?. ,

, P(z, y) and ? Q(x, x) with P(x, y) ? P(x, z), P(z , y) and ? Q(x, x ), respectively. Consequently, ordered resolution derives ? P(a, z 1 ), P(z 2 , a) which subsumes any further inference clauses ? P(a, FO-AR replaces P(x, y) ? P(x, z)

, Both examples belong to the Bernays-Schönfinkel fragment, so model evolution (Darwin) and Inst-Gen (iProver) can decide them as well, Hence, saturation of the approximation terminates immediately

, With a shallow approximation of P(x, y) ? P(g(x), z) into S(v) ? P(v, z) and P(x, y) ? S(g(x)), FO-AR (SPASS-AR) terminates after deriving ? S(g(a)) and S(x) ? S(g(x)). Again, model evolution (Darwin) and Inst-Gen (iProver) can also solve this example, superposition derives all clauses of the form ? P(g

, P( f ( f (x))) ? P(x)

, is already saturated under superposition. For FO-AR, the clause P(x) ? P( f ( f (x))) is replaced by S(x) ? P( f (x)) and P(x) ? S( f (x)). Then ordered resolution terminates after inferring S(a) ? and S( f (x)) ? P(x)

, The Inst-Gen and model evolution calculi, however, fail. In either, a satisfying model is represented by a finite set of literals, i.e, a model of the propositional approximation for Inst-Gen and the trail of literals in case of model evolution. Therefore, there necessarily exists a literal P( f n (x)) or ¬P( f n (x)) with a maximal n in these models

, Lastly consider an example of the form f (x) ? x ?; f ( f (x)) ? x ?; . . . ; f n (x) ? x ? which is trivially satisfiable, e.g., saturated by superposition, but any model has at least n + 1 domain elements. Therefore, adding these clauses to any satisfiable clause set containing f forces calculi that explicitly consider finite models to consider at least n + 1 elements. The performance of final model finders [16] typically degrades in the number of different domain elements to be considered. Combining each of these examples into one problem is then solvable by neither superposition, Inst-Gen, or model evolution and not practically solvable with increasing n via testing finite models. For example

. P(-f,

, P( f ( f (x)), y) ? P

, P(x, y) ? P( f ( f (x)), y)

). ?-x-?; and .. .. ,

, Vampire, iProver, and Darwin for more than one hour each without success. Only SPASS-AR solved it in less than one second. For iProver we added an artificial positive equation b ? c. For otherwise, iProver throws away all inequations while preprocessing. This is a satisfiability preserving operation, however, the afterwards found (finite) models are not models of the above clause set due to the collapsing of ground terms

). and P. (x-,-y,-z)-?-p,

, P(x, y , z) ? P(x, f 3 (y), z)

). and P. (x,-y,-z-)-?-p,

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