammer system, from the Van Emde Boas Trees entry in the Isabelle Archive of Formal Proofs.

- The THF versions do not use
`$ite`,`$let`, or tuples. - There are no common include files.
- The TF0 problems have upto 12792 axioms, the TF1 problems have upto 12528 axioms, the TX0 problems have upto 12579 axioms, the TX1 problems have upto 12488 axioms, the TH0 problems have upto 10296 axioms, and the TH1 problems have upto 10078 axioms.
- Not all of the problems are theorems.

- The

- An octa-core Intel(R) Xeon(R) E5-2620 v4 @ 2.10GHz, without hyperthreading.
- 128GB memory
- The CentOS Linux release 7.4.1708 (Core) operating system, Linux kernel 3.10.0-693.el7.x86_64.

One ATP system runs on one CPU at a time. Systems can use all the cores on the CPU, which is advantageous in divisions where a wall clock time limit is used.

Problems for the THF, TFA, FOF, FNT, and UEQ divisions are taken from the TPTP Problem Library. The TPTP version used for CASC is released only after the competition has started, so that new problems in the release have not been seen by the entrants. The problems have to meet certain criteria to be eligible for selection. The problems used are randomly selected from the eligible problems based on a seed supplied by the competition panel.

- The TPTP tags problems that are designed specifically to be suited or
ill-suited to some ATP system, calculus, or control strategy as
*biased*. They are excluded from the competition. - The problems must be syntactically non-propositional.
- The TPTP uses system performance data in the Thousands of Solutions from Theorem Provers (TSTP) solution library to compute problem difficulty ratings in the range 0.00 (easy) to 1.00 (unsolved). Difficult problems with a rating in the range 0.21 to 0.99 are eligible. Problems of lesser and greater ratings might also be eligible in some divisions if there are not enough problems with ratings in that range. Systems can be submitted before the competition so that their performance data is used in computing the problem ratings.
- The selection is constrained so that no division or category contains an excessive number of very similar problems.
- The selection is biased to select problems that are new in the TPTP version used, until 50% of the problems in each problem category have been selected, after which random selection from old and new problems continues. The number of new problems used depends on how many new problems are eligible and the limitation on very similar problems.

Problems for the SLH division were generated by Isabelle's Sledgehammer system. Appropriately difficult problems were chosen based on performance data similar to that in the TSTP. Here are some sample problems that have been extracted from the collection, for you to get a feel for the SLH division problems.

Problems for the LTB division are taken from various sources, with each problem category being based on one source. Entrants are expected to honestly not use publicly available problem sets for tuning before the competition. The process for selecting problems depends on the problem source.

*Number of Problems*

In the TPTP-based divisions, the minimal numbers of problems that must be
used in each division and category, to ensure sufficient confidence in the
competition results, are determined from the numbers of eligible problems
in each division and category. The competition organizer have to ensure
that there are sufficient computers available to run the ATP systems on
this minimal number of problems.
The minimal numbers of problems are used in determining the
time limit imposed on solution attempts.
The minimal numbers of problems to be used in each division of the competition
are determined from the number of computers available, the time allocated
to the competition, the number of ATP systems to be run on the computers over
the divisions, and the time limit imposed on solution attempts, according to
the following relationship:

NumberOfComputers * TimeAllocated NumberOfProblems = --------------------------------- NumberOfATPSystems * TimeLimitIt is a lower bound on the number of problems because it assumes that every system uses all of the time limit for each problem. Since some solution attempts succeed before the time limit is reached, more problems can be used. The number of problems used in each division and problem category is (roughly) proportional to the numbers of eligible problems, after taking into account the limitation on very similar problems, determined according to the judgement of the competition organizer.

In the LTB division the number of problems in each category is determined by the number of problems in the corresponding problem source. In this CASC the VBT problem category has 800 problems (with six versions of each problem).

- stripping out all comment lines, including the problem header
- randomly reordering the formulae/clauses
(
`include`directives are left before formulae, type declarations and definitions are left before the symbols' uses) - randomly swapping the arguments of associative connectives, and randomly reversing implications
- randomly reversing equalities

In the SLH and LTB division the formulae are not obfuscated, thus allowing the ATP systems to take advantage of natural structure that occurs in the problems.

In the TPTP-based divisions the problems are given to the ATP systems in increasing order of TPTP difficulty rating. In the SLH division the problems are given in a roughly estimated increasing order of difficulty. In the LTB division the problems in each batch are given in their natural order in the problem source.

**Batch Specification Files**

The problems for each problem category of the LTB division are listed in a
*batch specification* file, containing global data lines and one or
more batch specifications.
The global data lines are:

- A problem category line of the form

`division.category LTB.`*category_mnemonic*

For this CASC it is

`division.category LTB.VBT`

- The name of a
`.tgz`file (relative to the directory holding the batch specification file) that contains training data in the form of problems in TPTP format and one or more solutions to each problem in TSTP format, in a line of the form

`division.category.training_data`*tgz_file_name*

For this CASC it is

`division.category.training_data TrainingData/TrainingData.VBT.tgz`

The`.tgz`file expands in place to three directories:`Axioms`,`Problems`, and`Solutions`.`Axioms`contains all the axiom files that are used in the training and competition problems.`Problems`contains the training problems.`Solutions`contains a subdirectory for each of the`Problems`, containing TPTP format solutions to the problem. The language of a solution might not be the same as the language of the problem, e.g., a proof to a THF problem might be written in FOF, or the proof of a TFF problem might be written in THF.

- A header line
`% SZS start BatchConfiguration` - A specification of whether or not the problems in the batch must be
attempted in order is given, in a line of the form

`execution.order`*ordered/unordered*

If the batch is ordered the ATP systems may not start any attempt on a problem, including reading the problem file, before ending the attempt on the preceding problem. For this CASC it is

`execution.order unordered`

- A specification of what output is required from the ATP systems
for each problem, in a line of the form

`output.required`*space_separated_list*

where the available list values are the SZS values`Assurance`,`Proof`,`Model`, and`Answer`. For this CASC it is

`output.required Proof` - The wall clock time limit for each problem, in a line of the form

`limit.time.problem.wc`*limit_in_seconds*

A value of zero indicates no per-problem limit. For this CASC it is

`limit.time.problem.wc 0` - The overall wall clock time limit for the batch, in a line of the form

`limit.time.overall.wc`*limit_in_seconds* - A terminator line
`% SZS end BatchConfiguration` - A header line
`% SZS start BatchIncludes` -
`include`directives that are used in every problem. All the problems in the batch have these`include`directives, and can also have other`include`directives that are not listed here. For this CASC, see the additional notes below. - A terminator line
`% SZS end BatchIncludes` - A header line
`% SZS start BatchProblems` - Pairs of problem file names (relative to the directory holding the batch
specification file), and output file names where the output for the
problem must be written.
The output files must be written in the directory specified
as the second argument to the
`starexec_run`script (the first argument is the name of the batch specification file). For this CASC, see the additional notes below. - A terminator line
`% SZS end BatchProblems`

- In the
`BatchProblems`section, the multiple versions of each problem are specified using UNIX`*`globbing, e.g.,`VBT00001*.p`. The versions of each problem have extensions as follows: the TF0 version uses`_1`, the TF1 version uses`_2`, the TX0 version uses`_3`, the TX1 version uses`_4`, the TH0 version uses`^1`, and the TH1 version uses`^2`. - Proof output must identify which version of the problem was solved - see the section on output notification lines.
- In the
`BatchIncludes`section (not in problem files), multiple versions of included axiom files may be specified using UNIX`*`globbing, e.g.,`include('Axioms/VBT002*.ax')`could refer to all of`VBT001+1.ax`,`VBT001_1.ax`,`VBT001_2.ax`,`VBT001^1.ax`,`VBT001^2.ax`. For a given problem, systems may use only the axiom files whose version matches that of the problem file (there might be none), e.g., if the problem version is`+1`then use only the axiom files with the version`+1`. Using any other versions could lead to weird results.

In the SLH division a CPU time limit is imposed for each problem. The minimal time limit for each problem is 15s, and the maximal time limit for each problem is 90s. The time limit is chosen as a reasonable value within the range allowed, and is announced at the competition.

In the LTB division wall clock time limits are imposed. For each batch there might be a wall clock time limit for each problem, provided in the configuration section at the start of each batch. If there is a wall clock time limit for each problem, the minimal limit for each problem is 15s, and the maximal limit for each problem is 90s. For each batch there might be an overall wall clock time limit, provided in the configuration section at the start of each batch. The overall limit is proportional to the number of problems in the batch, e.g. (but not necessarily), the batch's per-problem time limit multiplied by the number of problems in the batch. Time spent before starting the first problem of a batch (e.g., preloading and analysing the batch axioms), and times spent between the end of an attempt on a problem and the starting of the next (e.g., learning from a proof just found), are not part of the times taken on the individual problems, but are part of the overall time taken. There are no CPU time limits.

The systems are ranked in the competition divisions according to the number of problems solved with an acceptable solution output. Ties are broken according to the average time taken over problems solved. Trophies are awarded to the competition divisions' winners.

The competition panel decides whether or not the systems' solutions are "acceptable". The criteria include:

- Derivations must be complete, starting at formulae from the problem, and ending at the conjecture (for axiomatic proofs) or a false formula (for proofs by contradiction, e.g., CNF refutations).
- For solutions that use translations from one form to another, e.g., translation of FOF problems to CNF, the translations must be adequately documented.
- Derivations must show only relevant inference steps.
- Inference steps must document the parent formulae, the inference rule used, and the inferred formula.
- Inference steps must be reasonably fine-grained, except in the SLH division where just a single inference step from the axioms to the conjecture is also an acceptable output.
- An unsatisfiable set of ground instances of clauses is acceptable for establishing the unsatisfiability of a set of clauses.
- Models must be complete, documenting the domain, function maps, and predicate maps. The domain, function maps, and predicate maps may be specified by explicit ground lists (of mappings), or by any clear, terminating algorithm.

- The
*state-of-the-art contribution*(SotAC) quantifies the unique abilities of each system (excluding the previous year's winners that are earlier versions of competing systems). For each problem solved by a system, its SotAC for the problem is the fraction of systems that do not solve the problem, and a system's overall SotAC is the average over the problems it solves but that are not solved by all the systems. - The
*core usage*measures the extent to which the systems take advantage of multiple cores. It is the average of the ratios of CPU time used to wall clock time used, over the problems solved. - The
*efficiency measure*balances the number of problems solved with the time taken. It is the average solution rate over the problems solved (the solution rate for one problem is the reciprocal of the time taken to solve it), multiplied by the fraction of problems solved. Efficiency is computed for both CPU time and wall clock time, to measure how efficiently the systems use one core and multiple cores respectively.

At some time after the competition all high ranking systems in the competition divisions are tested over the entire TPTP. This provides a final check for soundness (see the section on system properties regarding soundness checking before the competition). If a system is found to be unsound during or after the competition, but before the competition report is published, and it cannot be shown that the unsoundness did not manifest itself in the competition, then the system is retrospectively disqualified. At some time after the competition, the solutions from the winners are checked by the panel. If any of the solutions are unacceptable, i.e., they are sufficiently worse than the samples provided, then that system is retrospectively disqualified. All disqualifications are explained in the competition report.

Systems can be entered at only the division level, and can be entered into more than one division. A system that is not entered into a division is assumed to perform worse than the entered systems, for that type of problem - wimping out is not an option. Entering many similar versions of the same system is deprecated, and entrants may be required to limit the number of system versions that they enter. Systems that rely essentially on running other ATP systems without adding value are deprecated; the competition panel may disallow or move such systems to the demonstration division.

The division winners of the previous CASC are automatically entered into their demonstration divisions, to provide benchmarks against which progress can be judged. Prover9 1109a is automatically entered into the FOF demonstration division, to provide a fixed-point against which progress can be judged.

- Architecture. This section introduces the ATP system, and describes the calculus and inference rules used.
- Strategies. This section describes the search strategies used, why they are effective, and how they are selected for given problems. Any strategy tuning that is based on specific problems' characteristics must be clearly described (and justified in light of the tuning restrictions).
- Implementation. This section describes the implementation of the ATP system, including the programming language used, important internal data structures, and any special code libraries used. The availability of the system is also given here.
- Expected competition performance. This section makes some predictions about the performance of the ATP system for each of the divisions and categories in which it is competing.
- References.

The system description must be emailed to the competition organizer by the system description deadline. The system descriptions form part of the competition proceedings.

Proof/model samples are required as follows:

- THF:
`SET014^4` - TFA:
`DAT013=1` - FOF:
`SEU140+2` - FNT:
`NLP042+1`and`SWV017+1` - UEQ:
`BOO001-1` - LTB: All of the variants of
`VBT00001`that the system can attempt.

**Execution, Soundness, and Completeness**

- Systems must be fully automatic, i.e., all command line switches have to be the same for all problems in each division.
- Systems' performances must be reproducible by running the system again.
- Systems must be sound. At some time before the competition all the systems in the competition divisions are tested for soundness. Non-theorems are submitted to the systems in the THF, TFA, FOF, EPR, UEQ, and LTB divisions, and theorems are submitted to the systems in the FNT and EPR divisions. Finding a proof of a non-theorem or a disproof of a theorem indicates unsoundness. If a system fails the soundness testing it must be repaired by the unsoundness repair deadline or be withdrawn.
- Systems do not have to be complete in any sense, including calculus, search control, implementation, or resource requirements.
- All techniques used must be general purpose, and expected to extend usefully to new unseen problems. The precomputation and storage of information about individual problems that might appear in the competition or their solutions is not allowed. (It's OK to store information about LTB training problems.) Strategies and strategy selection based on individual problems or their solutions are not allowed. If machine learning procedures are used to tune a system, the learning must ensure that sufficient generalization is obtained so that no there is no specialization to individual problems. The system description must explain any such tuning or training that has been done. The competition panel may disqualify any system that is deemed to be problem specific rather than general purpose. If you are in doubt, contact the competition organizer.

- In all divisions except LTB the solution output must be to
`stdout`. In the LTB division the solution output must be to the named output file for each problem, in the directory specified as the second argument to the`starexec_run`script. If multiple attempts are made on a problem in an unordered batch, each successive output file must overwrite the previous one. - In the LTB division the systems must print SZS notification lines to
`stdout`when starting and ending work on a problem (including any cleanup work, such as deleting temporary files). For example% SZS status Started for CSR075+2.p ... (system churns away, progress output to file) % SZS status GaveUp for CSR075+2.p % SZS status Ended for CSR075+2.p

... and later in another attempt on that problem ...% SZS status Started for CSR075+2.p ... (system churns away, progress, result, and solution overwrites file) % SZS status Theorem for CSR075+2.p % SZS status Ended for CSR075+2.p

- For each problem, the system must output a distinguished string
indicating what solution has been found or that no conclusion has been
reached.
Systems must use the
SZS ontology and standards for this.
For example
% SZS status Theorem for SYN075+1.p

or% SZS status GaveUp for SYN075+1.p

In the LTB division this line must be the last line output before the`Ended`notification line. The line must also be output to the output file. - When outputting a solution, the start and end of the solution must
be delimited by distinguished strings.
Systems must use the
SZS ontology and standards for this.
For example
% SZS output start CNFRefutation for SYN075+1.p ... % SZS output end CNFRefutation for SYN075+1.p

The string specifying the problem status must be output before the start of a solution. Use of the TPTP format for proofs and finite interpretations is encouraged. - Solutions may not have irrelevant output (e.g., from other threads running in parallel) interleaved in the solution.

- Systems that run on the competition computers must be
interruptible by a
`SIGXCPU`signal so that CPU time limits can be imposed, and interruptable by a`SIGALRM`signal so that wall clock time limits can be imposed. For systems that create multiple processes the signal is sent first to the process at the top of the hierarchy, then one second later to all processes in the hierarchy. The default action on receiving these signals is to exit (thus complying with the time limit, as required), but systems may catch the signals and exit of their own accord. If a system runs past a time limit this is noticed in the timing data, and the system is considered to have not solved the problem. - If a system terminates of its own accord it may not leave any
temporary or intermediate output files.
If a system is terminated by a
`SIGXCPU`or`SIGALRM`it may not leave any temporary or intermediate output files anywhere other than in`/tmp`. - For practical reasons excessive output from an ATP system is not allowed. A limit, dependent on the disk space available, is imposed on the amount of output that can be produced.

For systems running on entrant supplied computers in the demonstration
division, entrants must email a `.tgz` file containing the source code
and any files required for building the executable system to the competition
organizer by the system delivery deadline.

After the competition all competition division systems' source code is made publicly available on the CASC web site. In the demonstration division the entrant specifies whether or not the source code is placed on the site. An open source license is encouraged.

Entrants are encouraged to make a public release of their systems ASAP after the competition, so that users can enjoy the latest capabilities.

A system has solved a problem iff it outputs its termination string within the time limit, and a system has produced a solution iff it outputs its end-of-solution string within the time limit. The result and timing data is used to generate an HTML file, and a web browser is used to display the results.

The execution of demonstration division systems is supervised by their entrants.

- Check: You can
login to StarExec. If not,
apply for an account in the TPTP community.
- Check: You can access the TPTP space. If not,
email the competition organizer.
- Check: You can create and upload a
StarExec installation package.
The competition organizer have examplar StarExec installation packages
that you can use as a starting point - email the competition organizer
to get one that is appropriate for your ATP system.
- Check: You can create a job and run it, and your ATP system gets the
correct result.
Use the SZS post processor.
- Check: Your ATP system can solve a problem that has
`include`directives. Because of the way StarExec runs jobs, your ATP system must implement the TPTP requirement that "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." - Check: You can email your StarExec installation package to the competition organizer for testing.