##  [Clause Width](/clause-width-0) 

 Definition

The maximum number of literals appearing in any single clause of a clause set; often called clause size in combinatorial analyses.

 

 

 

 

 

 





## Principle

Principle

Use the largest clause length as a bound on local disjunction complexity; resolution and combinatorial bounds frequently scale with width.

 

 

 

 

 





## Demonstration

Demonstration

In CNF { (x1 ∨ x2), (¬x1 ∨ x3 ∨ x4), (x2) } the clause widths are 2, 3, and 1, so the Clause Width is 3.

 

 

 

 

## Misapplication

Misapplication

Replacing Clause Width with average clause size or median width mischaracterizes worst-case combinatorial behavior used in many proofs.

 

 

 

 

 





## Consequence

Consequence

A small fixed Clause Width can yield stronger complexity guarantees (for example, bounded-width resolution lower bounds) and can permit specialized algorithms.

 

 

 

 

## Reversal

Reversal

Considering minimum clause width instead focuses on clauses that are easiest to satisfy but ignores the clauses that determine worst-case search branching.

 

 

 

 

 





## Boundary

Boundary

Defined for clausal (CNF) representations; it does not directly transfer to non-clausal formulas without clause conversion and excludes measures of literal multiplicity inside clauses.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension with Literal Count and Clause Density: two instances with the same Clause Width may behave very differently if one has many wide clauses repeated while the other has only a single wide clause.

 

 

 

 

 





## Synthesis

Synthesis

Clause Width captures the single-clause worst-case disjunction size within a clause set; it is a local structural bound used to control combinatorial and proof complexity analyses.