A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. In GULP-PRODE'94 Joint Conference on Declarative Programming, pages 372--397, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....One could define P so that it explicitly contains the abstract domain Pos . Even when this is not the case, it should be noted that, as soon as the parameter P includes the set sharing domain of Jacobs and Langen [45] then it will subsume the groundness information captured by the domain Def [15, 24]. 21 The following example shows that, when computing on rational trees, finite tree dependencies may provide groundness information that is not captured by the usual approaches. Example 32. Consider the program: p(a, Y) p(X, a) q(X, Y) p(X, Y) X = f(X, Z) The abstract semantics of ....

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163--192, 1998.

....) 7 One could define P so as it explicitly contains the abstract domain Pos. Even when this is not the case, it should be noted that, as soon as the parameter P includes the setsharing domain of Jacobs and Langen [28] then it will subsume the groundness information captured by the domain Def [9, 14]. 11 Proof. #G (#1 # #2 ) # # # RSubst # # ## # # # : #1 # #2 ) # gval(#) # = 1 # = # # # RSubst # # # # # ## # # # : #i # 1, 2 : # i # gval(#) # = 1 # = # # # RSubst # # ## # # # : #1 # gval(#) # = 1 # # # # # RSubst # # ## # # # : #2 ....

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163--192, 1998.

....2) 7 One could de ne P so as it explicitly contains the abstract domain Pos . Even when this is not the case, it should be noted that, as soon as the parameter P includes the setsharing domain of Jacobs and Langen [28] then it will subsume the groundness information captured by the domain Def [9, 14]. 11 Proof. G ( 1 2 ) 2 RSubst 8 2 # : 1 2 ) gval( 1 = 2 RSubst 8 2 # : 8i 2 f1; 2g : i gval( 1 ) 2 RSubst 8 2 # : 1 gval( 1 2 RSubst 8 2 # : 2 gval( 1 = G( 1 ) G( 2 ) The following is a ....

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163-192, 1998.

....behavior in the analysis process is indistinguishable with respect to the pair sharing property. As a final remark, the technique we use to extract from SS the component that is relevant in order to compute pair sharing is very similar to the one introduced by Cortesi, File, and Winsborough in [7], even though the formal definitions are slightly di#erent. 4 Star union is not needed The theory developed in the previous section has at least one practical consequence: in the definition of the abstract unification for domain X, the star union operator can be safely replaced by the ....

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. In Proceedings of GULP-PRODE'94, pages 372--397, Penscola, Spain, September 1994.

....that PSD also captures groundness. Finally, letting k = n, we observe that TSD n = SH . In [1] the PSD domain is shown to be as good as SH for both representing and propagating pair sharing. It is also proved that any weaker domain does not satisfy these properties, so that PSD is the quotient [6] of SH with respect to the pair sharing property PS . In the view of recent results on abstract domain completeness [9] this also means that PSD is the least fully complete extension (lfce) of PS with respect to SH . From a purely theoretical point of view, the quotient of an abstract ....

....is the least fully complete extension (lfce) of PS with respect to SH . From a purely theoretical point of view, the quotient of an abstract interpretation with respect to a property of interest and the least fully complete extension of an upper closure operator are not equivalent. It is known [6] that the quotient may not exist, while the lfce is always de ned. However, it is also known [10] that when the quotient exists it is exactly the same as the lfce. Moreover, it should be noted that the quotient will exist as long as we consider a semantics where at least one of the domain ....

[Article contains additional citation context not shown here]

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163{ 192, 1998.

....Vars of variables of interest is required. For example, in the Ph.D. thesis of Langen (Langen 1990) this set is implicitly de ned, for each clause being analyzed, as the nite set of variables occurring in that clause. A clearer approach has been introduced in (Cortesi, Fil e and Winsborough 1994, Cortesi, Fil e and Winsborough 1998) and also adopted in (Bagnara et al. 1997, Bagnara et al. 2001, Cortesi and Fil e 1999) where the set of variables of interest is given explicitly as a component of the abstract domain. During the analysis process, this set is elastic. That is, it expands (e.g. when solving clause s bodies) and ....

....element a new component, representing the set of variables of interest. It is shown that SS is as good as SS for both representing and propagating pair sharing and it is also proved that any weaker domain does not satisfy these properties, so that SS is the quotient (Cortesi et al. 1994, Cortesi et al. 1998) of SS with respect to the pair sharing property PS . We now generalize and strengthen the results in (Bagnara et al. 1997, Bagnara et al. 2001) and show that, for each k 2 f1; ng, TSD k is the quotient of SH with respect to the reduced product TS 1 u u TS k . These results are ....

[Article contains additional citation context not shown here]

Cortesi, A., File, G. and Winsborough, W. (1998). The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202(1&2): 163-192.

....behavior in the analysis process is indistinguishable with respect to the pair sharing property. As a final remark, the technique we use to extract from SS the component that is relevant in order to compute pair sharing is very similar to the one introduced by Cortesi, Fil e, and Winsborough in [7], even though the formal definitions are slightly different. 4 Star union is not needed The theory developed in the previous section has at least one practical consequence: in the definition of the abstract unification for domain X, the star union operator can be safely replaced by the ....

A. Cortesi, G. Fil'e, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. In Proceedings of GULP-PRODE'94, pages 372--397, Pe~n'iscola, Spain, September 1994.

....that PSD also captures groundness. Finally, letting k = n, we observe that TSD n = SH . In [1] the PSD domain is shown to be as good as SH for both representing and propagating pair sharing. It is also proved that any weaker domain does not satisfy these properties, so that PSD is the quotient [6] of SH with respect to the pair sharing property PS . In the view of recent results on abstract domain completeness [9] this also means that PSD is the least fully complete extension (lfce) of PS with respect to SH . From a purely theoretical point of view, the quotient of an abstract ....

....is the least fully complete extension (lfce) of PS with respect to SH . From a purely theoretical point of view, the quotient of an abstract interpretation with respect to a property of interest and the least fully complete extension of an upper closure operator are not equivalent. It is known [6] that the quotient may not exist, while the lfce is always defined. However, it is also known [10] that when the quotient exists it is exactly the same as the lfce. Moreover, it should be noted that the quotient will exist as long as we consider a semantics where at least one of the domain ....

[Article contains additional citation context not shown here]

A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163-- 192, 1998.

....of variables of interest is required. For example, in the Ph.D. thesis of Langen (Langen 1990) this set is implicitly defined, for each clause being analyzed, as the finite set of variables occurring in that clause. A clearer approach has been introduced in (Cortesi, File and Winsborough 1994, Cortesi, File and Winsborough 1998) and also adopted in (Bagnara et al. 1997, Bagnara et al. 2001, Cortesi and File 1999) where the set of variables of interest is given explicitly as a component of the abstract domain. During the analysis process, this set is elastic. That is, it expands (e.g. when solving clause s bodies) and ....

....element a new component, representing the set of variables of interest. It is shown that SS # is as good as SS for both representing and propagating pair sharing and it is also proved that any weaker domain does not satisfy these properties, so that SS # is the quotient (Cortesi et al. 1994, Cortesi et al. 1998) of SS with respect to the pair sharing property PS . We now generalize and strengthen the results in (Bagnara et al. 1997, Bagnara et al. 2001) and show that, for each k # 1, n , TSD k is the quotient of SH with respect to the reduced product TS 1 ##TS k . These results are proved at the ....

[Article contains additional citation context not shown here]

Cortesi, A., File, G. and Winsborough, W. (1998). The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202(1&2): 163--192.

....goal dependent groundness (and de niteness) analysis. We found that Def was as precise as Pos for all our realistic Prolog and CLP(R) benchmarks. We build on this and demonstrate how Def can be implemented eciently and coded succinctly in Prolog. Our starting point is the work of Cortesi et al. [15, 16] that shows that Share , which is a domain whose elements are sets of sets of variables, can be used to encode Def . We develop this to show: how the meet and join operations of Def can be computed straightforwardly based on this encoding, without the closure operation of Share [22] that has a ....

....= fv; x; zg, and occ( b ; x) occ( b ; y) occ( b ; z) The reader is encouraged to verify that # b 2 sh X (S b ) 3 Quotienting Share X to obtain Def X In this section we construct a homomorphism from Share X to Def X . We recall the well known connection between Share X and Def X [13, 14, 15, 16]. For the elements of Share X , we de ne an abstraction X which interprets a sharing abstraction as representing a set of models and hence a Boolean function. De nition 6 X . The (abstraction) map X : Share X Def X is de ned as follows: X (S) model 1 X (fX n ( S 0 ) j S 0 Sg) ....

[Article contains additional citation context not shown here]

A. Cortesi, G. File, and W. Winsborough. The Quotient of an Abstract Interpretation for Comparing Static Analyses. Theoretical Computer Science, 202(1{ 2):163-192, 1998.

....if they are bound to terms that contain a common variable. The accuracy of Share stems, in part, from its ability to track groundness dependencies. This is because ground variables cannot share with other variables. Interestingly, Share can encode groundness dependencies to the accuracy of Def [12, 13]. In other words, Share is equivalent to the reduced product domain Def Theta Sharing [11] in which Def just traces groundness dependencies and Sharing just traces possible sharing. This paper develops the connection between Share and Def to show how the meet operation of Def (as well ....

....words, Share is equivalent to the reduced product domain Def Theta Sharing [11] in which Def just traces groundness dependencies and Sharing just traces possible sharing. This paper develops the connection between Share and Def to show how the meet operation of Def (as well as the join [12, 13]) can be computed straightforwardly with a Share based set of sets representation. Previous studies of Def and Share have either focused on domain comparison (and quotienting) 12, 13] or domain decomposition [11] and, to the best of our knowledge, have not explored Share as a medium for efficient ....

[Article contains additional citation context not shown here]

A. Cortesi, G. Fil'e, and W. Winsborough. The Quotient of an Abstract Interpretation for Comparing Static Analyses. In Joint Conference on Declarative Programming (GULP-PRODE '94), pages 372--397, 1994. An extended version will appear in Theoretical Computer Science.

....behavior in the analysis process is indistinguishable with respect to the pair sharing property. As a final remark, the technique we use to extract from SS the component that is relevant in order to compute pair sharing is very similar to the one introduced by Cortesi, Fil e, and Winsborough in [6], even though the formal definitions are slightly different. 4 Star union is not needed The theory developed in the previous section has at least one practical consequence: in the definition of the abstract unification for domain X, the starunion operator can be safely replaced by the ....

A. Cortesi, G. Fil'e, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. In M. Alpuente, R. Barbuti, and I. Ramos, editors, Proceedings of the "1994 Joint Conference on Declarative Programming (GULP-PRODE '94)", pages 372--397, Pe~n'iscola, Spain, September 1994.

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A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. In GULP-PRODE'94 Joint Conference on Declarative Programming, pages 372--397, 1994.

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A. Cortesi, G. File, W. Winsborough, The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202 (1&2) (1998) 163--192.

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A. Cortesi, G. File, W. Winsborough, The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202 (1&2) (1998) 163--192.

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A. Cortesi, G. File, W. Winsborough, The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202 (1&2) (1998) 163--192.

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A. Cortesi, G. File, W. Winsborough, The quotient of an abstract interpretation for comparing static analyses, Theoretical Computer Science 202 (1&2) (1998) 163--192.

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A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163--192, 1998.

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A. Cortesi, G. File, and W. Winsborough. The quotient of an abstract interpretation for comparing static analyses. Theoretical Computer Science, 202(1&2):163-192, 1998.

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