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Nội dung chi tiết: Polynomial interpretations as a basis fo

Polynomial interpretations as a basis fo

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo -KampReport cw 412, Revised version. August 2007Katholieke Universiteit Leuven Department of Computer Science Celestijnenlaan 200A - B-3001 Heverlee (

Belgium)Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen. Danny De Schreye Jurgen Giesl, and Peter Sc Polynomial interpretations as a basis fo

hneider-KampReport CWJJ2, Revised version, August 2007Department of Computer Science. K.Ư.LeuvenAbstractThis paper introduces a new technique for term

Polynomial interpretations as a basis fo

inat ion analysis of definite logic programs (LPs) liased on polynomial interpretations. The principle of this technique is to map each function and p

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo nomial interpretations can 1st seen as a direct generalisation of the traditional techniques in termination analysis of LPs, where (semi-)linear norms

and level mappings are used. Our extension generalises these to arbitrary polynomials. We extend a number of standard concepts and results on termina Polynomial interpretations as a basis fo

tion analysis to the context of polynomial interpretations. We propose a constraint based approach for automatically generating polynomial interpretat

Polynomial interpretations as a basis fo

ions that satisfy the termination conditions. Based on tills approach, we implement a new tool, called Polytool, for automatic termination analysis of

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo tion Analysis of Logic ProgramsNíanh Thang Nguyen1. Danny De Schreye1, Jurgen Gicsl2, and Peter Schneider- K amp21 Department of Computer Science, K.U

.Lcuven Cclcstijnciihuui 200A. B-3001, llcvcrlce, Belgium {ManhThang-Nguycn, Danny.DcSchrcyeJiScs.kulcnvcn.ac.be2 Department of Computer Science. RWTH Polynomial interpretations as a basis fo

Aachen Ahornstr. 55, D-520S6 Aachen, Germany {gicsl, pskjiiinformatik.iwth-aachen.dcAbstract. This paper introduces a new technique for termination a

Polynomial interpretations as a basis fo

nalysis of definite logic programs based on polynomial interpretations, rhe principle of this technique is to map each function and predicate symbol t

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo tions carr be seen as a direct generalisation of the traditional techniques in termination analysis of Li’s, where (semi-)lincar norms and level mappi

ngs are used. Our extension generalises these to arbitrary polynomials. We extend a number of standard concepts and results on termination analysis to Polynomial interpretations as a basis fo

the context of polynomial interpretations. We propose a constraint based approach for automatically generating polynomial interpretations that satisf

Polynomial interpretations as a basis fo

y the termination conditions. Based orr this approach, we implement a new tool, namely Polytool, for automatic termination analysis of logic programs.

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo rogram correctness. A termination proof is mostly based on a mapping from computational states to some well-founded ordered set. Termination is guaran

teed if the mapped values of the encountered states during a computation, under this mapping, decrease w.r.t. the ordering.For Logic Programming (LP). Polynomial interpretations as a basis fo

termination analysis Ls done by mapping terms ami atoms to a well-founded set of natural numbers by means of norms and level mappings. Proving termin

Polynomial interpretations as a basis fo

ation is based on the search for a suitable norm and level mapping such that the mapped value of the initial predicate call is bounded and of the runn

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo on verifying the decrease in size of (mutually) recursive predicate calls,which correspond to the loops that the execution passes through.Until now. m

ost termination techniques in LP are based on the use of semi-linear norms and level mappings, which measure the size of each term or atom as a linear Polynomial interpretations as a basis fo

combination of its subterms. For example, the Hasta La Vista system '31] infers one specific semi-linear norm ami the TerminWeb analyser [34] uses a

Polynomial interpretations as a basis fo

combinat ion of several semi-linear norms for termination analysis. A restriction of semi-linear norms is that a lot of examples require more powerful

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo erivative of a function in some variable u. This example was first introduced in [13] (see also [10]).Example 1 (der).d(

DY): -d(der(X), DX),d(tUr(Y). DY).d(der(X ♦ Y), X *DY + Y*DX)>. -d(dcr(A). DX).d(der(Y). DY).d(der(der(X)),DDX): -d(der(X). DX),d(dẹr(DX), DDX).We are Polynomial interpretations as a basis fo

interested in proving termination of this prognun w.r.t. the query set 5 = {d(t;, te)|tj is a ground term, and tg is a free variable}. We consider th

Polynomial interpretations as a basis fo

e first argument of d/2 as an input argument ami the second as an output.Doing this on the basis of a semi-linear norm and level mapping is impossible

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo at there exists such a semi-linear norm |.| and level mapping |.| of general forms such that: II «■ = c.- II''-t M = ứ + tfllh II + jf/IIM. ló * M = f

s +/ÍIIGII + fi iLdl.II Jer(f )|| =4- /;'||í||, |d(QUs)| = d0 + df |t/|| + dgỊỊf.gịị where t, tj, tg are termsandami nt arc non-negative inte-gers. Ap Polynomial interpretations as a basis fo

plying the genend constraint based method in [12] shows a contradiction: the system of inequalities that is set up from the acceptability condition is

Polynomial interpretations as a basis fo

unsolvable. A complete proof can be found in [25]. Of course this only proves that one particular approach is unable to prove termination on the basi

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo Using polynomial interpretations as a basis for ordering terms in TRSs was first introduced by Lankford in [22]. It is currently one of the best know

n and most widely used techniques in TRS termination analysis.We develop the approach within an LP context. We redefine and extend several known conce Polynomial interpretations as a basis fo

pts ami results from LP termination analysis to polynomial interpretations. We show how polynomial interpretations can be seen as a direct generalisat

Polynomial interpretations as a basis fo

ion of currently used techniques in LP termination based on (semi-)linear norms and linear level-mappings. As one would expect, the generalisation is

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

Polynomial interpretations as a basis fo d ‘abstract level mapping’ [35]. We show that under this approach, examples as the2

Polynomial Interpretations as a Basis for Termination Analysis of Logic ProgramsManh Thang Nguyen, Danny De Schreye Jurgen Giesl, and Peter Schneider-

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