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A new order-theoretic characterisation of the polytime computable functions
Martin Avanzini1, Naohi Eguchi1, Georg Moser1
1Institute of Computer Science, University of Innsbruck, Austria.
Theoretical Computer Science
|September 29, 2015
Summary
We introduce a new method to analyze program runtime complexity. This order-theoretic characterization directly links program structure to polynomial time bounds, offering precise complexity analysis for recursive functions.
Area of Science:
- Theoretical Computer Science
- Algorithm Analysis
- Formal Methods
Background:
- Understanding and bounding the runtime complexity of programs is crucial for efficient software development.
- Existing methods for complexity analysis can be intricate and difficult to automate.
- Term rewrite systems are a fundamental model for computation and program analysis.
Purpose of the Study:
- To develop a novel order-theoretic characterization for polytime computable functions.
- To introduce a new syntactic method for automatically analyzing the innermost runtime complexity of term rewrite systems.
- To establish a direct correspondence between program properties and their asymptotic worst-case complexity.
Main Methods:
- Definition of the 'small polynomial path order' ([Formula: see text]).
- Utilizing this termination order to analyze term rewrite systems.
- Establishing conditions for polynomial time bounds based on recursion depth.
Main Results:
- The small polynomial path order ([Formula: see text]) provides a new characterization of polytime computable functions.
- A direct, automatic, and syntactic method for analyzing innermost runtime complexity is presented.
- For systems compatible with [Formula: see text] and recursion depth *d*, runtime complexity is polynomially bounded by degree *d*.
- This polynomial bound is shown to be tight.
Conclusions:
- The proposed order-theoretic approach offers a powerful tool for program complexity analysis.
- The method establishes a tight, direct link between syntactic program properties and runtime complexity.
- This facilitates fully automatic and verifiable complexity assessments for recursive programs.
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