Package

at.logic.gapt.proofs

resolution

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package resolution

Created by sebastian on 7/13/15.

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Type Members

  1. trait AppliedSubstitution extends AnyRef

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  2. trait BinaryResolutionProof[V <: OccSequent] extends BinaryAGraph[V] with BinaryAGraphProof[V] with ResolutionProof[V] with BinaryProof[V]

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  3. type Clause[+A] = Sequent[A]

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  4. type FOLClause = Sequent[FOLAtom]

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  5. type HOLClause = Sequent[HOLAtom]

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  6. trait InstantiatedVariable extends AnyRef

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  7. trait NullaryResolutionProof[V <: OccSequent] extends LeafAGraph[V] with NullaryAGraphProof[V] with ResolutionProof[V] with NullaryProof[V]

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  8. type OccClause = Sequent[FormulaOccurrence]

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  9. case class RalException[V <: LabelledOccSequent](message: String, rp: List[RobinsonResolutionProof], ralp: List[RalResolutionProof[V]], exp: List[LambdaExpression]) extends Exception with Product with Serializable

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  10. case class ResolutionException(msg: String, proofs: List[RobinsonResolutionProof], clauses: List[HOLClause]) extends Exception with Product with Serializable

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    Given a formula f and a clause a in CNF(-f), PCNF computes a proof of s o a (see logic.at/ceres for the definition of o) Note about checking containment up to variables renaming: we compute the variable renaming from the lk proof to the resolution proof for a specific clause.

    Given a formula f and a clause a in CNF(-f), PCNF computes a proof of s o a (see logic.at/ceres for the definition of o) Note about checking containment up to variables renaming: we compute the variable renaming from the lk proof to the resolution proof for a specific clause. We cannot apply it to the formula in s as it might be quantified over this variables so we apply it to the resulted lk proof. We must apply it as otherwise the substitution in the resolution to lk transformation will not be applied to these clauses. In the weakenings application at the end of this method we try to apply it to the formulas as well as if it is quantified over these variables, it will be also quantified in the proof so no damage done.

  11. trait ResolutionProof[V <: OccSequent] extends AGraphProof[V]

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  12. class ResolutionRuleCreationException extends ResolutionRuleException

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  13. class ResolutionRuleException extends RuleException

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  14. implicit class RichClause[+A] extends AnyRef

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  15. implicit class RichOccClause extends RichClause[FormulaOccurrence]

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  16. abstract class RobinsonToRal extends AnyRef

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  17. class TseitinCNF extends AnyRef

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  18. trait UnaryResolutionProof[V <: OccSequent] extends UnaryAGraph[V] with UnaryAGraphProof[V] with ResolutionProof[V] with UnaryProof[V]

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Value Members

  1. object Clause

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  2. object FOLClause

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  3. object ForgetfulParamodulate

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  4. object ForgetfulResolve

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  5. object ForgetfulResolveIndexed

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  6. object HOLClause

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  7. object InitialSequent

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  8. object InitialType extends NullaryRuleTypeA with Product with Serializable

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  9. object InstantiateElimination

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  10. object OccClause

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  11. object PCNF

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  12. object Paramodulants

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  13. object ResolutionSubstitution

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    Eliminates the insantiate rule from a RobinsonResolutionProof

  14. object RobinsonToExpansionProof

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  15. object RobinsonToLK extends Logger

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  16. object RobinsonToRal extends RobinsonToRal

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    Created by marty on 9/9/14.

  17. object TseitinCNF

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  18. package andrews

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  19. object computeSkolemTerm

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  20. object containedVariables

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  21. object createContext

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  22. object factorDuplicateLiterals

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  23. object findDerivationViaResolution

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  24. object fixDerivation extends Logger

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    Sometimes, we have a resolution refutation R of a set of clauses C and want a refutation R' of a set C' such that C implies C'.

    Sometimes, we have a resolution refutation R of a set of clauses C and want a refutation R' of a set C' such that C implies C'.

    This algorithm tries to obtain R' by trying to replace clauses c from C in R by derivations of C from C' in the following way:

    • If c is in C' or c is an instance of reflexivity, do nothing.
    • If c is subsumed by some c' in C', derive c from c' by factoring.
    • Otherwise, try to derive c from C' by paramodulation and symmetry (prover9 often needs this, and the check is usually fast),
    • Otherwise, try to derive c from C' by propositional resolution.

    If none of this works, we issue a warning and keep the clause c. If no warning is issued and the algorithm terminates, the result is the desired R'.

    In general, if R is a derivation of a clause c, the result R' of fixDerivation(R) is a derivation of a subclause of c.

  25. object initialSequents

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  26. object mapInitialClauses

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    Applies a function to each initial clause in a resolution proof, replacing the initial clause with a new proof.

    Applies a function to each initial clause in a resolution proof, replacing the initial clause with a new proof. The resulting proof may prove a smaller clause than the original one.

  27. object numberOfResolutionsAndParamodulations

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  28. package ral

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  29. package robinson

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  30. object tautologifyInitialClauses

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