Design and Organization

This document contains information about the:

• Changes
• Divisions
  • Competition Divisions
  • Demonstration Division
• Infrastructure
  • Computers
  • Problems
  • Resource Limits
• System Evaluation
• System Entry
  • System Description
  • Sample Solutions
• System Requirements
  • System Properties
  • System Delivery
  • System Execution
  • System Checks

Disclaimer

• The TFN division has been put into a hiatus state.

• The SLH division has been added.

Competition Divisions

• The THF division: Typed 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 TFA division: Typed First-order with Arithmetic theorems (axioms with a provable conjecture). The TFA division has three problem categories:
  • The TFI category: TFA with only Integer arithmetic
  • The TFR category: TFA with only Rational arithmetic
  • The TFE category: TFA with only rEal arithmetic

• 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 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 EPR division: Effectively PRopositional clause normal form theorems and non-theorems (clause sets). Effectively propositional means that the problems are syntactically non-propositional but are known to be reducible to propositional problems, e.g., CNF problems that have no functions with arity greater than zero. The EPR division has two problem categories:
  • The EPT category: Effectively Propositional Theorems (unsatisfiable clause sets)
  • The EPS category: Effectively Propositional non-theorems (Satisfiable clause sets)

• The SLH division: Typed first order theorems without arithmetic (axioms with a provable conjecture), generated by Isaballe's SLedgeHammer system and submitted to the SystemOnTPTP service.

• The LTB division: First-order form theorems (axioms with a provable conjecture) from Large Theories, presented in Batches. A large theory has many functors and predicates, and many axioms of which typically only a few are required for the proof of a theorem. Problems in a batch all use a common core set of axioms, and the problems in a batch are given to the ATP system all at once. Each problem category is accompanied by a set of training problems and their solutions, taken from the same source as the competition problems, that can be used for tuning and training during (typically at the start of) the competition. In CASC-26 the LTB division has one problem category, which remained a secret until the day of CASC (to ensure there was no pre-tuning).

Demonstration Division

Computers

• Four (a quad-core chip) Intel(R) Xeon(R) E5-2609, 2.40GHz CPUs
• 128GB memory
• The Red Hat Enterprise Linux Server release 7.2 (Maipo) operating system, kernel 3.10.0-327.10.1.el7.x86_64.

Problems

• The TPTP tags problems that designed specifically to be suited or ill-suited to some ATP system, calculus, or control strategy as biased, and they are excluded from the competition.
• The problems are 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. Performance data from systems submitted by the system submission deadline is used for computing the problem ratings for the TPTP version used for the competition.
• 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 actual percentage 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 collected from submissions from Isabelle's Sledgehammer subsystem to the SystemOnTPTP service. The problems were collected over a long period with sampling that ensures diversity. 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. The problems in each category have a large number of common included axiom files (the "common core set of axioms"). The batch presentation allows the ATP systems to load and process the common core set of axioms just once, and to share logical and control results between proof searches. The training problems and solutions facilitate and promote learning from previous proofs. In CASC-26, the problems in the one problem category have between 8000 and 90000 axioms, of which about 8000 are commonly included. They have less than 10 predicates, but from about 1500 to 25000 function symbols. The training set has 1000 problems, each of which is accompamied by one or two solutions.

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 organizers 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 limits imposed on each solution attempt.

In the TPTP-based and SLH divisions, the lower bound on the total number of problems to be used is determined from the number of computers available, the time allocated to the competition, the number of ATP systems to be run on the competition computers over all the divisions, and the time limit per problem, according to the following relationship:

                   NumberOfComputers * TimeAllocated
NumberOfProblems = ---------------------------------
                     NumberOfATPSystems * TimeLimit

In the LTB division the number of problems in each problem category is determined by the number of problems in the corresponding problem set. In CASC-26, the one problem category has 1500 problems.

Problem Preparation
The problems are given to the ATP systems in TPTP format, with include directives. In order to ensure that no system receives an advantage or disadvantage due to the specific presentation of the problems in the TPTP, the problems in the TPTP-based divisions are preprocessed to:

• strip out all comment lines, including the problem header
• randomly reorder the formulae/clauses (include directives are left before formulae, type declarations and definitions are kept before the symbols' uses)
• randomly swap the arguments of associative connectives, and randomly reverse implications
• randomly reverse equalities

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
  
• 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
  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.

Resource Limits

In the SLH division, a wall clock time limit is imposed for each problem. The minimal wall clock time limit per problem is 15s, and the maximal wall clock time limit per problem is 90s. The wall clock time limit is chosen as a reasonable value within the range allowed, based on performance data similar to that in the TSTP, and is announced at the competition. There are no CPU time limits (i.e., using all cores makes sense).

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. The minimal wall clock time limit per problem is 15s, and the maximal wall clock time limit per problem is 90s. For each batch there is 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., 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, from the performance data. The THF, TFA, FOF, FNT, and LTB divisions are ranked according to the number of problems solved with an acceptable proof/model output. The EPR and SLH divisions are ranked according to the number of problems solved, but not necessarily accompanied by a proof or model (but systems that do output proofs/models are highlighted in the presentation of results). Ties are broken according to the average time taken over problems solved (CPU time or wall clock time, depending on the type of limit in the division). Winners are announced and prizes are awarded.

The competition panel decides whether or not the systems' proofs and models 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 proofs 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.
• 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. For each problem solved by a system, its SOTAC for the problem is the inverse of the number of systems that solved the problem. A system's overall SOTAC is its average SOTAC over the problems it solves.
• The core usage is the average of the ratios of CPU time to wall clock time used, over the problems solved. This measures the extent to which the systems take advantage of multiple cores.
• The efficiency measure balances the number of problems solved with the time taken. It is the average of the inverses of the times for problems solved multiplied by the fraction of problems solved. This can be interpreted intuitively as the average of the solution rates for problems solved, 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 how efficiently systems use 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 proofs and models from the winners (of divisions ranked by the numbers of proofs/models output) are checked by the panel. If any of the proofs or models are unacceptable, i.e., they are significantly 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 competition 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 divisions, to provide benchmarks against which progress can be judged. Prover9 2009-11A is automatically entered into the FOF division, to provide a fixed-point against which progress can be judged.

System Description

• 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 in each of the divisions and categories in which it is competing.
• References.

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

Sample Solutions

Proof/model samples are required as follows:

• THF: SET014^4
• TFA: DAT013=1
• FOF and LTB: SEU140+2
• FNT: NLP042+1 and SWV017+1

System Properties

Execution, Soundness, and Completeness

1. Systems must be fully automatic, i.e., all command line switches have to be the same for all problems/problem categories in each division.
2. Systems' performance must be reproducible by running the system again.
3. 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, SLH, 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. For systems running on computers supplied by the entrant in the demonstration division, the entrant must perform the soundness testing and report the results to the competition organizers.
4. Systems do not have to be complete in any sense, including calculus, search control, implementation, or resource requirements.
5. 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 or their solutions. The system description must fully 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.

1. In all divisions except LTB all solution output must be to stdout. In the LTB division all 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.
2. 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, result and solution 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, result and solution appended to file)
     % SZS status Theorem for CSR075+2.p
     % SZS status Ended for CSR075+2.p

3. 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 ending notification line (the line must also be output to the output file).
4. When outputting proofs/models, the start and end of the proof/model 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 proof/model. Use of the TPTP format for proofs and finite interpretations is encouraged.

1. 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 that problem.
2. 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.
3. 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.

System Delivery

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 organizers 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.

System Execution

A system has solved a problem iff it outputs its termination string within the time limit, and a system has produced a proof/model iff it outputs its end-of-proof/model 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 the demonstration division systems is supervised by their entrants.

System Checks

• 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 organizers.

• Check: You can create and upload a StarExec installation package. The competition organizers have examplar StarExec installation packages that you can use as a starting point - email the competition organizers 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.