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Characterizations of the Basic Feasible Functionals of Finite Type (Extended Abstract)
 Feasible Mathematics: A Mathematical Sciences Institute Workshop
, 1990
"... Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finit ..."
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Stephen A. Cook and Bruce M. Kapron Department of Computer Science University of Toronto Toronto, Canada M5S 1A4 1 Introduction Functionals are functions which take natural numbers and other functionals as arguments and return natural numbers as values. The class of "feasible" functionals of finite type was introduced in [6] via the typed lambda calculus, and used to interpret certain formal systems of arithmetic: systems capturing the notion of "feasibly constructive proof" (we equate feasibility with polynomial time computability) . Here we name the functionals of [6] the basic feasible functionals and justify the designation by presenting results which include two programming style characterizations of the class. We also give examples of both feasible and infeasible functionals, and argue that the notion plays a natural role in complexity theory. Type 2 functionals take numbers and ordinary numerical functions as arguments. When these argument functions are 01 valued (i.e. sets) ...
A Logical Calculus for Polynomialtime Realizability
 Journal of Methods of Logic in Computer Science
, 1991
"... A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduct ..."
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A logical calculus, not unlike Gentzen's sequent calculus for intuitionist logic, is described which is sound for polynomialtime realizability as defined by Crossley and Remmel. The sequent calculus admits cut elimination, thus giving a decision procedure for the propositional fragment. 0 Introduction In [4], a restricted notion of realizability is introduced, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized by realizers that give a measure of the resources required to establish the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all free variables universally quantified simultaneously. (There is a difference between the quantifiers 8h¸; ji and 8¸8j.) And third, the realizers code polynomialtime ("ptime") functions, rather than arbitrary recurs...
Graded Multicategories of Polynomialtime Realizers (Extended Abstract)
 Department of Mathematics Department of Mathematics and Computer Science McGill University John Abbott College Monash University 805 Sherbrooke St
, 1989
"... Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered ..."
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Preliminary Version Abstract We present a logical calculus which imposes a grading on a sequentstyle calculus to account for the runtime of the programmes represented by the sequents. This system is sound for a notion of polynomialtime realizability. An extension of the grading is also considered, giving a notion of "dependant grades", which is also sound. Furthermore, we define a notion of closed graded multicategory, and show how the structure of polynomialtime realizers has that structure. 0 Introduction In [4], a restricted notion of realizability is defined, a special case of which is polynomialtime realizability: this is like Kleene's original realizability, save for three features. First, closed atomic formulae are realized only by realizers that express a reason for the "truth" (or provability) of the formula, unlike Kleene's system which only reflects the fact that the formula is provable. Second, open formulae are treated as the corresponding closed formulae with all fre...