Towards computing distances between programs via Scott domains

Partial metrics and co-continuous valuations

“…Thus [3] only requires symmetry and the ordinary triangle inequality for the upper bounds of relaxed metrics.…”

“…This is, essentially, the approach of Section 5 of [3], where Info(x) and Neginfo(x) can be understood as subsets of a domain basis. However: there was a number of remaining open problems.…”

“…However: there was a number of remaining open problems. In particular, while [3] builds partial metrics on all continuous Scott domains with countable bases, the reliance of [3] on finite weights of non-compact basic elements does not allow to obtain some natural partial metrics on real-line based domains: and also introduces some unpleasant restrictions on domains which should be satisfied in order to obtain an induced classical metric on Total(A). [5, 11, 18:2 l and references therein).…”

“…In [3] Bukatin and Scott generalized this relationship by considering valuations on powersets of bases, instead of valuations on the domains themselves, as in [15]. They also explained the computational intuition of partial metrics by generalizing them to relaxed metrics, which take values in the interval numbers.…”

Weak condition for generalized f-weakly Picard mappings on partial metric spaces

“…In the context of computer science where a computable function can also be proved to be a contraction, the partial metric extension of the contraction fixed point theorem can be used to prove that the unique fixed point, which is the programs output, will be totally computed [18]. Further applications of partial metrics to problems in theoretical computer science were discussed in [11,12,25,26,28,29].…”