As a corollary ( i . e ., its contrapositive ), the compactness theorem says that every unsatisfiable first-order theory has a finite unsatisfiable subset.
42.
A sufficient condition of tractability is that a non-uniform problem is tractable if the set of its unsatisfiable instances can be expressed by a Boolean Datalog query.
43.
P ( x ) \ } is unsatisfiable, but a closed tableau is never obtained if one unwisely keeps applying the rule for universal quantifiers to \ forall x.
44.
I am not spending more time and money on certifying public record and disclosing private facts ascertained in judicial and extrajudicial discovery in the service of whimsical and unsatisfiable editorial agenda.
45.
Both conditions of connectedness lead to a complete first-order calculus : if a set of clauses is unsatisfiable, it has a closed connected ( strongly or weakly ) tableau.
46.
The resulting inference rule is refutation-complete, in that a set of clauses is unsatisfiable if and only if there exists a derivation of the empty clause using resolution alone.
47.
An unsatisfiable core is called a " minimal unsatisfiable core ", if every proper subset ( allowing removal of any arbitrary clause or clauses ) of it is satisfiable.
48.
An unsatisfiable core is called a " minimal unsatisfiable core ", if every proper subset ( allowing removal of any arbitrary clause or clauses ) of it is satisfiable.
49.
The above two rules for universal and existential quantifiers are correct, and so are the propositional rules : if a set of formulae generates a closed tableau, this set is unsatisfiable.
50.
As a result, completeness does not automatically imply the existence of a feasible policy of application of rules that always leads to a closed tableau for every given unsatisfiable set of formulae.
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