Horn (Schwarzbach)

In the context of static analysis and program verification, a Horn clause, as used by Schwarzbach and others, refers to a logical clause of the form:

A ← B1 ∧ B2 ∧ ... ∧ Bn

Where:

• A is an atomic formula (a predicate applied to terms), often called the head of the clause.
• B1, B2, ..., Bn are atomic formulas, forming the body of the clause.
• ← represents logical implication (read as "if"). The clause states that if all B1, B2, ..., Bn are true, then A must also be true.
• ∧ represents logical conjunction ("and").

A Horn clause can also be written as:

A ∨ ¬B1 ∨ ¬B2 ∨ ... ∨ ¬Bn

where ∨ represents logical disjunction ("or") and ¬ represents logical negation ("not").

Horn clauses are significant in program analysis because they provide a relatively simple and computationally tractable fragment of first-order logic. Many program analysis problems, such as dataflow analysis, interprocedural analysis, and security analysis, can be effectively encoded as systems of Horn clauses. Solutions to these systems of clauses then provide information about the program's behavior.

In Schwarzbach's work, Horn clauses may be used within frameworks like constraint-based analysis, where program properties are expressed as constraints in the form of Horn clauses. Solving these constraints reveals information about potential program behaviors, such as possible values of variables, reachable program points, or security vulnerabilities. The use of Horn clauses enables efficient static analysis techniques to be applied, automatically inferring program properties without requiring program execution. Specialized solvers, like those based on symbolic execution or abstract interpretation, are often employed to find solutions to the Horn clause systems generated during analysis.