S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Type 2 Complexity Classes Preliminary Report Chung Chih Li # James S. Royer # 15 March 2001 Abstract There are now a number of things called higher type complexity classes. The most promenade of these is the class of basic feasible functionals [CU93, CK90], a fairly conservative higher type analogue the (type 1) polynomial time computable functions. There is however currently no satisfactory general notion of what a higher type complexity class should be. In this paper we propose one such notion for type 2 functionals and begin an investigation of ....

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop, (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....of Buss [Bus86] defined an equivalent class of type 2 functionals, as well as analogous functionals of type 3 and above. This class of functionals, which they called the (type 2) basic feasible functionals, was developed by Cook and co workers in a series of papers; see, for example, [Coo91, CK90, KC91, KC96]. Kapron and Cook s 1991 paper [KC91, KC96] is of particular importance here, as that paper introduced the first natural machine characterization of the type 2 basic feasible functionals, stated as Theorem 6 below. Definition 4 (Kapron and Cook [KC91, KC96] A second order polynomial over type 1 ....

....shred operations of footnote 14. These functionals in some respects resemble the effective continuous functionals. How close is this resemblance Can one obtain a language characterization of this class along the lines of Cook and Kapron s characterizations of the basic feasible functionals [CK90] or of Plotkin s PCF [Plo77] As noted in footnote 14, there is an even more general class of polynomial time shred operations on PR N. Is there a most general polynomial time shred operation and, if so, can one prove some analog of the MyhillShepherdson Theorem for this class I am curious ....

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop, (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....provide tools that may be handy along the way. 21 April 2000 Separating Notions of Higher Type Polynomial Time 11 APPENDICES A. Bounded Typed Loop Programs Our o#cial definition of the basic feasible functionals is through our version of Cook and Kapron s BTLP (Bounded Typed Loop Programs) [CK90]. In BTLP all variables come equipped with a type. To make the type of a variable explicit, we may decorate the variable with the type as a superscript or, in declarations, add : the name of the type after the variable. The allowable types in BTLP are the simple types over N. The grammar of ....

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....overview of this paper. 2. A Summary of Part I Part I centered around three characterizations of BFF 2 , the type 2 basic feasible functionals. The first of these three characterizations is through Cook and Kapron s type 2 bounded typed loop programs (abbreviated, BTLP 2 ) programming formalism [CK89, CK90]. This is a simply typed, imperative programming formalism with a loop construct based on Cobham s limited recursion on notation [Cob65] BTLP 2 is representative of several restricted programming languages that formalize a type 2 analogue of PF through a careful lift of a programming formalism ....

....of PF, variables of simple types, and a type 2 recursor R that corresponds to Cobham s limited recursion on notation. The underlying semantic domain of PV is Full. The class of PV computable functionals (at all simple types) was later named the basic feasible functionals by Cook and Kapron [CK89, CK90]. In that work they established several programming formalism characterizations of the BFFs, including the result that the (general) BTLP computable functionals are exactly the BFFs. They also showed that the BFFs satisfy a Ritchie Cobham property at all simple types (see footnote 1 in Part I) ....

[Article contains additional citation context not shown here]

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....overview of this paper. 2. A Summary of Part I Part I centered around three characterizations of BFF 2 , the type 2 basic feasible functionals. The first of these three characterizations is through Cook and Kapron s type 2 bounded typed loop programs (abbreviated, BTLP 2 ) programming formalism [CK89, CK90]. This is a simply typed, imperative programming formalism with a loop construct based on Cobham s limited recursion on notation [Cob65] BTLP 2 is representative of several restricted programming languages that formalize a type 2 analogue of PF through a careful lift of a programming formalism ....

....of PF, variables of simple types, and a type 2 recursor R that corresponds to Cobham s limited recursion on notation. The underlying semantic domain of PV is Full. The class of PV computable functionals (at all simple types) was later named the basic feasible functionals by Cook and Kapron [CK89, CK90]. In that work they established several programming formalism characterizations of the BFFs, including the result that the (general) BTLP computable functionals are exactly the BFFs. They also showed that the BFFs satisfy a Ritchie Cobham property at all simple types (see footnote 1 in Part I) ....

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Proceedings of the 30nd Annual IEEE Symposium on the Foundations of Computer Science, 1989, pp. 154--159.

....Preliminary Version 4 2. A Summary of Part I Part I centered around three characterizations of BFF 2 , the type 2 basic feasible functionals. The first of these three characterizations was through Cook and Kapron s type 2 bounded typed loop programs (abbreviated, BTLP 2 ) programming formalism [CK89, CK90]. This is a simply typed, imperative programming formalism with a loop construct based on Cobham s limited recursion on notation [Cob65] BTLP 2 is representative several restricted programming languages that formalize a type 2 analogue of PF though a careful lift of a programming formalism ....

....PF, variables of simple types, and a type 2 recursor R that corresponds to Cobham s limited recursion on notation. The underlying semantic domain of PV # is Full. The class of PV # computable functionals (at all simple types) was later named the basic feasible functionals by Cook and Kapron [CK89, CK90]. In that work they established several programming formalism characterizations of the BFFs, including the result that the (general) BTLP computable functionals are exactly the BFFs. They also showed that the BFFs satisfy a Ritchie Cobham property at all simple types (see footnote 1 in Part I) ....

[Article contains additional citation context not shown here]

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....Preliminary Version 4 2. A Summary of Part I Part I centered around three characterizations of BFF 2 , the type 2 basic feasible functionals. The first of these three characterizations was through Cook and Kapron s type 2 bounded typed loop programs (abbreviated, BTLP 2 ) programming formalism [CK89, CK90]. This is a simply typed, imperative programming formalism with a loop construct based on Cobham s limited recursion on notation [Cob65] BTLP 2 is representative several restricted programming languages that formalize a type 2 analogue of PF though a careful lift of a programming formalism ....

....PF, variables of simple types, and a type 2 recursor R that corresponds to Cobham s limited recursion on notation. The underlying semantic domain of PV # is Full. The class of PV # computable functionals (at all simple types) was later named the basic feasible functionals by Cook and Kapron [CK89, CK90]. In that work they established several programming formalism characterizations of the BFFs, including the result that the (general) BTLP computable functionals are exactly the BFFs. They also showed that the BFFs satisfy a Ritchie Cobham property at all simple types (see footnote 1 in Part I) ....

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Proceedings of the 30nd Annual IEEE Symposium on the Foundations of Computer Science, 1989, pp. 154--159.

....machine models. 25 September 2000 On Characterizations of the BFFs, Part I Draft 5 resulting class of functionals nicely satisfied their goals [CU89, CU93] This class of functionals (at all simple types) was named the basic feasible functionals (abbreviated, BFF) in Cook and Kapron s work [CK89, CK90], where it was shown that Mehlhorn s L( exactly corresponds to the type 2 BFFs. The basic in basic feasible functionals is meant to suggest that any natural higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this ....

....natural higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this analogue. The robustness of the basic feasible functionals The issue of the naturalness of BFFs as higher type analogues of PF was taken up by Cook and Kapron [CK89, CK90]. They establish several functionalgebra characterizations of the BFFs and showed that the BFFs satisfy a Ritchie Cobham property 2 at all simple types. The naturalness question was also a central issue in Cook s [Coo91] where he stated two serious reservations about the type 2 BFFs. Notation: ....

[Article contains additional citation context not shown here]

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

....machine models. 25 September 2000 On Characterizations of the BFFs, Part I Draft 5 resulting class of functionals nicely satisfied their goals [CU89, CU93] This class of functionals (at all simple types) was named the basic feasible functionals (abbreviated, BFF) in Cook and Kapron s work [CK89, CK90], where it was shown that Mehlhorn s L( exactly corresponds to the type 2 BFFs. The basic in basic feasible functionals is meant to suggest that any natural higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this ....

....natural higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this analogue. The robustness of the basic feasible functionals The issue of the naturalness of BFFs as higher type analogues of PF was taken up by Cook and Kapron [CK89, CK90]. They establish several functionalgebra characterizations of the BFFs and showed that the BFFs satisfy a Ritchie Cobham property 2 at all simple types. The naturalness question was also a central issue in Cook s [Coo91] where he stated two serious reservations about the type 2 BFFs. Notation: ....

[Article contains additional citation context not shown here]

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Proceedings of the 30nd Annual IEEE Symposium on the Foundations of Computer Science, 1989, pp. 154--159.

.... to Cobham s limited recursion on notation (see Definition 1 below) They showed that the resulting class of functionals nicely satisfied their goals [CU89, CU93] This class of functionals (at all simple types) was named the basic feasible functionals (abbreviated, BFF) in Cook and Kapron s work [CK89, CK90], where it was shown that Mehlhorn s L( exactly corresponds to the type 2 BFFs. The basic in basic feasible functionals is meant to suggest that any natural higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this ....

....higher type analogue of PF should include the BFFs, but it leaves open the possibility that the BFFs may be too small to be this analogue. The robustness of the basic feasible functionals The issue of the naturalness of BFFs as higher type analogues of PF was taken up by Cook and Kapron in [CK89, CK90]. They establish several functionalgebra characterizations of the BFFs and showed that the BFFs satisfy a Ritchie Cobham property 2 at all simple types. The naturalness question was also a central issue in Cook s [Coo91] where he stated two serious reservations about the type 2 BFFs. Notation: ....

[Article contains additional citation context not shown here]

S. Cook and B. Kapron, Characterizations of the basic feasible functions of finite type, Feasible Mathematics: A Mathematical Sciences Institute Workshop, (S. Buss and P. Scott, eds.), Birkhauser, 1990, pp. 71--95.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC