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**1 - 4**of**4**### Time-complexity semantics for feasible affine recursions (extended abstract)

, 2007

"... Abstract. The authors ’ ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs characterize the type-level ≤ 2 basic feasible functions (ATR-types are confined to levels 0, 1, and 2). A limitation of the original version of AT ..."

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Abstract. The authors ’ ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs characterize the type-level ≤ 2 basic feasible functions (ATR-types are confined to levels 0, 1, and 2). A limitation of the original version of ATR is that the only directly expressible recursions are tail-recursions. Here we extend ATR so that a broad range of affine recursions are directly expressible. In particular, the revised ATR can fairly naturally express the classic insertion- and selection-sort algorithms, thus overcoming a sticking point of most prior implicit-complexity-based formalisms. The paper’s main work is in extending and simplifying the original time-complexity semantics for ATR to develop a set of tools for extracting and solving the higher-type recurrences arising from feasible affine recursions. 1. Two algorithms in search of a type-system As Hofmann [9] has noted, a problem with implicit characterizations of complexity classes is that they often fail to capture many natural algorithms—usually because the complexity-theoretic types used to control primitive recursion impose draconian restrictions on programming. Here is an example. In Bellantoni and Cook’s [3] and Leivant’s [11] well-known characterizations of the polynomial-time computable functions, a recursively-computed value is prohibited from driving another

### Two algorithms in search of a type-system

, 710

"... Abstract. The authors ’ ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable (ATR types are confined to levels 0, 1, and 2). ..."

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Abstract. The authors ’ ATR programming formalism is a version of call-by-value PCF under a complexity-theoretically motivated type system. ATR programs run in type-2 polynomial-time and all standard type-2 basic feasible functionals are ATR-definable (ATR types are confined to levels 0, 1, and 2). A limitation of the original version of ATR is that the only directly expressible recursions are tail-recursions. Here we extend ATR so that a broad range of affine recursions are directly expressible. In particular, the revised ATR can fairly naturally express the classic insertion- and selection-sort algorithms, thus overcoming a sticking point of most prior implicit-complexity-based formalisms. The paper’s main work is in refining the original time-complexity semantics for ATR to show that these new recursion schemes do not lead out of the realm of feasibility. 1.