We address the problem of complementing higher-order patterns without repetitions of existential variables. Differently from the first-order case, the complement of a pattern cannot, in general, be described by a pattern, or even by a finite set of patterns. We therefore generalize the simply-typed λ-calculus to include an internal notion of strict function so that we can directly express that a term must depend on a given variable. We show that, in this more expressive calculus, finite sets of patterns without repeated variables are closed under complement and intersection. Our principal application is the transformational approach to negation in higher-order logic programs.
Lassez87 proposed the seminal algorithm for computing relative complements and introduced the now familiar restriction to linear terms. We quote the definition of the @math '' algorithm for the (singleton) complement problem given in @cite_22 which we generalize in Definition . Given a finite signature @math and a linear term @math they define: [ ] The relative complement problem is then solved by composing the above complement operation with term intersection implemented via first-order unification.
@cite_22
2082994045
Suppose the complex matrix M is partitioned into a @math array of blocks; let @math . The generalized Schur complement of A in M is defined to be @math , where @math is the Moore–Penrose inverse of A. The relationship of the ranks of M, A, and @math is determined. A new proof, under certain conditions, of Sylvester’s determinantal formula is given. A quotient formula like one previously proved for the Schur complement is obtained. Finally, several known inequalities for positive semidefinite Hermitian matrices are generalized. Adapting and extending the techniques developed in recent work with Vasy for the study of the Cauchy horizon of cosmological spacetimes, we obtain boundedness, regularity and decay of linear scalar waves on subextremal Reissner-Nordstr "om and (slowly rotating) Kerr spacetimes, without any symmetry assumptions; in particular, we provide simple microlocal and scattering theoretic proofs of analogous results by Franzen. We show polynomial decay of linear waves relative to a Sobolev space of order slightly above @math . This complements the generic @math blow-up result of Luk and Oh. We consider the fundamental problem of solving quadratic systems of equations in @math variables, where @math , @math and @math is unknown. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data @math and @math as soon as the ratio @math between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems in which we only have @math and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size. In this paper we consider particular equational formulae where equality = AC is the congruence induced by a set of associative-commutative axioms. The formulae we are interested in have the form t ? ACt1 ??t ? ACtn and are usually known as complement problems. To solve a complement problem is to find an instance of t which is not an instance of any ti modulo associativity-commutativity. We give a decision procedure based on tree automata which solves these formulae when all the ti1s are linear. We show that this solution also gives a decision of inductive reducibility modulo associativity and commutativity in the linear case and we give several extensions of this approach. Then, we define a new class of tree automata, called conditional tree automata which recognize sets of generalized terms, i.e terms written as multisets, and where the application of a rule depends on the satisfiability of a formula of Presburger's arithmetic. This class of tree automata allows one to solve non-linear complement problems when all occurrences of a non-linear variable occur under the same node (in flattened terms). This solution also provides a procedure to decide inductive reducibility modulo associativity-commutativity in the same case.