Provably Total Primitive Recursive Functions Theories with Induction 1st Edition by Andres Cordon Franco, Alejandro Fernandez Margarit, F F lix Lara Martin ISBN 9783540301240

Provably Total Primitive Recursive Functions: Theories with Induction 1st Edition by Andres Cordon Franco, Alejandro Fernandez Margarit, F F lix Lara Martin – Ebook PDF Instant Download/Delivery. 9783540301240
Full download Provably Total Primitive Recursive Functions: Theories with Induction 1st Edition after payment

Product details:

ISBN 10: 
ISBN 13: 9783540301240
Author: Andres Cordon Franco, Alejandro Fernandez Margarit, F F lix Lara Martin

A natural example of a function algebra is <span id="MathJax-Element-1-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="R”>�(T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kind of function algebras is obtained. This provides a useful tool for studying the class of primitive recursive functions in <span id="MathJax-Element-2-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="R”>�(T). We prove that this is the class of p.t.c.f. of the theory axiomatized by the induction scheme restricted to (parameter free) Δ₁(T)–formulas (i.e. Σ₁–formulas which are equivalent in T to Π₁–formulas).

Moreover, if T is a sound theory and proves that exponentiation is a total function, we characterize the class of primitive recursive functions in <span id="MathJax-Element-3-Frame" class="MathJax_SVG" style="box-sizing: inherit; display: inline-block; font-style: normal; font-weight: normal; line-height: normal; font-size: 18px; text-indent: 0px; text-align: left; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;" tabindex="0" role="presentation" data-mathml="R”>�(T) as a function algebra described in terms of bounded recursion (and composition). Extensions of this result are related to open problems on complexity classes. We also discuss an application to the problem on the equivalence between (parameter free) Σ₁–collection and (uniform) Δ₁–induction schemes in Arithmetic.

The proofs lean upon axiomatization and conservativeness properties of the scheme of Δ₁(T)–induction and its parameter free version.

Provably Total Primitive Recursive Functions: Theories with Induction 1st Table of contents:

• Preliminaries
  2.1 Computability and Recursive Functions
  2.2 Primitive Recursive Functions: Formal Definition
  2.3 Total vs. Partial Functions in Recursive Theory
  2.4 Theories of Induction: Classical and Modern Approaches
  2.5 Formal Systems and Proof Theory

• Primitive Recursive Functions and Totality
  3.1 Definition of Total Functions
  3.2 Key Properties of Primitive Recursive Functions
  3.3 Totality of Primitive Recursive Functions: Criteria and Characterizations
  3.4 Closure Properties and Extensions of Primitive Recursion
  3.5 Relations Between Totality and Inductive Definitions

• Induction in Formal Systems
  4.1 Inductive Definitions and Their Role in Mathematics
  4.2 Induction in the Context of Recursive Functions
  4.3 Theories of Inductive Reasoning: Formalizing Inductive Principles
  4.4 Inductive Theories for Proving Totality of Primitive Recursive Functions
  4.5 Comparison of Inductive Systems

• Provability of Totality
  5.1 Formal Proofs of Totality in Recursive Functions
  5.2 Proof Systems for Recursive Functions: Axiomatic Approaches
  5.3 Provability of Total Primitive Recursive Functions in Theories with Induction
  5.4 Limits of Provability: Incompleteness and Computability

• Key Results in Primitive Recursive Function Theory
  6.1 Fundamental Theorems on Primitive Recursion
  6.2 Totality of Recursive Functions and Inductive Proofs
  6.3 Proof Techniques for Totality in Primitive Recursive Functions
  6.4 Relationship Between Induction and Recursive Definitions

• Applications and Case Studies
  7.1 Applications in Mathematical Logic and Proof Theory
  7.2 Total Functions in Formal Verification
  7.3 Case Study 1: Recursive Definitions in Set Theory
  7.4 Case Study 2: Inductive Reasoning in Computability
  7.5 Real-World Applications of Total Primitive Recursive Functions

• Complexity and Decidability in Primitive Recursive Systems
  8.1 Computational Complexity of Primitive Recursive Functions
  8.2 Decidability of Theories with Inductive Definitions
  8.3 Challenges in Extending Primitive Recursive Functions
  8.4 Complexity of Proving Totality in Recursive Theories

• Challenges and Open Problems
  9.1 Limitations in Current Theories
  9.2 Open Questions in Inductive Theories of Total Recursive Functions
  9.3 Extending Provable Totality to More Complex Functions
  9.4 Future Directions for Research

People also search for Provably Total Primitive Recursive Functions: Theories with Induction 1st:

provably total primitive recursive functions
    
primitive recursive functions
    
primitive recursive
    
primitive recursive relation
    
primitive recursion vs general recursion