I know that it's easier to prove if a horn-formula is satisfiable. My Question is: Why is it easier with a horn formula rather than a normal CNF?
CNF vs Horn Satisfiability
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Presence or absence of Horn satisfiability can be shown in linear time. Here is a good introduction with some examples. The solution can be found by unit propagation without backtracking.
Pseudocode from a UC Berkeley lecture note:
Satisfiability for general CNF expressions is a classic NP-complete problem. No polynomial time algorithms are known for CNF satisfiability (except if P=NP).