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
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
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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
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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
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final
def
!=(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
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final
def
##(): Int
- Definition Classes
- AnyRef → Any
-
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
-
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
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- @inline()
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final
def
==(arg0: Any): Boolean
- Definition Classes
- AnyRef → Any
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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.
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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
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def
clone(): AnyRef
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
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
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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
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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 to Ensuring[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 to Ensuring[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 to Ensuring[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 to Ensuring[ResolutionProof] performed by method Ensuring in scala.Predef.
- Definition Classes
- Ensuring
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final
def
eq(arg0: AnyRef): Boolean
- Definition Classes
- AnyRef
-
def
equals(that: Any): Boolean
- Definition Classes
- DagProof → Equals → AnyRef → Any
-
def
finalize(): Unit
- Attributes
- protected[java.lang]
- Definition Classes
- AnyRef
- Annotations
- @throws( classOf[java.lang.Throwable] )
-
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()
-
final
def
getClass(): Class[_]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
-
val
hashCode: Int
- Definition Classes
- DagProof
-
def
introducedDefinitions: Map[HOLAtomConst, Expr]
Definitions introduced by the bottom-most inference rule.
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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?
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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()
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final
def
notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
-
def
premises: Seq[Sequent[Formula]]
The upper sequents of the rule.
The upper sequents of the rule.
- Definition Classes
- SequentProof
-
def
productIterator: Iterator[Any]
- Definition Classes
- Product
-
def
productPrefix: String
- Definition Classes
- Product
- def skolemFunctions: SkolemFunctions
-
def
stepString(subProofLabels: Map[Any, String]): String
- Attributes
- protected
- Definition Classes
- ResolutionProof → SequentProof → DagProof
- def stringifiedConclusion(implicit sig: BabelSignature): String
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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(): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws( ... )
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final
def
wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
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- @throws( ... )
-
final
def
wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @native() @throws( ... )
-
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
This is the API documentation for GAPT.
The main package is gapt.