A Denotational Semantics for Prolog

analysis is nite, which was not the case in our previous proposal [13]. The overheadneeded for reaching an improved precision essentially consists in the use of observability andkernel constraints and of their success and failure conditions.We can now consider the already mentioned paper [11]. A big di erence arises lookingfor instance at the way they handle clause composition and the cut. Their de nitions arethe following ones.Consider two sequences S1 and S2 without cut information. S1 stands for theresult of ci and S2 stands for the (combined) result of ci+1; : : : ; cn. If executionof ci terminates, then su x ci+1; : : : ; cn is executed. Otherwise ci+1; : : : ; cn isnot executed. Therefore, the combined result S1 S2 = S1 :: S2 if S1 is nite (i.e.neither incomplete nor in nite) and S1 S2 = S1 otherwise. The de nition canbe extended to sequences with cut information. If no cut is executed in ci (...),the previous reasoning applies. Otherwise, su x ci+1; : : : ; cn is not executed (...).So, the combined result (...) is de ned by hS1; cfi S2 = S1 S2 if cf = nocutand hS1; cfi S2 = S1 if cf = cut.Compare this with our de nitions 10 and 11. The di erence is that we delay the decisiondivergence/no divergence or cut/no cut to the actual evaluation of the denotation in a givenconstraint store. They take instead this decision just when they compose the denotations.It is not clear how their approach could lead to a goal-independent denotation. In any casetheir approach makes impossible to compute the denotations of the most general goals only,like we do in this paper. This is important because it leads to a simpler and more e cientcomputation of the semantics.While a general approach for de ning success and failure conditions is available (equations1 and 2), one should not think that the problem of nding success and failure conditionsis de nitely solved. Actually, we only shifted the problem in the de nition of the abstractdomain, which must be devised in such a way that equations 1 and 2 do not give trivialapproximations (like >, for instance). Future work will be spent in devising such domains,and in showing the feasibility of the approach with its implementation.References[1] R. Barbuti, M. Codish, R. Giacobazzi, and G. Levi. Modelling Prolog Control. Journal of Logic andComputation, 3:579{603, 1993.[2] R. Barbuti, M. Codish, R. Giacobazzi, and M. Maher. Oracle Semantics for PROLOG. Informationand Computation, 1995.[3] E. Borger. A Logical Operational Semantics of Full Prolog. In E. Borger, H. Kleine, H. Buning, andM. Richter, editors, CSL 89. 3rd workshop on Computer Science Logic, volume 440 of Lecture Notesin Computer Science, pages 36{64. Springer-Verlag, 1990.[4] A. Bossi, M. Bugliesi, and M. Fabris. Fixpoint Semantics for PROLOG. In D. S. Warren, editor, Proc.Tenth Int'l Conf. on Logic Programming, pages 374{389. The MIT Press, 1993.[5] P. Cousot and R. Cousot. Abstract Interpretation and Applications to Logic Programs. Journal ofLogic Programming, 13(2 & 3):103{179, 1992.[6] A. de Bruin and E. de Vink. Continuation Semantics for Prolog with Cut. In J. Diaz and F. Orejas,editors, Proc. CAAP 89, volume 351 of Lecture Notes in Computer Science, pages 178{192. Springer-Verlag, 1989.[7] S. K. Debray and P. Mishra. Denotational and Operational Semantics for Prolog. Journal of LogicProgramming, 5:61{91, 1988.[8] Melvin Fitting. A Deterministic Prolog Fixpoint Semantics. Journal of Logic Programming, 2(2):111{118, 1985.[9] D. Jacobs and A. Langen. Static Analysis of Logic Programs for Independent AND Parallelism. Journalof Logic Programming, 13(2 & 3):291{314, 1992.[10] N. D. Jones and A. Mycroft. Stepwise Development of Operational and Denotational Semantics forPROLOG. In StenAke Tarnlund, editor, Proc. Second Int'l Conf. on Logic Programming, pages281{288, 1984.[11] B. Le Charlier, Rossi S., and P. Van Hentenryck. An Abstract Interpretation Framework which Accu-rately Handles PROLOG Search Rule and the Cut. In M. Bruynooghe, editor, Proceedings of the 1994Int'l Symposium on Logic Programming, pages 157{171. The MIT Press, 1994.[12] G. Levi and D. Micciancio. Analysis of pure PROLOG programs. In M.I. Sessa, editor, ProceedingsGULP-PRODE '95, 1995 Joint Conference on Declarative Programming, pages 521{532, 1995.[13] G. Levi and F. Spoto. Accurate Analysis of Prolog with Cut. In P. Lucio, M. Martelli, and M. Navarro,editors, Proceeding APPIA-GULP-PRODE'96, pages 481{492, 1996.