Feasible Mathematics II

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.

Preface.- On the Existence of modulo p Cardinality Functions.- Predicative Recursion and The Polytime Hierarchy.- Are there Hard Examples for Frege Systems?.- On Godel's Theorems on Lengths of Proofs II: Lower Bounds for Recognizing k Symbol Provability.- Feasibly Categorical Abelian Groups.- First Order Bounded Arithmetic and Small Boolean Circuit Complexity Classes.- Parameterized Computational Feasibility.- On Proving Lower Bounds for Circuit Size.- Effective Properties of Finitely Generated R.E. Algebras.- On Frege and Extended Frege Proof Systems.- Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time.- Bounded Arithmetic and Lower Bounds in Boolean Complexity.- Ordinal Bounds for Programs.- Turing Machine Characterizations of Feasible Functionals of All Finite Types.- The Complexity of Feasible Interpretability.