General Boolean Expressions Research Articles

CTFTP: A Test Case Generation Strategy for General Boolean Expressions Based on Ordered Binary Label-Driven Petri Nets

Boolean expression testing requires certain types of tests for each Boolean expression in program specification or implementation. Fault-based testing essentially uses a subset of the exhaustive test set to detect certain special types of faults. A fault-based Boolean expression testing strategy called constraint true and false test point (CTFTP) is proposed. The test consists of two test case generation strategies, namely a unique constraint true point (UCTP) strategy and a near constraint false point (NCFP) strategy. An ordered binary label-driven Petri net model is presented to analyze the interaction between Boolean transitions and Boolean literals and yield the test paths of a singular term for the irredundant disjunctive normal forms (IDNFs). On the basis of the test paths, we develop a configuration-based IDNF test generation algorithm, which is employed to obtain the UCTP, NCFP, and CTFTP test sets for the IDNFs. The proposed test generation algorithm based on literal substitution is applied to extend the CTFTP strategy and generate a test suite for general Boolean expressions, which are evaluated using TCAS II specifications. Experimental results show that the CTFTP strategy can detect the same seven types of faults similar to the MUMCUT strategy when testing IDNFs, but only a subset of the MUMCUT test set is required. Five types of faults of general Boolean expressions can also be detected using CTFTP strategies.

Evaluating and Comparing Fault-Based Testing Strategies for General Boolean Specifications: A Series of Experiments

A great amount of fault-based testing strategies have been proposed to generate test cases for detecting certain types of faults in Boolean specifications. However, most of the previous studies on these strategies were focused on the Boolean expressions in the disjunctive normal form (DNF), even the irredundant DNF (IDNF)-little work has been conducted to comprehensively investigate their performance on general Boolean specifications. In this study, we conducted a series of experiments to evaluate and compare 18 fault-based testing strategies using over 4000 randomly generated fault-seeded Boolean expressions. In the experiments, a testing strategy is regarded as effective and efficient if it can detect most of the seeded faults using a small number of test cases. Our experimental results show that if a testing strategy is highly effective and efficient when testing the Boolean expressions in the IDNF, it also shows high effectiveness and efficiency on general Boolean expressions. It is found that one family of fault-based testing strategies, namely MUMCUT, normally deliver the best performance among all the 18 strategies. Our study provides an in-depth understanding and insight of fault-based testing for general Boolean expressions.

The Unique Horn-Satisfiability problem and quadratic Boolean equations

The unique satisfiability problem for general Boolean expressions has attracted interest in recent years in connection with basic complexity issues [12,13]. We investigate here Unique Horn-Satisfiability, i.e. the subclass of Unique-Sat restricted to Horn expressions. We introduce two operators,reduction andshrinking, each transforming a given Horn expression into another Horn expression involving strictly fewer variables and preserving the unique satisfiability property, if present. Uniquely satisfiable Horn expressions are then characterized as those Horn expressions which can be converted into a formula composed of an empty set of clauses on an empty set of free variables through finitely many applications of the shrink-and-reduce operator. Finally, an algorithm for testing whether a given irreducible Horn formula is uniquely satisfiable is described. Data structures for its implementation are discussed, leading toO(mn) complexity for the general case (m=number of clauses,n=number of variables), hence to linear complexity for dense formulae.