NDLI: Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

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Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Wunderlich, Henning Niggl, Karl-Heinz
Copyright Year	2006
Abstract	In earlier work of Kristiansen and Niggl the polynomial-time computable functions were characterized by stack programs of $\mu$-measure 0, and the linear-space computable functions by loop programs of $\mu$-measure 0. Until recently, an open problem was how to extend these characterizations to programs with user-friendly basic instructions, such as assignment statements, and with mixed data structures.It is shown how to strengthen the above characterizations to imperative programs built from arbitrary basic instructions by sequencing and by if-then-else and for-do statements. These programs operate on variables, each of which may represent any data structure such as stacks, registers, trees, or graphs.The paper presents a new method of certifying "polynomial size boundedness" of such imperative programs under the natural assumption that the basic instructions used are polynomially size bounded, too. The certificate for a program ${\tt P}$ with variables among ${\tt X}_{1}, \ldots, {\tt X}_{n}$ will be an $(n+1)\times (n+1)$ matrix $M({\tt P})$ over the finite set $\{ 0, 1, \infty\}$.It is shown that certified string programs (i.e., stack programs, but with any polynomial-time computable basic instructions) exactly compute the functions in {\sc fptime}. Accordingly, certified general loop programs (using any linear-space computable basic instructions) exactly compute the functions in {\sc flinspace}.Furthermore, it is shown that certified power string programs (i.e., string programs, but built from polynomial-space computable basic instructions and extended by power loop statements) exactly compute the polynomial-space computable functions in {\sc fpspace}.In addition, examples of certified "natural" (implementations of) algorithms, such as insertion-sort or binary addition and multiplication, are given.
Starting Page	1122
Ending Page	1147
Page Count	26
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539704445597
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	5
Volume Number	35
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-27
Access Restriction	Subscribed
Subject Keyword	Specification and verification linear space Analysis of algorithms and problem complexity polynomial space imperative programming languages Complexity of computation polynomial time static program analysis property testing implicit computational complexity Complexity classes
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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