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:
- Input rules for formulas of the end-sequent
- AndL, AllR, etc. for the clausification. A combination of DefIntro and Taut is used to introduce definitions for subformulas.
- Resolution, Paramod, Subst. Factor, Refl, Taut for the refutation of the resulting clauses. Clauses are split by successively applying AvatarSplit, reducing the clause to an empty clause. The clause components can then be used using AvatarComponent.
- Each empty clause of a splitting branch is followed by an AvatarContradiction inference, moving the assertion back to the clause.
- A Resolution refutation of the AvatarContradiction clauses.
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
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Abstract Value Members
- 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
- abstract def canEqual(that: Any): Boolean
- Definition Classes
- Equals
- abstract def conclusion: Sequent[Formula]
The conclusion of the rule.
The conclusion of the rule.
- Definition Classes
- SequentProof
- abstract def immediateSubProofs: Seq[ResolutionProof]
The immediate subproofs of this rule.
The immediate subproofs of this rule.
- Definition Classes
- DagProof
- 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
- 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
- abstract def productArity: Int
- Definition Classes
- Product
- abstract def productElement(n: Int): Any
- Definition Classes
- Product
Concrete Value Members
- final def !=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
- final def ##(): Int
- Definition Classes
- AnyRef → Any
- def +(other: String): String
- Implicit
- This member is added by an implicit conversion from ResolutionProof toany2stringadd[ResolutionProof] performed by method any2stringadd in scala.Predef.
- Definition Classes
- any2stringadd
- def ->[B](y: B): (ResolutionProof, B)
- Implicit
- This member is added by an implicit conversion from ResolutionProof toArrowAssoc[ResolutionProof] performed by method ArrowAssoc in scala.Predef.
- Definition Classes
- ArrowAssoc
- Annotations
- @inline()
- final def ==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
- final def asInstanceOf[T0]: T0
- Definition Classes
- Any
- 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.
- 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
- def clone(): AnyRef
- Attributes
- protected[lang]
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.CloneNotSupportedException]) @native() @IntrinsicCandidate()
- 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
- def definitions: Map[HOLAtomConst, Expr]
All definitions introduced by any subproof.
All definitions introduced by any subproof.
throws java.lang.Exception if inconsistent definitions are used
- 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
- def ensuring(cond: (ResolutionProof) => Boolean, msg: => Any): ResolutionProof
- Implicit
- This member is added by an implicit conversion from ResolutionProof toEnsuring[ResolutionProof] performed by method Ensuring in scala.Predef.
- Definition Classes
- Ensuring
- def ensuring(cond: (ResolutionProof) => Boolean): ResolutionProof
- Implicit
- This member is added by an implicit conversion from ResolutionProof toEnsuring[ResolutionProof] performed by method Ensuring in scala.Predef.
- Definition Classes
- Ensuring
- def ensuring(cond: Boolean, msg: => Any): ResolutionProof
- Implicit
- This member is added by an implicit conversion from ResolutionProof toEnsuring[ResolutionProof] performed by method Ensuring in scala.Predef.
- Definition Classes
- Ensuring
- def ensuring(cond: Boolean): ResolutionProof
- Implicit
- This member is added by an implicit conversion from ResolutionProof toEnsuring[ResolutionProof] performed by method Ensuring in scala.Predef.
- Definition Classes
- Ensuring
- final def eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- def equals(that: Any): Boolean
- Definition Classes
- DagProof → Equals → AnyRef → Any
- def formatted(fmtstr: String): String
- Implicit
- This member is added by an implicit conversion from ResolutionProof toStringFormat[ResolutionProof] performed by method StringFormat in scala.Predef.
- Definition Classes
- StringFormat
- Annotations
- @inline()
- final def getClass(): Class[_ <: AnyRef]
- Definition Classes
- AnyRef → Any
- Annotations
- @native() @IntrinsicCandidate()
- val hashCode: Int
- Definition Classes
- DagProof
- def introducedDefinitions: Map[HOLAtomConst, Expr]
Definitions introduced by the bottom-most inference rule.
- final def isInstanceOf[T0]: Boolean
- Definition Classes
- Any
- def isProof: Boolean
Is this a proof of the empty clause with empty assertions, and consistent definitions?
- def longName: String
The name of this rule (in words).
The name of this rule (in words).
- Definition Classes
- DagProof
- def mainFormulas: Seq[Formula]
The list of main formulas of the rule.
The list of main formulas of the rule.
- Definition Classes
- SequentProof
- def name: String
The name of this rule (in symbols).
The name of this rule (in symbols).
- Definition Classes
- DagProof
- final def ne(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
- final def notify(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @IntrinsicCandidate()
- final def notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @IntrinsicCandidate()
- def premises: Seq[Sequent[Formula]]
The upper sequents of the rule.
The upper sequents of the rule.
- Definition Classes
- SequentProof
- def productElementName(n: Int): String
- Definition Classes
- Product
- def productElementNames: Iterator[String]
- Definition Classes
- Product
- def productIterator: Iterator[Any]
- Definition Classes
- Product
- def productPrefix: String
- Definition Classes
- Product
- def skolemFunctions(implicit ctx: Context): SkolemFunctions
- def skolemSymbols: Set[Const]
- def stepString(subProofLabels: Map[Any, String]): String
- Attributes
- protected
- Definition Classes
- ResolutionProof → SequentProof → DagProof
- def stringifiedConclusion(implicit sig: BabelSignature): String
- 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
- def subProofs: Set[ResolutionProof]
Set of all (transitive) sub-proofs including this.
Set of all (transitive) sub-proofs including this.
- Definition Classes
- DagProof
- final def synchronized[T0](arg0: => T0): T0
- Definition Classes
- AnyRef
- def toString(): String
- Definition Classes
- DagProof → AnyRef → Any
- 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
- final def wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException])
- final def wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException]) @native()
- final def wait(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException])
Deprecated Value Members
- def finalize(): Unit
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- protected[lang]
- Definition Classes
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- @throws(classOf[java.lang.Throwable]) @Deprecated @deprecated
- Deprecated
(Since version ) see corresponding Javadoc for more information.
- def →[B](y: B): (ResolutionProof, B)
- Implicit
- This member is added by an implicit conversion from ResolutionProof toArrowAssoc[ResolutionProof] performed by method ArrowAssoc in scala.Predef.
- Definition Classes
- ArrowAssoc
- Annotations
- @deprecated
- Deprecated
(Since version 2.13.0) Use
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This is the API documentation for GAPT.
The main package is gapt.