Unit propagation (UP) or Boolean Constraint propagation or the one-literal rule (OLR) is a procedure of automated theorem proving that can simplify a set of (usually propositional) clauses.
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Definition
The procedure is based on unit clauses, i.e. clauses that are composed of a single literal. Because each clause needs to be satisfied, we know that this literal must be true. If a set of clauses contains the unit clause
- every clause (other than the unit clause itself) containing
l is removed (the clause is satisfied ifl is); - in every clause that contains
¬ l this literal is deleted (¬ l can not contribute to it being satisfied).
The application of these two rules lead to a new set of clauses that is equivalent to the old one.
For example, the following set of clauses can be simplified by unit propagation because it contains the unit clause
Since
The resulting set of clauses
Unit propagation and resolution
The second rule of unit propagation can be seen as a restricted form of resolution, in which one of the two resolvents must always be a unit clause. As for resolution, unit propagation is a correct inference rule, in that it never produces a new clause that was not entailed by the old ones. The difference between unit propagation and resolution are:
- resolution is a complete refutation procedure while unit propagation is not; in other words, even if a set of clause is contradictory, unit propagation may not generate an inconsistency;
- the two clauses that are resolved cannot in general be removed after the generated clause is added to the set; on the contrary, the non-unit clause involved in a unit propagation can be removed when its simplification is added to the set;
- resolution does not in general include the first rule used in unit propagation.
Resolution calculi that include subsumption can model rule one by subsumption and rule two by a unit resolution step, followed by subsumption.
Unit propagation, applied repeatedly as new unit clauses are generated, is a complete satisfiability algorithm for sets of propositional Horn clauses; it also generates a minimal model for the set if satisfiable: see Horn-satisfiability.
Using a partial model
The unit clauses that are present in a set of clauses or can be derived from it can be stored in form of a partial model (this partial model may also contain other literals, depending on the application). In this case, unit propagation is performed based on the literals of the partial model, and unit clauses are removed if their literal is in the model. In the example above, the unit clause
Complexity
The direct implementation of unit propagation takes time quadratic in the total size of the set to check, which is defined to be the sum of the size of all clauses, where the size of each clause is the number of literals it contains.
Unit propagation can however be done in linear time by storing, for each variable, the list of clauses in which each literal is contained. For example, the set above can be represented by numbering each clause as follows:
and then storing, for each variable, the list of clauses containing the variable or its negation:
This simple data structure can be built in time linear in the size of the set, and allows finding all clauses containing a variable very easily. Unit propagation of a literal can be performed efficiently by scanning only the list of clauses containing the variable of the literal. More precisely, the total running time for doing unit propagation for all unit clauses is linear in the size of the set of clauses.