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SizeChange Termination, Monotonicity Constraints and Ranking Functions
"... Abstract. Sizechange termination involves deducing program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x> y ′ a ..."
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Abstract. Sizechange termination involves deducing program termination based on the impossibility of infinite descent. To this end we may use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x> y ′ and x ≥ y ′ are allowed, we have sizechange graphs, for which both theory and practice are now more evolved then for general monotonicity constraints. This work shows that it is possible to transfer some theory from the domain of sizechange graphs to the general case, complementing and extending previous work on monotonicity constraints. Significantly, we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singlyexponential time, which is better than what has been published so far even for sizechange graphs. We also consider the integer domain, where general monotonicity constraints are essential because the domain is not wellfounded. 1
A SATBased Approach to Size Change Termination with Global Ranking Functions
"... Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable t ..."
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Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a welldefined and significant subset of SCT instances, designed to be amenable to a SATbased solution. We have implemented the approach using a stateoftheart Boolean satisfaction solver. Experimentation indicates that the approach is a viable alternative to the complete SCT decision procedure based on closure computation and local ranking functions. Our approach has the extra benefit of producing an explicit witness to prove termination in the form of a global ranking function. 1
A complexity tradeoff in rankingfunction termination proofs
 Acta Informatica
, 2009
"... To prove that a program terminates, we can employ a ranking function argument, where program states are ranked so that every transition decreases the rank. Alternatively, we can use a set of ranking functions with the property that every cycle in the program’s flowchart can be ranked with one of th ..."
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To prove that a program terminates, we can employ a ranking function argument, where program states are ranked so that every transition decreases the rank. Alternatively, we can use a set of ranking functions with the property that every cycle in the program’s flowchart can be ranked with one of the functions. This “local ” approach has gained interest recently on the grounds that local ranking functions would be simpler and easier to find. The current study is aimed at better understanding the tradeoffs involved, in a precise quantitative sense. We concentrate on a convenient setting, SizeChange Termination framework (SCT). In SCT, programs are replaced by an abstraction whose termination is decidable. Moreover, sufficient classes of ranking functions (both global and local) are known. Our results show a tradeoff: either exponentially many local functions of certain simple forms, or an exponentially complex global function may be required for proving termination.