A New Look at the Automatic Synthesis of Linear Ranking Functions$

function. This is the case, for instance, of [13], where generation consists in theinstantiation of template functions and testing employs an algorithm based on avariant of Farkas’ Lemma. Non-linear constraints generated by the method describedin [24] are handled by first resorting to semidefinite programming solversand then validating the obtained results by using some other tools, since thesesolvers are typically based on interior point algorithms and hence may incur intounsafe rounding errors. Note that, in principle, the very same observation wouldapply to the case of linear constraints, if the corresponding linear programmingproblem is solved using an interior point method or even a floating-point basedimplementation of the simplex algorithm; however, there exist implementationsof the simplex algorithm based on exact arithmetic, so that linear programmingproblems can be numerically solved incurring no rounding errors at all and witha computational overhead that is often acceptable. 34. The Approach of Sohn and Van Gelder, GeneralizedAs far as we know, the first approach to the automatic synthesis of rankingfunctions is due to Kirack Sohn and Allen Van Gelder [7, 10]. Even thoughSohn and Van Gelder’s work concerned termination of logic programs, we willshow that the key ideas of their approach can be applied, with only rathersimple modifications, to the synthesis of ranking functions for any programmingparadigm, thus going beyond what subsequent authors acknowledged.4.1. Ranking Functions for Binary, Directly Recursive CLP(N) ProgramsWe refer the interested reader to [3] for a more complete reconstruction ofthe approach by Sohn and Van Gelder. For the other readers, suffices it to saythat termination analysis of logic programs can be mapped onto terminationanalysis of CLP(N) programs, the termination of which implies the terminationof the original program. 4 In turn, the termination analysis of general CLP(N)programs can be approximated by termination analysis of directly recursive, binaryCLP(N) clauses. In order to show how ranking functions can be computedfrom these, we deal first with a single clausep(¯x) :− c[¯x, ¯x ′ ], p(¯x ′ ), (5)where p is a predicate symbol, ¯x and ¯x ′ are disjoint n-tuples of variables, andc[¯x, ¯x ′ ] is a linear constraint involving variables in ¯x ∪ ¯x ′ . 5 The meaning of sucha clause is that, if p is called on some tuple of integers ¯x, then there are twocases:3 In contrast, an exact solver for non-linear constraints would probably require a trulysymbolic computation, incurring a much more significant computational overhead.4 See, e.g., [26] for an introduction to CLP —constraint logic programming— languages,and [16] for the mapping of logic programs to CLP(N) programs.5 As usual, we abuse notation by confusing a tuple with the set of its elements.10