Entrants' System Descriptions

• E 3.1
• Prover9 1109a
• Twee 2.4.2
• Vampire 4.8

Strategies

For CASC-29, E implements a two-stage multi-core strategy-scheduling automatic mode. The total CPU time available is broken into several (unequal) time slices. For each time slice, the problem is classified into one of several classes, based on a number of simple features (number of clauses, maximal symbol arity, presence of equality, presence of non-unit and non-Horn clauses, possibly presence of certain axiom patterns...). For each class, a schedule of strategies is greedily constructed from experimental data as follows: The first strategy assigned to a schedule is the the one that solves the most problems from this class in the first time slice. Each subsequent strategy is selected based on the number of solutions on problems not already solved by a preceding strategy. The strategies are then scheduled onto the available cores and run in parallel.

About 140 different strategies have been thoroughly evaluated on all untyped first-order problems from TPTP 7.3.0. We have also explored some parts of the heuristic parameter space with a short time limit of 5 seconds. This allowed us to test about 650 strategies on all TPTP problems, and an extra 7000 strategies on UEQ problems from TPTP 7.2.0. About 100 of these strategies are used in the automatic mode, and about 450 are used in at least one schedule.

Implementation

    https://www.eprover.org

Expected Competition Performance

Prover9 has available positive ordered (and nonordered) resolution and paramodulation, negative ordered (and nonordered) resolution, factoring, positive and negative hyperresolution, UR-resolution, and demodulation (term rewriting). Terms can be ordered with LPO, RPO, or KBO. Selection of the "given clause" is by an age-weight ratio.

Proofs can be given at two levels of detail: (1) standard, in which each line of the proof is a stored clause with detailed justification, and (2) expanded, with a separate line for each operation. When FOF problems are input, proof of transformation to clauses is not given.

Completeness is not guaranteed, so termination does not indicate satisfiability.

Strategies

Given a problem, Prover9 adjusts its inference rules and strategy according to syntactic properties of the input clauses such as the presence of equality and non-Horn clauses. Prover9 also does some preprocessing, for example, to eliminate predicates.

For CASC Prover9 uses KBO to order terms for demodulation and for the inference rules, with a simple rule for determining symbol precedence.

For the FOF problems, a preprocessing step attempts to reduce the problem to independent subproblems by a miniscope transformation; if the problem reduction succeeds, each subproblem is clausified and given to the ordinary search procedure; if the problem reduction fails, the original problem is clausified and given to the search procedure.

Implementation

    http://www.cs.unm.edu/~mccune/prover9/

Expected Competition Performance

Twee features ground joinability testing [MN90] and a connectedness test [BD88], which together eliminate many redundant inferences in the presence of unorientable equations. The ground joinability test performs case splits on the order of variables, in the style of [MN90], and discharges individual cases by rewriting modulo a variable ordering.

Strategies

• Select and normalise the lowest-scored critical pair, and if it is not redundant, add it as a rewrite rule to the active set.
• Normalise the active rules with respect to each other.
• Normalise the goal with respect to the active rules.

Each critical pair is scored using a weighted sum of the weight of both of its terms. Terms are treated as DAGs when computing weights, i.e., duplicate subterms are counted only once per term.

For CASC, to take advantage of multiple cores, several versions of Twee run in parallel using different parameters (e.g., with the goal-directed transformation on or off).

Implementation

The passive set is represented compactly (12 bytes per critical pair) by storing only the information needed to reconstruct the critical pair, not the critical pair itself. Because of this, Twee can run for an hour or more without exhausting memory.

Twee uses an LCF-style kernel: all rules in the active set come with a certified proof object which traces back to the input axioms. When a conjecture is proved, the proof object is transformed into a human-readable proof. Proof construction does not harm efficiency because the proof kernel is invoked only when a new rule is accepted. In particular, reasoning about the passive set does not invoke the kernel.

Twee can be downloaded as open source from:

    https://nick8325.github.io/twee

Expected Competition Performance

Strategies

• Choices of saturation algorithm:
  • Limited Resource Strategy [RV03]
  • DISCOUNT loop
  • Otter loop
  • Instantiation using the Inst-Gen calculus
  • MACE-style finite model building with sort inference
• Splitting via AVATAR [Vor14]
• A variety of optional simplifications.
• Parameterized reduction orderings.
• A number of built-in literal selection functions and different modes of comparing literals [HR+16].
• Age-weight ratio that specifies how strongly lighter clauses are preferred for inference selection. This has been extended with a layered clause selection approach [GS20].
• Set-of-support strategy with extensions for theory reasoning.
• For theory-reasoning:
  • Ground equational reasoning via congruence closure.
  • Addition of theory axioms and evaluation of interpreted functions [RSV21].
  • Use of Z3 with AVATAR to restrict search to ground-theory-consistent splitting branches [RB+16].
  • Specialised theory instantiation and unification [RSV18].
  • Extensionality resolution with detection of extensionality axioms

Implementation

Expected Competition Performance