A program denotes computations in some universe of objects. Abstract interpretation of programs consists in using that denotation to describe computations in another universe of abstract objects, so that the results of abstract execution give some information on the actual computations. An intuitive example (which we borrow from Sintzoff [72]) is the rule of signs. The text -1515 * 17 may be understood to denote computations on the abstract universe {(+), (-), (±)} where the semantics of arithmetic operators is defined by the rule of signs. The abstract execution -1515 * 17 → -(+) * (+) → (-) * (+) → (-), proves that -1515 * 17 is a negative number. Abstract interpretation is concerned by a particular underlying structure of the usual universe of computations (the sign, in our example). It gives a summary of some facets of the actual executions of a program. In general this summary is simple to obtain but inaccurate (e.g. -1515 + 17 → -(+) + (+) → (-) + (+) → (±)). Despite its fundamentally incomplete results abstract interpretation allows the programmer or the compiler to answer questions which do not need full knowledge of program executions or which tolerate an imprecise answer, (e.g. partial correctness proofs of programs ignoring the termination problems, type checking, program optimizations which are not carried in the absence of certainty about their feasibility, …).

/pdf/abstract-interpretation-a-unified-lattice-model-for-static-2jfaix0dre.pdf

Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints

/pdf/automatic-discovery-of-linear-restraints-among-variables-of-4lyseyeyff.pdf

Automatic discovery of linear restraints among variables of a program

interpretation is a theory of semantics approximation that is used for the construction of semantic-based program analysis algorithms (sometimes called “data flow analysis”), the comparison of formal semantics (e.g., construction of a denotational semantics from an operational one), design of proof methods, etc.

Automatic program analysers are used for determining statistically conservative approximations of dynamic properties of programs. Such properties of the run-time behavior of programs are useful for debugging (e.g., type inference), code optimization (e.g., compile-time garbage collection, useless occur-check elimination), program transformation (e.g., partial evaluation, parallelization), and even program correctness proofs (e.g., termination proof).

After a few simple introductory examples, we recall the classical framework for abstract interpretation of programs. Starting from a ground operational semantics formalized as a transition system, classes of program properties are first encapsulated in collecting semantics expressed as fixpoints on partial orders representing concrete program properties. We consider invariance properties characterizing descendants of the initial states (corresponding to top/down or forward analyses), ascendant states of the final states (corresponding to bottom/up or backward analyses) as well as a combination of the two. Then we choose specific approximate abstract properties to be gathered about program behaviors and express them as elements of a poset of abstract properties. The correspondence between concrete and abstract properties is established by a concretization and abstraction function that is a Galois connection formalizing the loss of information. We can then constructively derive the abstract program properties from the collecting semantics by a formal computation leading to a fixpoint expression in terms of abstract operators on the domain of abstract properties. The design of the abstract interpreter then involves the choice of a chaotic iteration strategy to solve this abstract fixpoint equation. We insist on the compositional design of this abstract interpreter, which is formalized by a series of propositions for designing Galois connections (such as Moore families, decomposition by partitioning, reduced product, down-set completion, etc.). Then we recall the convergence acceleration methods using widening and narrowing allowing for the use of very expressive infinite domains of abstract properties.

We show that this classical formal framework can be applied in extenso to logic programs. For simplicity, we use a variant of SLD-resolution as the ground operational semantics. The first example is groundness analysis, which is a variant of Mellish mode analysis. It is extended to a combination of top/down and bottom/up analyses. The second example is the derivation of constraints among argument sizes, which involves an infinite abstract domain requiring the use of convergence accelaration methods. We end up with a short thematic guide to the literature on abstract interpretation of logic programs.

Abstract interpretation and application to logic programs

ACSAR (Automatic Checker of Safety properties based on Abstraction Refinement) is a software model checker for C programs in the spirit of Blast [6], F-Soft [7], Magic [5] and Slam [1]. It is based on the counterexample-guided abstraction refinement (CEGAR) paradigm. Its specificity lies in the way it overcomes a problem common to all tools based on this paradigm. The problem arises from creating more and more spurious counterexamples by unfolding the same (while- or for-) loop over and over again; this leads to an infinite or at least too large sequence of refinement steps. The idea behind ACSAR is to abstract not just states but also the state changes induced by structured program statements, including for- and while-statements. The use of the new abstraction allows one to shortcut such a “transfinite” sequence of refinement steps.

/pdf/acsar-software-model-checking-with-transfinite-refinement-4a11u0db37.pdf

ACSAR: software model checking with transfinite refinement

The method of creative telescoping

Results which allow either the computatmn of symbolic solutions to first-order-linear difference equauons or the determination that solutions of a certain form do not exist are presented. Starting with a field of constants, larger fields may be constructed by the formal adjunctlon of symbols whtch behave hke solutions to first-order-linear equations (with a few restrictions) It IS in these extension fields that the difference equations may be posed and m which the solutions are requested. The principal apphcatmn of these results is In finding formulas for a broad class of finite sums or In showing the nonexistence of such formulas.

Summation in Finite Terms

Several optimizations of programs can be performed when in certain regions of a program equality relationships hold between a linear combination of the variables of the program and a constant. This paper presents a practical approach to detecting these relationships by considering the problem from the viewpoint of linear algebra. Key to the practicality of this approach is an algorithm for the calculation of the "sum" of linear subspaces.

/pdf/affine-relationships-among-variables-of-a-program-50qfiponbk.pdf

Affine relationships among variables of a program

The issues of how a programming language might aid in keeping track of physical units (feet, sec, etc.) are discussed. A method is given for the introduction of relationships among units (a watt is volts*amps, a yard is three feet) and subsequent automatic conversion based upon these relationships. Various proposals for syntax are considered.

Incorporation of units into programming languages

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Summation in Finite Terms

Affine relationships among variables of a program

Incorporation of units into programming languages