scala.tools.nsc.transform.patmat.Solving
CNF
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package Solving
Conjunctive normal form (of a Boolean formula).
Conjunctive normal form (of a Boolean formula). A formula in this form is amenable to a SAT solver (i.e., solver that decides satisfiability of a formula).
Plaisted transformation: used for conversion of a propositional formula into conjunctive normal form (CNF) (input format for SAT solver).
Plaisted transformation: used for conversion of a propositional formula into conjunctive normal form (CNF) (input format for SAT solver). A simple conversion into CNF via Shannon expansion would also be possible but it's worst-case complexity is exponential (in the number of variables) and thus even simple problems could become untractable. The Plaisted transformation results in an _equisatisfiable_ CNF-formula (it generates auxiliary variables) but runs with linear complexity. The common known Tseitin transformation uses bi-implication, whereas the Plaisted transformation uses implication only, thus the resulting CNF formula has (on average) only half of the clauses of a Tseitin transformation. The Plaisted transformation uses the polarities of sub-expressions to figure out which part of the bi-implication can be omitted. However, if all sub-expressions have positive polarity (e.g., after transformation into negation normal form) then the conversion is rather simple and the pseudo-normalization via NNF increases chances only one side of the bi-implication is needed.
Test two objects for inequality.
Test two objects for inequality.
true if !(this == that), false otherwise.
Equivalent to x.hashCode except for boxed numeric types and null.
Equivalent to x.hashCode except for boxed numeric types and null.
For numerics, it returns a hash value which is consistent
with value equality: if two value type instances compare
as true, then ## will produce the same hash value for each
of them.
For null returns a hashcode where null.hashCode throws a
NullPointerException.
a hash value consistent with ==
The expression x == that is equivalent to if (x eq null) that eq null else x.equals(that).
The expression x == that is equivalent to if (x eq null) that eq null else x.equals(that).
true if the receiver object is equivalent to the argument; false otherwise.
Cast the receiver object to be of type T0.
Cast the receiver object to be of type T0.
Note that the success of a cast at runtime is modulo Scala's erasure semantics.
Therefore the expression 1.asInstanceOf[String] will throw a ClassCastException at
runtime, while the expression List(1).asInstanceOf[List[String]] will not.
In the latter example, because the type argument is erased as part of compilation it is
not possible to check whether the contents of the list are of the requested type.
the receiver object.
ClassCastException if the receiver object is not an instance of the erasure of type T0.
Create a copy of the receiver object.
Create a copy of the receiver object.
The default implementation of the clone method is platform dependent.
a copy of the receiver object.
not specified by SLS as a member of AnyRef
Tests whether the argument (that) is a reference to the receiver object (this).
Tests whether the argument (that) is a reference to the receiver object (this).
The eq method implements an equivalence relation on
non-null instances of AnyRef, and has three additional properties:
x and y of type AnyRef, multiple invocations of
x.eq(y) consistently returns true or consistently returns false.x of type AnyRef, x.eq(null) and null.eq(x) returns false.null.eq(null) returns true. When overriding the equals or hashCode methods, it is important to ensure that their behavior is
consistent with reference equality. Therefore, if two objects are references to each other (o1 eq o2), they
should be equal to each other (o1 == o2) and they should hash to the same value (o1.hashCode == o2.hashCode).
true if the argument is a reference to the receiver object; false otherwise.
The equality method for reference types.
Called by the garbage collector on the receiver object when there are no more references to the object.
Called by the garbage collector on the receiver object when there are no more references to the object.
The details of when and if the finalize method is invoked, as
well as the interaction between finalize and non-local returns
and exceptions, are all platform dependent.
not specified by SLS as a member of AnyRef
A representation that corresponds to the dynamic class of the receiver object.
A representation that corresponds to the dynamic class of the receiver object.
The nature of the representation is platform dependent.
a representation that corresponds to the dynamic class of the receiver object.
not specified by SLS as a member of AnyRef
The hashCode method for reference types.
Test whether the dynamic type of the receiver object is T0.
Test whether the dynamic type of the receiver object is T0.
Note that the result of the test is modulo Scala's erasure semantics.
Therefore the expression 1.isInstanceOf[String] will return false, while the
expression List(1).isInstanceOf[List[String]] will return true.
In the latter example, because the type argument is erased as part of compilation it is
not possible to check whether the contents of the list are of the specified type.
true if the receiver object is an instance of erasure of type T0; false otherwise.
Equivalent to !(this eq that).
Equivalent to !(this eq that).
true if the argument is not a reference to the receiver object; false otherwise.
Wakes up a single thread that is waiting on the receiver object's monitor.
Wakes up a single thread that is waiting on the receiver object's monitor.
not specified by SLS as a member of AnyRef
Wakes up all threads that are waiting on the receiver object's monitor.
Wakes up all threads that are waiting on the receiver object's monitor.
not specified by SLS as a member of AnyRef
Simplifies propositional formula according to the following rules: - eliminate double negation (avoids unnecessary Tseitin variables) - flatten trees of same connectives (avoids unnecessary Tseitin variables) - removes constants and connectives that are in fact constant because of their operands - eliminates duplicate operands - convert formula into NNF: all sub-expressions have a positive polarity which makes them amenable for the subsequent Plaisted transformation and increases chances to figure out that the formula is already in CNF
Simplifies propositional formula according to the following rules: - eliminate double negation (avoids unnecessary Tseitin variables) - flatten trees of same connectives (avoids unnecessary Tseitin variables) - removes constants and connectives that are in fact constant because of their operands - eliminates duplicate operands - convert formula into NNF: all sub-expressions have a positive polarity which makes them amenable for the subsequent Plaisted transformation and increases chances to figure out that the formula is already in CNF
Complexity: DFS over formula tree
See http://www.decision-procedures.org/slides/propositional_logic-2x3.pdf
Creates a String representation of this object.
Creates a String representation of this object. The default representation is platform dependent. On the java platform it is the concatenation of the class name, "@", and the object's hashcode in hexadecimal.
a String representation of the object.