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## Filters

##
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On the Termination of Integer Loops

2012
*
ACM Transactions on Programming Languages and Systems
*

In this paper we study

doi:10.1145/2400676.2400679
fatcat:kjkf2dnscvdk7l2v435zq2zuka
*the*decidability*of**termination**of*several variants*of*simple*integer**loops*, without branching in*the**loop*body and with affine constraints as*the**loop*guard (and possibly a precondition ... For*the*common case*of**integer*linear-constraint*loops*with rational coefficients we have not succeeded in proving either decidability or undecidability*of**termination*, but we show that a Petri net can ... Acknowledgement We thank Pierre Ganty for discussions*on*Petri nets. We also thank William Gasarch for motivating us to add Section 7. ...##
###
On the Termination of Integer Loops
[chapter]

2012
*
Lecture Notes in Computer Science
*

In this paper we study

doi:10.1007/978-3-642-27940-9_6
fatcat:jf6tscspefgfnevchfhej3fbba
*the*decidability*of**termination**of*several variants*of*simple*integer**loops*, without branching in*the**loop*body and with affine constraints as*the**loop*guard (and possibly a precondition ... For*the*common case*of**integer*linear-constraint*loops*with rational coefficients we have not succeeded in proving either decidability or undecidability*of**termination*, but we show that a Petri net can ... Acknowledgement We thank Pierre Ganty for discussions*on*Petri nets. We also thank William Gasarch for motivating us to add Section 7. ...##
###
On Termination of Integer Linear Loops
[chapter]

2014
*
Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms
*

A fundamental problem in program verification concerns

doi:10.1137/1.9781611973730.65
dblp:conf/soda/OuakninePW15
fatcat:lz326n4wszd67i7iydxy5brmmm
*the**termination**of*simple linear*loops**of**the*form: where x is a vector*of*variables, u, a, and c are*integer*vectors, and A and B are*integer*matrices ... Assuming*the*matrix A is diagonalisable, we give a decision procedure for*the*problem*of*whether, for all initial*integer*vectors u, such a*loop**terminates*. ... Acknowledgements*The*authors would like to thank Elias Koutsoupias and Ventsislav Chonev for their advice and feedback. ...##
###
Termination of Linear Loops over the Integers

2019
*
International Colloquium on Automata, Languages and Programming
*

*The*question is whether such a

*loop*

*terminates*

*on*all

*integer*initial values. ... We consider

*the*problem

*of*deciding

*termination*

*of*single-path while

*loops*with

*integer*variables, affine updates, and affine guard conditions. ... I C A L P 2 0 1 9 118: 4 94

*Termination*

*of*Linear

*Loops*over

*the*

*Integers*I C A L P 2 0 1 9 118: 6 96

*Termination*

*of*Linear

*Loops*over

*the*IntegersDefine a linear preorder

*on*I := {0, . . . , k max − ...

##
###
Termination Analysis of Integer Linear Loops
[chapter]

2005
*
Lecture Notes in Computer Science
*

*Integer*division and modulo arithmetic must be soundly abstracted away so that

*the*analysis over

*the*abstracted

*loop*is sound for

*the*original

*loop*. Consequently,

*the*analysis loses precision. ... Usually, ranking function synthesis and invariant generation over a

*loop*with

*integer*variables involves abstracting

*the*

*loop*to have real variables. ... Acknowledgments We thank Sriram Sankaranarayanan and

*the*reviewers for their insightful comments. ...

##
###
Termination of Linear Loops over the Integers
[article]

2019
*
arXiv
*
pre-print

*The*question is whether such a

*loop*

*terminates*

*on*all

*integer*initial values. ... We consider

*the*problem

*of*deciding

*termination*

*of*single-path while

*loops*with

*integer*variables, affine updates, and affine guard conditions. ...

*The*

*termination*problem for simple linear

*loops*

*on*

*integers*is reducible to

*the*special case

*of*

*the*problem for non-degenerate update functions. Proof. ...

##
###
Automatic Termination Analysis of Programs Containing Arithmetic Predicates
[article]

2000
*
arXiv
*
pre-print

In summary,

arXiv:cs/0011036v1
fatcat:kz25racdabgm7bukfgtazqfv2q
*the*proposed approach is based*on*combining a finite abstraction*of**the**integers*with*the*technique*of**the*query-mapping pairs, and is essentially capable*of*dividing a*termination*proof into ...*The*method consists*of**the*following steps: First, a finite abstract domain for representing*the*range*of**integers*is deduced automatically. ... Let S be a set*of*clauses in P , consisting*of*an*integer**loop*and all*the*clauses for predicates*on*which*the*predicates*of**the**integer**loop*depend. ...##
###
Automatic Termination Analysis of Programs Containing Arithmetic Predicates

1999
*
Electronical Notes in Theoretical Computer Science
*

In summary,

doi:10.1016/s1571-0661(04)00113-6
fatcat:tgpwr355svaehpr3f5z2jjffpy
*the*proposed approach is based*on*combining a nite abstraction*of**the**integers*with*the*technique*of**the*query-mapping pairs, and is essentially capable*of*dividing a*termination*proof into ...*The*method consists*of**the*following steps: First, a nite abstract domain for representing*the*range*of**integers*is deduced automatically. ... Let S 1 be a set*of*clauses in P 1 , consisting*of*an*integer**loop*and*the*clauses*of**the*predicates*on*which*the**integer**loop*predicates depend. ...##
###
Unrestricted Termination and Non-termination Arguments for Bit-Vector Programs
[chapter]

2015
*
Lecture Notes in Computer Science
*

to a

doi:10.1007/978-3-662-46669-8_8
fatcat:vkaqrin44zcrboih7y4hddhary
*termination*argument for*the*same program operating*on*fixed-width machine*integers*. ... This class*of*functions is insufficient for proving*termination**of*many*terminating*programs, and furthermore a*termination*argument for a program operating*on*mathematical*integers*does not always lead ... -If L 2 is non-*terminating*, then N 2 ∧ G 2 is its recurrence set. Composing a*Loop*with*the*Rest*of**the*Program Sometimes*the**termination*behaviour*of*a*loop*depends*on**the*rest*of**the*program. ...##
###
On the linear ranking problem for integer linear-constraint loops

2013
*
Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '13
*

In this paper we study

doi:10.1145/2429069.2429078
dblp:conf/popl/Ben-AmramG13
fatcat:aveql7koy5bqhme6qdlofj7bua
*the*complexity*of**the*Linear Ranking problem: given a*loop*, described by linear constraints over a finite set*of**integer*variables, is there a linear ranking function for this*loop*... However, when they range over*the**integers*, whether for single-path or multipath*loops*,*the*complexity*of**the*Linear Ranking problem has not yet been determined. We show that it is coNP-complete. ...*The*work*of*Samir Genaim was funded in part by*the*EU project ICT-231620 HATS, and*the*Spanish projects TIN-2008-05624 DOVES and TIN-2012-38137 VIVAC. ...##
###
Unrestricted Termination and Non-Termination Arguments for Bit-Vector Programs
[article]

2014
*
arXiv
*
pre-print

to a

arXiv:1410.5089v1
fatcat:mvqkuujswzdk5izmk57v7t22za
*termination*argument for*the*same program operating*on*fixed-width machine*integers*. ... This class*of*functions is insufficient for proving*termination**of*many*terminating*programs, and furthermore a*termination*argument for a program operating*on*mathematical*integers*does not always lead ...*termination*behaviour*of*a*loop*depends*on**the*rest*of**the*program. ...##
###
Termination of Integer Linear Programs
[chapter]

2006
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/11817963_34
fatcat:bzbi5xnksffj5ehkqz4xwn2hma
*termination**of*a simple class*of*linear*loops*over*the**integers*is decidable. ... Namely we show that*termination**of*deterministic linear*loops*is decidable over*the**integers*in*the*homogeneous case, and over*the*rationals in*the*general case. ... At*the*limit,*the*orbit*of*(x, y) will alternate between*the*directions (−1 − √ 17, 4) and (1 + √ 17, −4). Hence*the**loop**terminates**on*all*integers*. ...##
###
On Termination of Integer Linear Loops
[article]

2014
*
arXiv
*
pre-print

A fundamental problem in program verification concerns

arXiv:1407.1891v2
fatcat:nvcj3o5p3fez3bfh4jas2yw3mu
*the**termination**of*simple linear*loops**of**the*form x := u ; while Bx >= b do x := Ax + a where x is a vector*of*variables, u, a, and c are*integer*... Assuming*the*matrix A is diagonalisable, we give a decision procedure for*the*problem*of*whether, for all initial*integer*vectors u, such a*loop**terminates*. ... Acknowledgements*The*authors would like to thank Elias Koutsoupias and Ventsislav Chonev for their advice and feedback. ...##
###
Invariant Based Programming
[chapter]

2006
*
Lecture Notes in Computer Science
*

*of*

*the*triangle

*on*

*the*hypotenuse h divides

*the*hypotenuse in

*the*proportion 3var a, b , c , h : real

*the*triangle is right, with hypotenuse c and catheters a and b, -h is

*the*height

*of*

*the*triangle

*on*... below. •

*The*inner

*loop*will eventually

*terminate*because n − j is decreased but is bounded from below. •

*The*outer

*loop*will

*terminate*because n − i is decreased and is bounded from below. •

*The*inner ...

##
###
Page 663 of American Journal of Mathematics Vol. 73, Issue 3
[page]

1951
*
American Journal of Mathematics
*

A

*loop*-edge can be part*of*only*one**loop*, but a*loop*-point may have an indefinite number*of*dependent*loop*-edges, and may besides be*one**terminal*point*of*an indefinite number*of*non-*loop*edges. ... Every such set*of**integers*may be regarded as a partition*of**the**integer*SP where SP = p, + po +--+ +--+ pn. ...
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