Integration Testing from Structured First-Order Specifications via Deduction Modulo

Abstract

Abstract. Testing from first-order specifications has mainly been studied for flat specifications, that are specifications of a single software module. However, the specifications of large software systems are generally built out of small specifications of individual modules, by enriching their union. The aim of integration testing is to test the composition of modules assuming that they have previously been verified, i.e. assuming their correctness. One of the main method for the selection of test cases from first-order specifications, called axiom unfolding, is based on a proof search for the different instances of the property to be tested, thus allowing the coverage of this property. The idea here is to use deduction modulo as a proof system for structured first-order specifications in the context of integration testing, so as to take advantage of the knowledge of the correctness of the individual modules.Testing is a very common practice in the software validation process. The principle of testing is to execute the software system on a subset of its possible inputs in order to detect failures. A failure is detected if the system behaves in a non-conformant way with respect to its specification.The testing process is usually decomposed into three phases: the selection of a relevant subset of the set of all the possible inputs of the system, called a test set; the submission of this test set to the system; the decision of the success or the failure of the test set submission, called the oracle problem. We focus here on the selection phase, which is the crucial point for the relevance and the efficiency of the testing process. In the approach called black-box testing, tests are selected from a (formal or informal) specification of the system, without any knowledge about the implementation.Our work follows the framework defined by Gaudel, Bernot and Marre [1], for testing from specifications expressed in a logical formalism. One approach to selection consists first in dividing an exhaustive test set into subsets, and then in choosing one test case in each of these subsets, thus building a finite test set which covers the initial exhaustive test set. One of the most studied selection method for testing from equational (and then first-order) specifications is known as axiom unfolding [1][2][3][4]. Its principle is to divide the initial exhaustive test set according to criteria derived from the axioms of the specification, using the well-known and efficient proof techniques associated to first-order logic.

Some General Results About Proof Normalization

“…); the confluence property of rewrite systems which establishes that derivability can be proved by "valleys" for many logics dealing with transitive relations (equational [20,8], preorder [42,52], special relations [51]), 1 or by using the Curry-Howard's isomorphism, normalization results in typed λ-calculi as initiated by D. Prawitz [48]. More recently, the authors also used such proof normalization results to show the correctness of procedures based on axiom unfolding which enable us to generate test data sets from various specification formalisms [1,2,7,43,45,44].…”