N.D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood, Chichester, U.K., 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in this domain consist of conjunctions of variables and equivalences between variables. For example, x 1 x 2 ) x 3 (x 1 x 4 ) We call this class of formulae EP os. Like P os, EP os is ordered by j= It is interesting to note that EP os is only slightly richer than its subdomain Con [15, 18] which consists of conjunctions of variables. However, as we shall see, EP os gives much greater precision than Con for groundness analysis of logic programs. Moreover, it shares with Con the important property that its longest chain has linear length. The proof of this result relies on the ....

N. Jones and H. Sndergaard. A Semantics-based Framework for the Abstract Interpretation of Prolog. In Abstract Interpretation of Declarative Languages, pages 123-142. Ellis Horwood Limited, 1987.

....(note that 15 (j j) I replaces all variable occurrences by Omega s) a binding x 7 ffi in (joej) I is ground whenever ffi is ground. We represent groundness information concerning a set of substitutions Phi by means of the subset X V of variables which are grounded in every 2 Phi [JS87], i.e. D # = V ) We define g : Val (j j) I ) V ) by g ( Phi) fx 2 V j 8 2 Phi; x) is groundg. Note that g is (vEM ; monotone: if Phi vEM Phi 0 , then 8 0 2 Phi 0 ; 9 2 Phi such that v 0 . On the other hand, if x 2 g ( Phi) then every 2 Phi grounds ....

N.D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of PROLOG. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pp. 123--142. Ellis Horwood, 1987.

.... partial evaluation [8, 17, 18, 19, 38, 37, 39] Some of the techniques used here, such as the encoding of computations via an and or tree [15] or the approximation of these computations using lattice theoretic notions, are common to these approaches, especially those based on top down analysis [23, 24, 34]. However, the work presented here differs from analogous work in traditional Logic Programming from two main perspectives: i) we are interested in the resource manipulation aspects of computations, which we formalize via sequent proofs in Linear Logic; hence, in our static analysis tool, we ....

N.D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood, Chichester, U.K., 1987.

....constructed by using a size relation as in [25] for terms or by defining a degree for substitutions as in [17] 4 , and is further based on the finiteness of the set of possible annotations for abstract variables. Consequently, when AEqs j, is used in a framework of abstract interpretation (e.g. [3, 4, 8, 15, 22, 28]) a terminating analysis is guaranteed. Example 2.2 Let AEqs 1 V = hfX = f(Y )g; i and AEqs 2 V = hfX = A 1 ; Y = A 2 g; ffA 1 ; A 2 ggi with V = fX; Y g, then AEqs 1 V AEqs 2 V and AEqs 2 V 6 AEqs 1 V . On the other hand, let AEqs 1 V 0 = hfX = f(Y )g; i and AEqs 2 V ....

N. D. Jones and H. Sndergaard. A semantic-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, Ellis Horwood Series in Computers and their Applications, pages 123--142. Ellis Horwood, Chichester, 1987.

....bindings for variables belonging to X 2 U are ground. This formally relates groundness and termination: groundness is the logic property which corresponds to the functional property of termination. In fact, 2 V is a well known abstract domain for groundness analysis in logic programming [JS87]. If C has constructors with positive arity, then h 1 t (f g) is the set of constructor rooted values (they correspond to terms having a constructorrooted head normal form) In this case, h 1 g (U) for a given open set U is a set of substitutions whose bindings for variables belonging to X 2 ....

N.D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of PROLOG. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pp. 123-142. Ellis Horwood, 1987. 40

....aliasing and ground dependencies could be represented with very limited accuracy. As far as ground dependencies are concerned, the domain of Chang was improved by Citrin [11] Jones and Sndergaard described an abstract domain constituted by sets of pairs of clause variables that might be aliased [44]. An approach that is essentially equivalent was introduced by Debray [29] Here each clause variable is mapped to the set of variables with which it might share. These domains, compared to those defined by Chang and Citrin, capture the independence of variables with much greater accuracy. The ....

....sharing group S # for S # # S. Downward closed sharing sets can be e#ciently represented by means of the sets of their maximal elements. The resulting domain, called # JL, is, essentially, PS# Ground , where # denotes the reduced product and Ground is the simplest domain for groundness [44,50]. In fact, in # JL, ground dependencies and pair sharing dependencies are lost. To compensate this loss of precision, Fecht combines # JL with Pos and then with Lin, where Lin is the usual, simple domain for linearity. 8 For the analyses based on these combined domains, Fecht reports huge ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, chapter 6, pages 123--142. Ellis Horwood Ltd, West Sussex, England, 1987. 35

....elements be given, for each X; Y 2 Vars of the same cardinality, by d X Y def = ae i f i ( X) ff i ( Y ) j 1 i # X g j : The resulting domain (a closed and Noetherian d.c.s. is the simplest one for definiteness analysis, and it was used in early groundness analyzers [31,29]. The name Con comes from the fact that elements of the form fX 1 ; X n g 22 are usually regarded as the conjunction X 1 Delta Delta Delta X n , meaning that X 1 , X n are definitely bound to a unique value. In this view Omega corresponds to logical conjunction. Con is a ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

....be given, for each X, Y # Vars # of the same cardinality, by d X Y def = # # # i ( X) # # i ( Y ) 1 # i # # X # . The resulting domain (a closed and Noetherian d.c.s. is the simplest one for definiteness analysis, and it was used in early groundness analyzers [32,30]. The name Con comes from the fact that elements of the form X 1 , X n 22 are usually regarded as the conjunction X 1 # #X n , meaning that X 1 , X n are definitely bound to a unique value. In this view # corresponds to logical conjunction. Con is a very weak domain ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

....2 Preliminaries Let U be a set. The set of all subsets of U will be denoted by (U ) The set of all finite subsets of U will be denoted by f (U ) The notation S f T stands for S 2 f (T ) 2. 1 Boolean Functions for Groundness Analysis After the early approaches to groundness analysis [14, 12], which suffered from serious precision drawbacks, the use of Boolean functions [10, 13] has become customary in the field. The reason is that Boolean functions allow to capture in a very precise way the groundness dependencies that are implicit in unification constraints such as z = f(g(x) y) ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, chapter 6, pages 123--142. Ellis Horwood Ltd, West Sussex, England, 1987.

.... simply by means of ROBDDs, as in [1, 8] is not the best thing we can do (whence the title) Here we propose an hybrid implementation where each Pos element is represented by a pair: the first component is the set of true variables (just as in the domain used in early groundness analyzers [7, 6]) the second component is a ROBDD. In each element of this new representation there is no redundancy: the ROBDD component does not contain any information about true variables. This solution uses the more e#cient representation for each kind of information: surely ground variables are best ....

N. D. Jones and H. Sndergaard. A Semantics-based Framework for the Abstract Interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

....which represents groundness information, is isomorphic to a domain of conjunctions of Boolean variables. The isomorphism tuples 1 maps each element of Con to the set of variables that are possibly non ground. From the domain tuples 1 (Con) by set complementation, we obtain the classical domain G (Jones and S ndergaard 1987) for representing the set of variables that are de nitely ground (so that we have TS 1 def = Con G) 10 E. Za anella, P. M. Hill and R. Bagnara De nition 6 (The pair sharing domain PS . The upper closure operator PS : SH SH and the corresponding domain PS are de ned as PS def = ....

Jones, N. D. and Sndergaard, H. (1987). A semantics-based framework for the abstract interpretation of Prolog, in S. Abramsky and C. Hankin (eds), Abstract Interpretation of Declarative Languages, Ellis Horwood Ltd, West Sussex, England, chapter 6, pp. 123{ 142.

....show a rst non trivial solution to the problem, while Sect. 7 introduces the hybrid domain. The results of the experimental evaluation are reported in Sect. 8. Sect. 9 concludes with some nal remarks. 2 Boolean Functions for Groundness Analysis After the early approaches to groundness analysis [17, 15], which su ered from serious precision drawbacks, using Boolean functions has become customary in the eld. The reason is that Boolean functions allow to capture in a very precise way the groundness dependencies which are implicit in uni cation constraints such as z = f(g(x) y) the corresponding ....

....a quite complicated interaction between domains. Another application of groundness analysis with fast access to ground variables is for aliasing analysis. The most popular domain for this kind of analysis is Sharing [13] Without going into details, its strength over the previous approaches [15, 12] comes from the fact that it keeps track of groundness dependencies. In fact, Sharing has, as far as groundness information is concerned, the same power of Def. When Pos is used for groundness, using Sharing for aliasing at the same time is a waste: Sharing spends time and space for keeping track ....

[Article contains additional citation context not shown here]

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123-142. Ellis Horwood Ltd, 1987.

....aliasing and ground dependencies could be represented with very limited accuracy. As far as ground dependencies are concerned, the domain of Chang was improved by Citrin [11] Jones and S ndergaard described an abstract domain constituted by sets of pairs of clause variables that might be aliased [44]. An approach that is essentially equivalent was introduced by Debray [29] Here each clause variable is mapped to the set of variables with which it might share. These domains, compared to those de ned by Chang and Citrin, capture the independence of variables with much greater accuracy. The same ....

....the sharing group S 0 for S 0 S. Downward closed sharing sets can be eciently represented by means of the sets of their maximal elements. The resulting domain, called # JL, is, essentially, PS Ground , where denotes the reduced product and Ground is the simplest domain for groundness [44,50]. In fact, in # JL, ground dependencies and pair sharing dependencies are lost. To compensate this loss of precision, Fecht combines # JL with Pos and then with Lin, where Lin is the usual, simple domain for linearity. 8 For the analyses based on these combined domains, Fecht reports huge ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, chapter 6, pages 123-142. Ellis Horwood Ltd, West Sussex, England, 1987. 35

....a first non trivial solution to the problem, while Sect. 7 introduces the hybrid domain. The results of the experimental evaluation are reported in Sect. 8. Sect. 9 concludes with some final remarks. 2 Boolean Functions for Groundness Analysis After the early approaches to groundness analysis [17, 15], which su#ered from serious precision drawbacks, using Boolean functions has become customary in the field. The reason is that Boolean functions allow to capture in a very precise way the groundness dependencies which are implicit in unification constraints such as z = f(g(x) y) the ....

....a quite complicated interaction between domains. Another application of groundness analysis with fast access to ground variables is for aliasing analysis. The most popular domain for this kind of analysis is Sharing [13] Without going into details, its strength over the previous approaches [15, 12] comes from the fact that it keeps track of groundness dependencies. In fact, Sharing has, as far as groundness information is concerned, the same power of Def. When Pos is used for groundness, using Sharing for aliasing at the same time is a waste: Sharing spends time and space for keeping track ....

[Article contains additional citation context not shown here]

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

....2 Preliminaries Let U be a set. The set of all subsets of U will be denoted by #(U ) The set of all finite subsets of U will be denoted by # f (U ) The notation S # f T stands for S # # f (T ) 2. 1 Boolean Functions for Groundness Analysis After the early approaches to groundness analysis [14, 12], which su#ered from serious precision drawbacks, the use of Boolean functions [10, 13] has become customary in the field. The reason is that Boolean functions allow to capture in a very precise way the groundness dependencies that are implicit in unification constraints such as z = f(g(x) y) ....

N. D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, chapter 6, pages 123--142. Ellis Horwood Ltd, West Sussex, England, 1987.

....which represents groundness information, is isomorphic to a domain of conjunctions of Boolean variables. The isomorphism tuples 1 maps each element of Con to the set of variables that are possibly non ground. From the domain tuples 1 (Con) by set complementation, we obtain the classical domain G (Jones and Sndergaard 1987) for representing the set of variables that are definitely ground (so that we have TS 1 def = Con # G) 10 E. Za#anella, P. M. Hill and R. Bagnara Definition 6 (The pair sharing domain PS . The upper closure operator # PS : SH # SH and the corresponding domain PS are defined as # PS ....

Jones, N. D. and Sndergaard, H. (1987). A semantics-based framework for the abstract interpretation of Prolog, in S. Abramsky and C. Hankin (eds), Abstract Interpretation of Declarative Languages, Ellis Horwood Ltd, West Sussex, England, chapter 6, pp. 123-- 142.

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N.D. Jones and H. Sndergaard. A semantics-based framework for the abstract interpretation of prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood, Chichester, U.K., 1987.

No context found.

N. Jones and H. Sondergaard. A semantics-based framework for the abstract interpretation of prolog. In Abstract Interpretation of Declarative Languages, chapter 6, pages 124--142. EllisHorwood, 1987.

No context found.

N. Jones and H. Sondergaard. A semantics-based framework for the abstract interpretation of prolog. In Abstract Interpretation of Declarative Languages, chapter 6, pages 124--142. EllisHorwood, 1987.

No context found.

N. Jones and H. Sondergaard. A semantics-based framework for the abstract interpretation of prolog. In Abstract Interpretation of Declarative Languages, chapter 6, pages 124--142. EllisHorwood, 1987.

No context found.

N.D. Jones, H. Sondergaard, A Semantics-Based Framework for the Abstract Interpretation of Prolog, Ellis Horwood, Chichester, UK, 1987, pp. 123--142.

No context found.

N. D. Jones and H. Sndergaard. A Semantics-based Framework for the Abstract Interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

No context found.

N.D.Jones and H.Sndergaard. A Semantics-based Framework for the Abstract Interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

No context found.

N.D. Jones and H. Sndergaard. A Semantics-based Framework for the Abstract Interpretation of Prolog. In S. Abramsky and C. Hankin, editors, Abstract Interpretation of Declarative Languages, pages 123--142. Ellis Horwood Ltd, 1987.

No context found.

N. D. Jones and H. Sondergaard, A Semantics-Based Framework for the Abstract Interpretation of Prolog, Ellis Horwood, Chichester, 1987, pp. 123--142.

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