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On the Termination of Integer Loops

Amir M. Ben-Amram, Samir Genaim, Abu Naser Masud
2012 ACM Transactions on Programming Languages and Systems  
In this paper we study 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.  ... 
doi:10.1145/2400676.2400679 fatcat:kjkf2dnscvdk7l2v435zq2zuka

On the Termination of Integer Loops [chapter]

Amir M. Ben-Amram, Samir Genaim, Abu Naser Masud
2012 Lecture Notes in Computer Science  
In this paper we study 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.  ... 
doi:10.1007/978-3-642-27940-9_6 fatcat:jf6tscspefgfnevchfhej3fbba

On Termination of Integer Linear Loops [chapter]

Joël Ouaknine, João Sousa Pinto, James Worrell
2014 Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms  
A fundamental problem in program verification concerns 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.  ... 
doi:10.1137/1.9781611973730.65 dblp:conf/soda/OuakninePW15 fatcat:lz326n4wszd67i7iydxy5brmmm

Termination of Linear Loops over the Integers

Mehran Hosseini, Joël Ouaknine, James Worrell, Michael Wagner
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 −  ... 
doi:10.4230/lipics.icalp.2019.118 dblp:conf/icalp/HosseiniO019 fatcat:kwhjzivi4jacvmwemhpjgrtbue

Termination Analysis of Integer Linear Loops [chapter]

Aaron R. Bradley, Zohar Manna, Henny B. Sipma
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.  ... 
doi:10.1007/11539452_37 fatcat:levdn3wtn5csjnrc5orzkdytym

Termination of Linear Loops over the Integers [article]

Mehran Hosseini, Joël Ouaknine, James Worrell
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.  ... 
arXiv:1902.07465v2 fatcat:5olzonedsrcdhc52jdcwyfvvge

Automatic Termination Analysis of Programs Containing Arithmetic Predicates [article]

Nachum Dershowitz, Naomi Lindenstrauss, Yehoshua Sagiv, Alexander Serebrenik
2000 arXiv   pre-print
In summary, 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.  ... 
arXiv:cs/0011036v1 fatcat:kz25racdabgm7bukfgtazqfv2q

Automatic Termination Analysis of Programs Containing Arithmetic Predicates

Nachum Dershowitz, Naomi Lindenstrauss, Yehoshua Sagiv, Alexander Serebrenik
1999 Electronical Notes in Theoretical Computer Science  
In summary, 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.  ... 
doi:10.1016/s1571-0661(04)00113-6 fatcat:tgpwr355svaehpr3f5z2jjffpy

Unrestricted Termination and Non-termination Arguments for Bit-Vector Programs [chapter]

Cristina David, Daniel Kroening, Matt Lewis
2015 Lecture Notes in Computer Science  
to a 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.  ... 
doi:10.1007/978-3-662-46669-8_8 fatcat:vkaqrin44zcrboih7y4hddhary

On the linear ranking problem for integer linear-constraint loops

Amir M. Ben-Amram, Samir Genaim
2013 Proceedings of the 40th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '13  
In this paper we study 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.  ... 
doi:10.1145/2429069.2429078 dblp:conf/popl/Ben-AmramG13 fatcat:aveql7koy5bqhme6qdlofj7bua

Unrestricted Termination and Non-Termination Arguments for Bit-Vector Programs [article]

Cristina David and Daniel Kroening and Matt Lewis
2014 arXiv   pre-print
to a 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.  ... 
arXiv:1410.5089v1 fatcat:mvqkuujswzdk5izmk57v7t22za

Termination of Integer Linear Programs [chapter]

Mark Braverman
2006 Lecture Notes in Computer Science  
We show that 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.  ... 
doi:10.1007/11817963_34 fatcat:bzbi5xnksffj5ehkqz4xwn2hma

On Termination of Integer Linear Loops [article]

Joël Ouaknine, João Sousa Pinto, James Worrell
2014 arXiv   pre-print
A fundamental problem in program verification concerns 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.  ... 
arXiv:1407.1891v2 fatcat:nvcj3o5p3fez3bfh4jas2yw3mu

Invariant Based Programming [chapter]

Ralph-Johan Back
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  ... 
doi:10.1007/11767589_1 fatcat:6eusfxcelrenvkmerzs64tiedu

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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