Feasible Mathematics II

1. Front Matter

   Pages i-viii

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2. On the Existence of modulo p Cardinality Functions

   • Miklós Ajtai

   Pages 1-14

3. Predicative Recursion and The Polytime Hierarchy

   • Stephen Bellantoni

   Pages 15-29

4. Are there Hard Examples for Frege Systems?

   • Maria Luisa Bonet, Samuel R. Buss, Toniann Pitassi

   Pages 30-56

5. On Gödel’s Theorems on Lengths of Proofs II: Lower Bounds for Recognizing k Symbol Provability

   • Samuel R. Buss

   Pages 57-90

6. Feasibly Categorical Abelian Groups

   • Douglas Cenzer, Jeffrey Remmel

   Pages 91-153

7. First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes

   • Peter Clote, Gaisi Takeuti

   Pages 154-218

8. Parameterized Computational Feasibility

   • Rodney G. Downey, Michael R. Fellows

   Pages 219-244

9. On Proving Lower Bounds for Circuit Size

   • Mauricio Karchmer

   Pages 245-255

10. Effective Properties of Finitely Generated R.E. Algebras

    • Bakhadyr Khoussainov, Anil Nerode

    Pages 256-283

11. On Frege and Extended Frege Proof Systems

    • Jan Krajíček

    Pages 284-319

12. Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time

    • Daniel Leivant

    Pages 320-343

13. Bounded Arithmetic and Lower Bounds in Boolean Complexity

    • Alexander A. Razborov

    Pages 344-386

14. Ordinal Bounds for Programs

    • Helmut Schwichtenberg, Stanley S. Wainer

    Pages 387-406

15. Turing Machine Characterizations of Feasible Functionals of All Finite Types

    • Anil Seth

    Pages 407-428

16. The Complexity of Feasible Interpretability

    • Rineke Verbrugge

    Pages 429-447

Perspicuity is part of proof. If the process by means of which I get a result were not surveyable, I might indeed make a note that this number is what comes out - but what fact is this supposed to confirm for me? I don't know 'what is supposed to come out' . . . . 1 -L. Wittgenstein A feasible computation uses small resources on an abstract computa­ tion device, such as a 'lUring machine or boolean circuit. Feasible math­ ematics concerns the study of feasible computations, using combinatorics and logic, as well as the study of feasibly presented mathematical structures such as groups, algebras, and so on. This volume contains contributions to feasible mathematics in three areas: computational complexity theory, proof theory and algebra, with substantial overlap between different fields. In computational complexity theory, the polynomial time hierarchy is characterized without the introduction of runtime bounds by the closure of certain initial functions under safe composition, predicative recursion on notation, and unbounded minimization (S. Bellantoni); an alternative way of looking at NP problems is introduced which focuses on which pa­ rameters of the problem are the cause of its computational complexity and completeness, density and separation/collapse results are given for a struc­ ture theory for parametrized problems (R. Downey and M. Fellows); new characterizations of PTIME and LINEAR SPACE are given using predicative recurrence over all finite tiers of certain stratified free algebras (D.

• algebra
• combinatorics
• complexity
• logic
• mathematics

• Department of Computer Science, Boston College, Chestnut Hill, USA

  Peter Clote

• Department of Mathematics, University of California, San Diego La Jolla, USA

  Jeffrey B. Remmel

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