General Domain Circumscription and its Effective Reductions

A Tractable Logic for Molecular Biology

“…Circumscription is an umbrella term for second-order characterisations of the minimal models of a first-order formula φ along an order. We combine general domain circumscription (GDC) [9,14] and parallel predicate circumscription [15].…”

“…The proof builds upon [9]. Given a model A of C (φ ) and B A, the internal structure of B can be "plugged in" the tuple D, M, V and verifies the left-hand side of the main implication in C (φ ); the righthand side implies that B cannot be strictly smaller than A.…”

“…We use an instanciation of unified circumscription, a new flavor of circumscription which generalises [9]. Previous works on taming circumscription require global syntactic properties of the formulas [9,16,5] and only consider satisfiability or FO-expressivity of circumscribed formulas. [8] uses circumscription for characterising weakest preconditions to reactions.…”

Declarative PTIME queries for relational databases using quantifier elimination

“…In the rst stage, we provide a PTIME (in the size of the input query) compilation process which uses a quanti er elimination algorithm called the DLS algorithm 6]. An extension for xpoint formulas is called the G-DLS algorithm 5,4,7]. The DLS algorithm takes as input a second-order formula and returns a logically equivalent rst-order formula, or terminates with failure, where failure does not mean there is not a reduction, but simply that the algorithm can not nd one.…”

“…The G-DLS algorithm is a generalization of the DLS algorithm and returns logically equivalent xpoint formulas for a wider class of inputs. Both algorithms can be combined into one algorithm which we denote by DLS (see 4,7]). Given the SHQL query, (Q), we pre x it with an existential quanti er and input the formula 9Q: (Q) to DLS .…”