Introduction. The concept of abstract interpretation has been introduced by Patrick and Radhia Cousot in [4, 5], in order to formalize static program analyses. Within this framework, our goal is to offer a unifying view on operators for enhancing and simplifying abstract domains. Enhancing and simplifying operators are viewed, respectively, as domain refinements and inverses of domain refinements. This new unifying viewpoint make

Francesco Ranzato

Gilberto Filé

Roberto Giacobazzi

The concept of abstract interpretation has been introduced by Patrick and Radhia Cousot in 1977, in order to formalize static program analyses. Within this framework, our goal is to offer a unifying view on operators for enhancing and simplifying abstract domains. Enhancing and simplifying operators are viewed, respectively, as domain refinements and inverses of domain refinements. This new unifying viewpoint makes both the understanding and the design of operators on abstract domains much simpler. Enhancing operators increase the expressiveness of an abstract domain: they comprise the Cousot and Cousot reduced product, disjunctive completion and reduced cardinal power, the Nielson tensor product, the open product and the pattern completion by Cortesi et al., and the functional dependencies by Giacobazzi and Ranzato (see here). Simplifying operators are used to reduce complex abstract domains into simpler ones with respect to the efficiency of the corresponding analysis as well as with respect to the proof of their correctness. Simplifying operators comprise the complementation of Cortesi et al. (see here) and the Giacobazzi and Ranzato least disjunctive basis

Gilberto Filé

GIACOBAZZI, Roberto

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A Unifying View of Abstract Domain Design

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A Unifying View of Abstract Domain Design
Gilberto Fil´ e∗ Roberto Giacobazzi∗∗ Francesco Ranzato∗
∗Dipartimento di Matematica Pura ed Applicata, Universit` a di Padova
Via Belzoni 7, 35131 Padova, Italy
{gilberto,franz}@hilbert.math.unipd.it
∗∗Dipartimento di Informatica, Universit` a di Pisa
Corso Italia 40, 56125 Pisa, Italy
giaco@di.unipi.it
Introduction. The concept of abstract interpretation has been introduced by Patrick
and Radhia Cousot in [4, 5], in order to formalize static program analyses. Within this
framework, our goal is to oﬀer a unifying view on operators for enhancing and simplifying
abstract domains. Enhancing and simplifying operators are viewed, respectively, as do-
main reﬁnements and inverses of domain reﬁnements. This new unifying viewpoint makes
both the understanding and the design of operators on abstract domains much simpler.
Enhancing operators increase the expressiveness of an abstract domain: they comprise the
Cousot and Cousot reduced product, disjunctive completion and reduced cardinal power
([5]), the Nielson tensor product ([9]), the open product and the pattern completion by
Cortesi et al. ([3]), and the functional dependencies by Giacobazzi and Ranzato ([7]).
Simplifying operators are used to reduce complex abstract domains into simpler ones with
respect to the eﬃciency of the corresponding analysis as well as with respect to the proof
of their correctness. Simplifying operators comprise the complementation of Cortesi et al.
([2]) and the Giacobazzi and Ranzato least disjunctive basis ([8]).
Domain Reﬁnements and their Inversion. Program analysis is deﬁned in abstract
interpretation as non-standard program evaluation. This non-standard semantics is ob-
tained from the standard one by substituting the actual domain of computation (called
concrete) and its basic operations with, respectively, an abstract domain and correspond-
ing abstract operations. Both the concrete and the abstract domain are required to be
complete lattices, where the ordering relations describe the relative precision of the de-
notations – the top elements representing no information. An abstraction of a concrete
domain C can be viewed as an upper closure operator on C, i.e., a function ρ : C → C
which associates with each x ∈ C, an object ρ(x) approximating x (i.e. x ≤ ρ(x)), and
which is both monotonic and idempotent (cf. [5]). Therefore, any domain A which is
isomorphic to the image ρ(C) by an upper closure operator ρ on C, can be considered a
proper abstraction of C. Once a concrete domain C has been ﬁxed, all its possible abstrac-
tions can be compared with each other with respect to their precision of representation.
This order corresponds in the most natural way to the usual functional pointwise order
1between closures (denoted by ⊑), which makes the set uco(C) of all closure operators on
C (or equivalently the set of all abstract interpretations of C) a complete lattice.
We introduce the notion of domain reﬁnement as a general scheme in order to for-
malize enhancing operators on abstract domains. A domain reﬁnement is a mapping
ℜ : uco(C) → uco(C) such that for any abstract domain A: ℜ(A) contains more infor-
mation than A (i.e., ℜ(A) ⊑ A), ℜ monotonically depends on the information contained
in its argument, namely it is monotonic, and a last very reasonable requirement is that ℜ
upgrades all at once, namely ℜ is idempotent. This clearly deﬁnes reﬁnements as lower
closure operators on uco(C).
A natural question that arises in this setting is whether it is possible to deﬁne the
inverse of a domain reﬁnement. For a given reﬁnement ℜ, the inverse ℜ−1 (if it exists) is
a function mapping any domain A into the (unique) most abstract domain ℜ−1(A) such
that ℜ(ℜ−1(A)) = ℜ(A). Whenever this happens, we say that ℜ−1(A) is the optimal
basis for the domain A and reﬁnement ℜ. The intuition is that the reﬁned domain ℜ(A)
can be systematically reconstructed by applying the reﬁnement ℜ to the more abstract,
and therefore simpler, domain ℜ−1(A). This domain ℜ−1(A) is in fact the most abstract
domain for which this condition holds.
Not all domain reﬁnements are invertible. A simple and meaningful example is pro-
vided by the following notion of completion by complements: a reﬁnement ℜ¬ which, un-
der certain hypotheses, upgrades a given abstract domain by adding denotations for the
lattice-theoretic complements of its elements. If Sign±, A1 and A2 are the domains for sign
analysis of integer variables depicted below, with their obvious meanings as upper closures
on the subsets of integers ℘(Z Z), ordered by inclusion, then ℜ¬(A1) = ℜ¬(A2) = Sign±,
but the common abstraction of A1 and A2, denoted A1 ⊔ A2, is {Z Z,∅} and ℜ¬({Z Z,∅}) =
{Z Z,∅}  = Sign±. Thus, ℜ−1
¬ does not exist.
•
• •
• Z Z
− +
∅
Sign±
￿
￿
@
@
￿
￿
@
@
•
•
•
∅
−
Z Z
A1
•
•
•
∅
+
Z Z
A2
•
•
∅
Z Z
ℜ¬(A1 ⊔ A2)
Two Examples. The most widely known domain reﬁnement is reduced product ([5]).
The reduced product of two abstract domains A and B is the domain obtained by com-
bining the informations in A and B via the concrete g.l.b. of their elements. Viewing
abstractions as uco’s, the reduced product is simply the g.l.b. in the complete lattice
uco(C), denoted A⊓B. Reduced product can be viewed as a reﬁnement: given a domain
A ∈ uco(C), λX.X ⊓ A is trivially a lower closure operator on uco(C). If A+, A− and
Sign are the domains depicted below, then Sign is the reduced product of A+ and A−, i.e.
Sign = A+ ⊓ A−. Note that the denotations 0 and ∅ are both derived from A+ and A−
by combining, through set intersection, 0+ and 0−, and, + and −, respectively.
•
•
•
+
0+
Z Z
A+
•
•
•
−
−0
Z Z
A−
•
•
•
•
•
•
• −
−0
∅
0
Z Z
0+
+
Sign
H H H
￿￿￿ ￿ ￿ @ @ ￿ ￿ @ @
￿ ￿ @ @
2The practical impact of reduced product has been experimentally shown by Codish et al.
in [1] for the analysis of logic programs. Reduced product can be inverted. In fact, it
corresponds precisely to the operation of complementation introduced in [2]. Given any
two domains A and B in uco(C), with A ⊑ B (i.e., B more abstract than A), under
non restrictive hypotheses, complementation of B in A gives as result the most abstract
domain A ∼ B, whose reduced product with B is exactly A, i.e. B ⊓ (A ∼ B) = A. It
is simple to observe that the existence of the complement implies the existence of the
inverse of the reduced product reﬁnement. If we consider the reﬁnement λX.X ⊓A, where
A ∈ uco(C), then the optimal basis for any domain B ∈ uco(C) is exactly (B ⊓ A)∼A.
As a simple example, consider the domains Sign, A+ and A− introduced above. A+ and
A− can be “subtracted” from Sign by complementation, obtaining Sign ∼A+ = A− and
Sign∼A− = A+.  A+,A−  is therefore a decomposition for Sign (i.e., Sign = A+ ⊓ A−).
Using complementation it is possible to decompose any domain just like we have done
for Sign. Such decompositions can both improve the space requirement for representing
abstract domains (compare  A+,A−  with Sign), and provide a divide et impera approach
to prove properties of complex abstract domains by exploiting properties of their factors.
A further example of invertible domain reﬁnement is disjunctive completion ℜ∨ ([5]).
An abstract domain is here upgraded by including denotations for concrete disjunctions
of its values. This is achieved by suitable powerset completions. It turns out that ℜ∨ is a
reﬁnement. Giacobazzi and Ranzato proved in [8] that, under non restrictive hypotheses
on the concrete domain, ℜ−1
∨ exists, and for a domain A, ℜ−1
∨ (A) depends on the set of the
join-irreducible elements of ℜ∨(A). The disjunctive completion ℜ∨(Sign) (with respect to
℘(Z Z)) and its optimal basis for ℜ∨ are depicted below, the optimal basis being a proper
abstraction of Sign.
∅
− 0 +
−0 0+
 = 0
Z Z
ℜ∨(Sign)
•
• •
•
•
•
•
•
l
l l
,
, ,
,
, ,
l
l l
,
, ,
l
l l
,
, ,
l
l l
•
• • •
• Z Z
− 0 +
∅
ℜ
−1
∨ (Sign)
,
, ,
l
l l
,
, ,
l
l l
Thus, both Sign and its optimal basis lead to the same disjunctive domain for sign-analysis,
but the latter is a less expensive domain. It turns out that 0+ = {0,+}, −0 = {0,−} and
 = 0 = {+,−} can all be systematically reconstructed by disjunctions from elements of the
optimal basis.
Research Directions. Many operators for domain design have been deﬁned in the
literature. It could be interesting to formalize them as reﬁnements and to ﬁnd their
optimal bases, in particular, for the operators of tensor product, open product, functional
dependencies and pattern completion.
Reﬁnements could be a basis for an environment for developing expressive abstract
domains. In such an environment, optimal bases represent the least eﬀort that a domain
designer has to provide in order to obtain a fully reﬁned domain. Such a situation is
considered by Cortesi et al. in [3] for the case of pattern completion in logic program
analysis. Using their environment, designers need only to contribute about 20% of the
3whole implementation. A case study of the cost savings when domains are automatically
upgraded from their optimal bases is one important direction for future research.
Domain reﬁnements and their inverses lead to an algebra of domains for program
analysis. A natural question that may arise in this setting is whether it is possible to
deﬁne abstract domains as solutions of equations on this algebra. This would provide
an equational-like presentation of abstract domains, similarly to what is well known in
standard domain theory, e.g. for denotational semantics. This algebra may also provide
systematic methods to design semantics for programming languages. Cousot and Cousot
proved in [6] that it is possible to relate diﬀerent semantic deﬁnitions, at diﬀerent levels of
abstraction, by abstract interpretation. The algebra of basic operators for domain reﬁne-
ment and their inverses would provide here new methods to derive semantic deﬁnitions.
References
[1] M. Codish, A. Mulkers, M. Bruynooghe, M. Garc´ ıa de la Banda, and M. Hermenegildo.
Improving abstract interpretations by combining domains. ACM Trans. on Program. Lang.
and Systems, 17(1):28–44, 1995.
[2] A. Cortesi, G. Fil´ e, R. Giacobazzi, C. Palamidessi, and F. Ranzato. Complementation in
abstract interpretation. In A. Mycroft, ed., Proc. of the 2nd Static Analysis Symp., Lecture
Notes in Comp. Sc. 983, pp. 100–117. Springer-Verlag, 1995.
[3] A. Cortesi, B. Le Charlier, and P. Van Hentenryck. Combinations of abstract domains for logic
programming. In Proc. of the 21st ACM Symp. on Principles of Programming Languages, pp.
227–239. ACM Press, 1994.
[4] P. Cousot and R. Cousot. Abstract interpretation: a uniﬁed lattice model for static analysis
of programs by construction or approximation of ﬁxpoints. In Proc. of the 4th ACM Symp.
on Principles of Programming Languages, pp. 238–252. ACM Press, 1977.
[5] P. Cousot and R. Cousot. Systematic design of program analysis frameworks. In Proc. of the
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[6] P. Cousot and R. Cousot. Inductive deﬁnitions, semantics and abstract interpretation. In
Proc. of the 19th ACM Symp. on Principles of Programming Languages, pp. 83–94. ACM
Press, 1992.
[7] R. Giacobazzi and F. Ranzato. Functional dependencies and Moore-set completions of abstract
interpretations and semantics. In J. Lloyd, ed., Proc. of the 1995 Int. Logic Programming
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[8] R. Giacobazzi and F. Ranzato. Compositional optimization of disjunctive abstract interpreta-
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[9] F. Nielson. Tensor products generalize the relational data ﬂow analysis method. In M. Arat´ o
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4

A unifying view of abstract domain design

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ACM Computing Surveys

The concept of abstract interpretation was introduced by Patrick and Radhia Cousot [1977, 1979] in order to formal- ize static program analyses. Within this framework, our goal is to offer a unify- ing view on operators for enhancing and simplifying abstract domains. Enhanc- ing and simplifying operators are viewed, respectively, as domain refine- ments and inverses of domain refine- ments. This new unifying viewpoint makes both the understanding and the design of operators on abstract domains much simpler

A unifying view of abstract domain design

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