t

at.logic.gapt.proofs.resolution

ResolutionProof

trait ResolutionProof extends SequentProof[Formula, ResolutionProof] with DagProof[ResolutionProof]

Proof using resolution.

Our resolution calculus integrates higher-order reasoning, structural clausification, and Avatar-style splitting as in [Vor14].

The judgments of this calculus are A-sequents. An A-sequent is a pair of a sequent of HOL formulas, and a conjunction of propositional literals---the assertion (that's where the "A" comes from).

Γ :- Δ <- A

We store the sequent as a HOLSequent in ResolutionProof#conclusion, and the negation of the assertion as a HOLClause in ResolutionProof#assertions (as the negation of a conjunction of literals is a clause). The judgment is interpreted as one of the following equivalent formulas:

assertions.toNegConjunction --> conclusion.toDisjunction
(conclusion ++ assertions).toDisjunction   // equivalent to the first one

Inferences such as Resolution or Paramod do not operate on the assertions. Unless specified otherwise, assertions are inherited by default:

Γ :- Δ, a <- A       a, Π :- Λ <- B
--------------------------------
      Γ, Π :- Δ, Λ <- A ∧ B

There is no factoring on assertions, duplicate assertions are automatically removed.

Reading from top to bottom, a resolution proof is usually structured as follows:

Substitutions are not absorbed into resolution, factoring, and paramodulation; they are explicitly represented using Subst.

[Vor14] Andrei Voronkov, AVATAR: The Architecture for First-Order Theorem Provers. CAV 2014: pp. 696-710

Source
resolution.scala
Ordering
  1. Alphabetic
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Inherited
  1. ResolutionProof
  2. SequentProof
  3. DagProof
  4. Product
  5. Equals
  6. AnyRef
  7. Any
Implicitly
  1. by any2stringadd
  2. by StringFormat
  3. by Ensuring
  4. by ArrowAssoc
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Visibility
  1. Public
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Abstract Value Members

  1. abstract def auxIndices: Seq[Seq[SequentIndex]]

    A list of lists of SequentIndices denoting the auxiliary formula(s) of the rule.

    A list of lists of SequentIndices denoting the auxiliary formula(s) of the rule. The first list contains the auxiliary formulas in the first premise and so on.

    Definition Classes
    SequentProof
  2. abstract def canEqual(that: Any): Boolean
    Definition Classes
    Equals
  3. abstract def conclusion: Sequent[Formula]

    The conclusion of the rule.

    The conclusion of the rule.

    Definition Classes
    SequentProof
  4. abstract def immediateSubProofs: Seq[ResolutionProof]

    The immediate subproofs of this rule.

    The immediate subproofs of this rule.

    Definition Classes
    DagProof
  5. abstract def mainIndices: Seq[SequentIndex]

    A list of SequentIndices denoting the main formula(s) of the rule.

    A list of SequentIndices denoting the main formula(s) of the rule.

    Definition Classes
    SequentProof
  6. abstract def occConnectors: Seq[SequentConnector]

    A list of occurrence connectors, one for each immediate subproof.

    A list of occurrence connectors, one for each immediate subproof.

    Definition Classes
    SequentProof
  7. abstract def productArity: Int
    Definition Classes
    Product
  8. abstract def productElement(n: Int): Any
    Definition Classes
    Product

Concrete Value Members

  1. final def !=(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  2. final def ##(): Int
    Definition Classes
    AnyRef → Any
  3. def +(other: String): String
    Implicit
    This member is added by an implicit conversion from ResolutionProof to any2stringadd[ResolutionProof] performed by method any2stringadd in scala.Predef.
    Definition Classes
    any2stringadd
  4. def ->[B](y: B): (ResolutionProof, B)
    Implicit
    This member is added by an implicit conversion from ResolutionProof to ArrowAssoc[ResolutionProof] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc
    Annotations
    @inline()
  5. final def ==(arg0: Any): Boolean
    Definition Classes
    AnyRef → Any
  6. final def asInstanceOf[T0]: T0
    Definition Classes
    Any
  7. val assertions: HOLClause

    Assertions of the proof.

    Assertions of the proof.

    Attention: this is interpreted as assertions.toNegConjunction --> conclusion.toDisjunction

    These assertions indicate the splitting assumptions this proof depends on.

  8. def auxFormulas: Seq[Seq[Formula]]

    A list of lists containing the auxiliary formulas of the rule.

    A list of lists containing the auxiliary formulas of the rule. The first list constains the auxiliary formulas in the first premise and so on.

    Definition Classes
    SequentProof
  9. def clone(): AnyRef
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  10. def dagLike: DagLikeOps[ResolutionProof]

    Operations that view the sub-proofs as a DAG, which ignore duplicate sub-proofs, see DagProof.DagLikeOps for a list.

    Operations that view the sub-proofs as a DAG, which ignore duplicate sub-proofs, see DagProof.DagLikeOps for a list.

    Definition Classes
    DagProof
  11. def definitions: Map[HOLAtomConst, Expr]

    All definitions introduced by any subproof.

    All definitions introduced by any subproof.

    Exceptions thrown

    Exception if inconsistent definitions are used

  12. def depth: Int

    Depth of the proof, which is the maximum length of a path you can take via immediateSubProofs.

    Depth of the proof, which is the maximum length of a path you can take via immediateSubProofs.

    Definition Classes
    DagProof
  13. def ensuring(cond: (ResolutionProof) ⇒ Boolean, msg: ⇒ Any): ResolutionProof
    Implicit
    This member is added by an implicit conversion from ResolutionProof to Ensuring[ResolutionProof] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  14. def ensuring(cond: (ResolutionProof) ⇒ Boolean): ResolutionProof
    Implicit
    This member is added by an implicit conversion from ResolutionProof to Ensuring[ResolutionProof] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  15. def ensuring(cond: Boolean, msg: ⇒ Any): ResolutionProof
    Implicit
    This member is added by an implicit conversion from ResolutionProof to Ensuring[ResolutionProof] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  16. def ensuring(cond: Boolean): ResolutionProof
    Implicit
    This member is added by an implicit conversion from ResolutionProof to Ensuring[ResolutionProof] performed by method Ensuring in scala.Predef.
    Definition Classes
    Ensuring
  17. final def eq(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  18. def equals(that: Any): Boolean
    Definition Classes
    DagProof → Equals → AnyRef → Any
  19. def finalize(): Unit
    Attributes
    protected[java.lang]
    Definition Classes
    AnyRef
    Annotations
    @throws( classOf[java.lang.Throwable] )
  20. def formatted(fmtstr: String): String
    Implicit
    This member is added by an implicit conversion from ResolutionProof to StringFormat[ResolutionProof] performed by method StringFormat in scala.Predef.
    Definition Classes
    StringFormat
    Annotations
    @inline()
  21. final def getClass(): Class[_]
    Definition Classes
    AnyRef → Any
  22. val hashCode: Int
    Definition Classes
    DagProof
  23. def introducedDefinitions: Map[HOLAtomConst, Expr]

    Definitions introduced by the bottom-most inference rule.

  24. final def isInstanceOf[T0]: Boolean
    Definition Classes
    Any
  25. def isProof: Boolean

    Is this a proof of the empty clause with empty assertions, and consistent definitions?

  26. def longName: String

    The name of this rule (in words).

    The name of this rule (in words).

    Definition Classes
    DagProof
  27. def mainFormulas: Seq[Formula]

    The list of main formulas of the rule.

    The list of main formulas of the rule.

    Definition Classes
    SequentProof
  28. def name: String

    The name of this rule (in symbols).

    The name of this rule (in symbols).

    Definition Classes
    DagProof
  29. final def ne(arg0: AnyRef): Boolean
    Definition Classes
    AnyRef
  30. final def notify(): Unit
    Definition Classes
    AnyRef
  31. final def notifyAll(): Unit
    Definition Classes
    AnyRef
  32. def premises: Seq[Sequent[Formula]]

    The upper sequents of the rule.

    The upper sequents of the rule.

    Definition Classes
    SequentProof
  33. def productIterator: Iterator[Any]
    Definition Classes
    Product
  34. def productPrefix: String
    Definition Classes
    Product
  35. def skolemFunctions: SkolemFunctions
  36. def stepString(subProofLabels: Map[Any, String]): String
    Attributes
    protected
    Definition Classes
    ResolutionProofSequentProofDagProof
  37. def stringifiedConclusion: String
  38. def subProofAt(pos: List[Int]): ResolutionProof

    Returns the subproof at the given position: p.subProofAt(Nil) is p itself; p.subProofAt(i :: is) is the ith subproof of p.subProofAt(is).

    Returns the subproof at the given position: p.subProofAt(Nil) is p itself; p.subProofAt(i :: is) is the ith subproof of p.subProofAt(is).

    Definition Classes
    DagProof
  39. def subProofs: Set[ResolutionProof]

    Set of all (transitive) sub-proofs including this.

    Set of all (transitive) sub-proofs including this.

    Definition Classes
    DagProof
  40. final def synchronized[T0](arg0: ⇒ T0): T0
    Definition Classes
    AnyRef
  41. def toString(): String
    Definition Classes
    DagProof → AnyRef → Any
  42. def treeLike: TreeLikeOps[ResolutionProof]

    Operations that view the sub-proofs as a tree, see DagProof.TreeLikeOps for a list.

    Operations that view the sub-proofs as a tree, see DagProof.TreeLikeOps for a list.

    Definition Classes
    DagProof
  43. final def wait(): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  44. final def wait(arg0: Long, arg1: Int): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  45. final def wait(arg0: Long): Unit
    Definition Classes
    AnyRef
    Annotations
    @throws( ... )
  46. def [B](y: B): (ResolutionProof, B)
    Implicit
    This member is added by an implicit conversion from ResolutionProof to ArrowAssoc[ResolutionProof] performed by method ArrowAssoc in scala.Predef.
    Definition Classes
    ArrowAssoc

Inherited from DagProof[ResolutionProof]

Inherited from Product

Inherited from Equals

Inherited from AnyRef

Inherited from Any

Inherited by implicit conversion any2stringadd from ResolutionProof to any2stringadd[ResolutionProof]

Inherited by implicit conversion StringFormat from ResolutionProof to StringFormat[ResolutionProof]

Inherited by implicit conversion Ensuring from ResolutionProof to Ensuring[ResolutionProof]

Inherited by implicit conversion ArrowAssoc from ResolutionProof to ArrowAssoc[ResolutionProof]

Ungrouped