In this paper, we consider processors which provide an idle instruction to the user for powering down processor units which are not required during portions of program execution. We describe algorithms which can be implemented in an energy-aware compiler to make efficient use of such an instruction. These algorithms are based on program static analysis and a combinatorial optimization formulation of the problem. We assume as input an assembly language program of the processor in question. The problem is to insert the idle instruction at different places in the assembly language program such that energy saving is maximized and the execution time of the resulting program is not increased beyond a user-specified value.

Let Q be a finite set of generalized quantifiers. By L/sup k/(Q) we denote the k-variable fragment of FO(Q), first order logic extended with Q. We show that for each k, there is a PFP(Q)-definable linear pre-order whose equivalence classes in any finite structure 21 are the L/sup k/(Q)-types in 21. For some special classes of generalized quantifiers Q, we show that such an ordering of L/sup k/(Q)-types is already definable in IFP(Q). As applications of the above results, we prove some generalizations of the Abiteboul-Vianu theorem. For instance, we show that for any finite set Q of modular counting quantifiers, P=PSPACE if, and only if, IFP(Q)=PFP(Q) over finite structures. On the other hand, we show that an ordering of L/sup k/(Q)-types is not always definable in IFP(Q). Indeed, we construct a single, polynomial time computable quantifier P such that the equivalence relation /spl equiv//sup k,P/, and hence ordering on L/sup k/(P)-types, is not definable in IFP(P).

We give examples of classes of rigid structures which are of unbounded rigidity but least fixed point (Partial fixed point) logic can express all Boolean PTIME (PSPACE) queries on these classes. This shows that definability of linear order in FO+LEP although sufficient for it to capture Boolean PTIME queries, is not necessary even on the classes of rigid structures. The situation however appears very different for nonzero-ary queries. Next, we turn to the study of fixed point logics on arbitrary classes of structures. We completely characterize the recursively enumerable classes of finite structures on which PFP captures all PSPACE queries of arbitrary arities. We also state in some alternative forms several natural necessary and some sufficient conditions for PFP to capture PSPACE queries on classes of finite structures. The conditions similar to the ones proposed above work for LFP and PTIME also in some special cases but to prove the same necessary conditions in general for LFP to capture PTIME seems harder and remains open.

We consider functionals of type 2 as transformers between functions of type 1. An intuitively feasible functional must preserve the complexity of the input function in some broad sense. We show that the well quasi-order functional, which has been proposed by S.A. Cook (1990) as being intuitively feasible, fails to preserve the class of Kalmar elementary functions. For the basic feasible functionals (BFF), we show that there are arbitrarily large complexity classes of type 1 functions, under the classical definition of a complexity class, which contain polynomial-time functions and are closed under composition but are not preserved by the BFF. However, for a more natural definition of a complexity class of type 1 functions, BFF is shown to preserve all such complexity classes. BFF is the largest known class with this property. We prove BFF to be the largest class of type 2 functionals which satisfies Cook's conditions and the Ritchie-Cobham property, and preserves all classes of type 1 computable functions that contain polynomial-time functions and are closed under composition and limited recursion on notation. These results give some evidence that basic feasible functionals may be the right notion of type 2 feasibility.>

The author shows a class of type-two feasible functionals, C/sub 2/, that satisfies Cook's conditions, (1990) and cannot be expressed as the lambda closure of type-one poly-time functions and any recursively enumerable set of type-two feasible functionals. Further, no class of total type-two functionals containing this class is representable as the lambda closure of a recursively enumerable set of type-two total computable functionals and type-one poly-time functions. The definition of C/sub 2/ provides a clear computational procedure for functionals of C/sub 2/. Using functionals of class C/sub 2/ a more general notion of polynomial-time reducibility between two arbitrary type-one functions can be introduced.>