Abstract
Logic is the basis for almost all formal methods in computer science. In this chapter, we introduce some of the most commonly used logics by examples. It serves as a reference for subsequent chapters. We start with propositional logic, introduce its syntax, semantics, and calculus. Then we extend our view of propositional logic as a so-called institution, discuss model transformations and modular specifications. Subsequently, we turn to first- and second-order logic and show how these can be obtained as natural extensions of propositional logic. Finally, we discuss non-classical logics: multimodal and deontic logics to deal with alternative and subjective viewpoints, respectively, and dynamic and temporal logics for reasoning about time. The chapter concludes with an elaborate example of how to transform an informal natural language description into a formal specification in linear temporal logic.
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20 December 2022
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Notes
- 1.
Here, we use the symbol \(\Vdash \) for the semantic consequence relation between formulae, which is different from the symbol \(\models \) for the satisfaction relation between a model and a formula. For universal validity, the notions coincide: \(\models \varphi \) if and only if \(\{\}\Vdash \varphi \).
- 2.
In this book, the words “sort” and “type” are mutually exchangeable. Historically, logicians prefer the word “type”, whereas computer scientists prefer to use the word “sort”.
- 3.
In Casl, the word “predicate” is used instead of the word “relation”. Thus, in algebraic specifications we will use the words “relation” and “predicate” interchangeably.
- 4.
Due to this axiom, the logic sometimes is called “modal logic K".
- 5.
Some authors and most tools use F and G as a notation instead of \(\Diamond \) and \(\square \). In some papers, F and G are used in addition to \(\Diamond \) and \(\square \), which then have a different meaning.
- 6.
Some authors write \(\textbf{X}\varphi \) instead of \(\bigcirc \varphi \).
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Schlingloff, BH., Roggenbach, M., Schneider, G., Cerone, A. (2022). Logics for Software Engineering. In: Formal Methods for Software Engineering. Texts in Theoretical Computer Science. An EATCS Series. Springer, Cham. https://doi.org/10.1007/978-3-030-38800-3_2
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