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Basic abstract interpretation theory

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any definition style, from a denotational definition to a ... abstract domain (Sign, , bot, top, lub, glb) 5. Concretization. concrete domain (P(C), , , C, ... – PowerPoint PPT presentation

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Title: Basic abstract interpretation theory


1
Basic abstract interpretation theory
2
The general idea
  • a semantics
  • any definition style, from a denotational
    definition to a detailed interpreter
  • assigning meanings to programs on a suitable
    concrete domain (concrete computations domain)
  • an abstract domain modeling some properties of
    concrete computations and forgetting about the
    remaining information (abstract computations
    domain)
  • we derive an abstract semantics, which allows us
    to execute the program on the abstract domain
    to compute its abstract meaning, i.e., the
    modeled property

3
Concrete and Abstract Domains
  • two complete partial orders
  • the partial orders reflect precision
  • smaller is better
  • concrete domain (P(C), ?, ?, C, ?, ?)
  • has the structure of a powerset
  • we will see later why
  • abstract domain (A, ?, bottom, top, lub, glb)
  • each abstract value is a description of a set
    of concrete values

4
The Sign Abstract Domain
  • concrete domain (P(Z), ?, ?, C, ?, ? )
  • sets of integers
  • abstract domain (Sign, ?, bot, top, lub, glb)

5
Concretization
  • concrete domain (P(C), ?, ?, C, ?, ?)
  • abstract domain (A, ?, bottom, top, lub, glb)
  • the meaning of abstract values is defined by a
    concretization function
  • ? A ? P(C)
  • ?a? A, ?(a) is the set of concrete computations
    described by a
  • thats why the concrete domain needs to be a
    powerset
  • the concretization function must be monotonic
  • ?a1,a2 ? A, a1 ? a2 implies ?(a1) ? ?(a2)
  • concretization preserves relative precision

6
Abstraction
  • concrete domain (P(C), ?, ?, C, ?, ?)
  • abstract domain (A, ?, bottom, top, lub, glb)
  • every element of P(C) should have a unique best
    (most precise) description in A
  • this is possible if and only if A is a Moore
    family
  • closed under glb
  • in such a case, we can define an abstraction
    function
  • a P(C) ? A
  • ?c? P(C), a(c) is the best abstract description
    of c
  • the abstraction function must be monotonic
  • ?c1,c2 ? P(C), c1 ? c2 implies a(c1) ? a(c2)
  • abstraction preserves relative precision

7
The example of Sign
  • ?Sign (x)
  • ?, if x bot
  • yygt0, if x
  • yy?0, if x 0
  • 0, if x 0
  • yy?0, if x 0-
  • yylt0, if x -
  • Z, if x top
  • ?Sign (y) glb of
  • bot , if y ?
  • - , if y ? yylt0
  • 0- , if y ? yy?0
  • 0 , if y 0
  • 0 , if y ? yy ? 0
  • , if y ? yygt0
  • top , if y ? Z

8
Galois connection
  • (P(C), ?, ?, C, ?, ? )
  • (A, ?, bottom, top, lub, glb)
  • ? A ? P(C) (concretization)
  • a P(C) ? A (abstraction)
  • ? , ? monotonic
  • there may be loss of information (approximation)
    in describing an element of P(C) by an element of
    A
  • Galois connection (insertion)
  • ?c? P(C). c ? ?(?(c))
  • ?a? A. ?(?(a)) ? a (?a? A. ?(?(a)) a)
  • ? , ? mutually determine each other

9
Concrete semantics
  • the concrete semantics is defined as the least or
    (greatest) fixpont of a concrete semantic
    evaluation function F defined on the domain C
  • this does not necessarily mean that the semantic
    definition style is denotational!
  • F is defined in terms of primitive semantic
    operations fi on C
  • the abstract semantic evaluation function is
    obtained by replacing in F each concrete
    operation fi by a suitable abstract operation
  • however, since the actual concrete domain is
    P(C), we need first to lift the concrete
    semantics lfp F to a collecting semantics defined
    on P(C)

10
Collecting semantics
  • lifting lfp F to the powerset (to get the
    collecting semantics) is simply a conceptual
    operation
  • collecting semantics lfp F
  • we dont need to define a brand new collecting
    semantic evaluation function on P(C)
  • we just need to reason in terms of liftings of
    all the primitive operations (and of F), while
    designing the abstract operations and
    establishing their properties
  • in the following, by abuse of notation, we will
    use the same notation for the standard and the
    collecting (conceptually lifted) operations

11
Abstract operations local correctness
  • an abstract operator fi? defined on A is locally
    correct wrt a concrete operator fi if
  • ?x1,..,xn ? P(C).
  • fi (x1,..,xn) ? ?(fi? (?(x1),..,?(xn)))
  • the concrete computation step is more precise
    than the concretization of the corresponding
    abstract computation step
  • a very weak requirement, which is satisfied, for
    example, by an abstract operator which always
    computes the worst abstract value top
  • the real issue in the design of abstract
    operations is therefore precision

12
Abstract operations optimality and completeness
  • correctness
  • ?x1,..,xn ? P(C).
  • fi (x1,..,xn) ? ?(fi? (?(x1),..,?(xn)))
  • optimality
  • ?y1,..,yn ? A .
  • fi? (y1,..,yn) a(fi (g(y1),..,g(yn)))
  • the most precise abstract operator fi? correct
    wrt fi
  • a theoretical bound and basis for the design,
    rather then an implementable definition
  • completeness (exactness or absolute precision)
  • ?x1,..,xn ? P(C).
  • a(fi (x1,..,xn)) fi? (?(x1),..,?(xn))
  • no loss of information, the abstraction of the
    concrete computation step is exactly the same as
    the result of the corresponding abstract
    computation step

13
Abstract operations on Sign TimesSign
14
Abstract operations on Sign PlusSign
15
The Sign example
  • Times and Plus are the usual operations lifted to
    P(Z)
  • both TimesSign and PlusSign are optimal (hence
    correct)
  • TimesSign is also complete (no approximation)
  • PlusSign is necessarily incomplete
  • ?Sign(Times(2,-3))
  • TimesSign(?Sign(2),?Sign(-3))
  • ?Sign(Plus(2,-3)) ?
  • PlusSign(?Sign(2),?Sign(-3))

16
From local to global correctness
  • the composition of locally correct abstract
    operations is locally correct wrt the composition
    of concrete operations
  • composition does not preserve optimality, i.e.,
    the composition of optimal operators may be less
    precise than the optimal abstract version of the
    composition
  • if we obtain F? (abstract semantic evaluation
    function) by replacing in F every concrete
    semantic operation by a corresponding (locally
    correct) abstract operation, the local
    correctness property still holds
  • ?x ? P(C). F (x) ? ?(F? (?(x)))
  • local correctness implies global correctness,
    i.e., correctness of the abstract semantics wrt
    the concrete one
  • lfp F ? ?(lfp F? ) gfp F ? ?(gfp F? )
  • a(lfp F ) ? lfp F? a(gfp F ) ? gfp F?
  • the abstraction of the concrete semantics is more
    precise than the abstract semantics

17
a (lfp F ) ? lfp F? why computing lfp F? ?
  • lfp F cannot be computed in finitely many steps
  • ? steps are in general required
  • lfp F? can be computed in finitely many steps, if
    the abstract domain is finite or at least
    noetherian
  • does not contain infinite increasing chains
  • interesting for static program analysis, where
    the fixpoint computation must terminate
  • most program properties considered in static
    analysis are undecidable
  • we accept a loss of precision (safe
    approximation) in order to make the analysis
    feasible

18
Where does the approximation come from?
  • incomplete abstract operators
  • more execution paths in the abstract semantics
  • the abstract state has no information to allow
    deterministic choices
  • conditionals, pattern matching, etc.
  • the set of resulting abstract states is
    transformed into a single abstract state by an
    abstract lub operation

19
Approximation in abstract Sign computations
  • concrete state x3
  • if xgt2 then y3 else y-5
  • concrete state x3, y3
  • abstract state x
  • if xgt2 then y3 else y-5
  • the abstract guard can be both true and false
  • we need to abstractly execute both paths
  • the resulting abstract states are merged by
    performing a lub on Sign
  • abstract state x,ytop

20
Approximation in type analysis
  • the following ML expression is not typed by the
    MLs type inference algorithm, because it always
    performs a lub operation in the conditional
  • if true then 3 else true
  • even when the guard is valid or unsatisfiable in
    the abstract state

21
Applications of Abstract Interpretation
  • comparative semantics
  • a technique to reason about semantics at
    different level of abstraction
  • non-noetherian abstract domain
  • abstraction without approximation (completeness)
  • ? (lfp F) lfp F?
  • static analysis effective computation of the
    abstract semantics
  • if the abstract domain is noetherian and the
    abstract operations are computationally feasible
  • if the abstract domain is non-noetherian or if
    the fixpoint computation is too complex
  • use widening operators
  • which effectively compute an (upper)
    approximation of lfp F?
  • one example later

22
The abstract interpretation framework
  • (P(C), ?, ?, C, ?, ? ) (concrete domain)
  • (A, ?, bottom, top, lub, glb) (abstract domain)
  • ? A ? P(C) monotonic (concretization function)
  • a P(C) ? A monotonic (abstraction function)
  • ?x?P(C). x ? ?(?(x))
  • ?y? A. ?(?(y)) ? y (Galois connection)
  • ? fi fi? ?x1,..,xn ? P(C).
  • fi (x1,..,xn) ? ?(fi? (?(x1),..,?(xn))) (local
    correctness)
  • critical choices
  • the abstract domain to model the property
  • the (possibly optimal) correct abstract operations

23
Other approaches and extensions
  • there exist weaker versions of abstract
    interpretation
  • without Galois connections (e.g., concretization
    function only)
  • based on approximation operators (widening,
    narrowing)
  • without explicit abstract domain (closure
    operators)
  • the theory provides also several results on
    abstract domain design
  • how to combine domains
  • how to improve the precision of a domain
  • how to transform an abstract domain into a
    complete one
  • ...
  • we will look at some of these results in the last
    lecture
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