Type-two Iteration with Bounded Query Revision

@inproceedings{Kapron2019TypetwoIW,
  title={Type-two Iteration with Bounded Query Revision},
  author={Bruce M. Kapron and Florian Steinberg},
  booktitle={DICE-FOPARA@ETAPS},
  year={2019}
}
Motivated by recent results of Kapron and Steinberg (LICS 2018) we introduce new forms of iteration on length in the setting of applied lambda-calculi for higher-type poly-time computability. In particular, in a type-two setting, we consider functionals which capture iteration on input length which bound interaction with the type-one input parameter, by restricting to a constant either the number of times the function parameter may return a value of increasing size, or the number of times the… 

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References

SHOWING 1-10 OF 17 REFERENCES

Type-two polynomial-time and restricted lookahead

This paper provides an alternate characterization of second-order polynomial-time computability, by refining the notion of oracle-poly- time computability introduced by Cook, and proves that all feasible problems can be solved within this class if one is allowed to separate a task into efficiently solvable subtasks.

A New Characterization of Type-2 Feasibility

The proof of this equivalence is not a simple generalization of the proof for type-1 functions; it depends on the fact that Mehlhorn's class is closed under a strong form of simultaneous limited recursion on notation and requires an analysis of the structure of oracle queries in time-bounded computations.

Feasible computation in higher types

We study the notion of feasible functional of finite type. For type level 2, we consider the oracle Turing machine model, and examine Mehlhorn's class of feasible functionals for this model. We give

Polynomial Running Times for Polynomial-Time Oracle Machines

This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider

Characterizations of the basic feasible functionals of finite type

  • S. CookB. M. Kapron
  • Computer Science
    30th Annual Symposium on Foundations of Computer Science
  • 1989
The authors define a simple typed while-programming language that generalizes the sort of simple language used in computability texts to define the familiar numerical computable functions and

Some applications of logic to feasibility in higher types

It is demonstrated that the class of basic feasible functionals has recursion theoretic properties which naturally generalise the corresponding properties of the classOf feasible functions, thus giving further evidence that the notion of feasibility of functionals mentioned above is correctly chosen.

A new recursion-theoretic characterization of the polytime functions

A recursion-theoretic characterization of FP which describes polynomial time computation independently of any externally imposed resource bounds, and avoids the explicit size bounds on recursion of Cobham.

Functional interpretations of feasibly constructive arithmetic

A notion of feasible function of finite type based on the typed lambda calculus is introduced which generalizes the familiar type 1 polynomial-time functions and an intuitionistic theory IPV’” is presented for reasoning about these functions.

Some desirable conditions for feasible functionals of type 2

  • Anil Seth
  • Mathematics
    [1993] Proceedings Eighth Annual IEEE Symposium on Logic in Computer Science
  • 1993
BFF is proved 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.

Polynomial and abstract subrecursive classes

This work defines abstract subrecursive reducibility relation and defines polynomial time computable operator in terms of these operators; the properties of these classes are investigated.