This monograph contains the results of our joint research over the last ten years on the logic of the fixed point operation. The intended au dience consists of graduate students and research scientists interested in mathematical treatments of semantics. We assume the reader has a good mathematical background, although we provide some prelimi nary facts in Chapter 1. Written both for graduate students and research scientists in theoret ical computer science and mathematics, the book provides a detailed investigation of the properties of the fixed point or iteration operation. Iteration plays a fundamental role in the theory of computation: for example, in the theory of automata, in formal language theory, in the study of formal power series, in the semantics of flowchart algorithms and programming languages, and in circular data type definitions. It is shown that in all structures that have been used as semantical models, the equational properties of the fixed point operation are cap tured by the axioms describing iteration theories. These structures include ordered algebras, partial functions, relations, finitary and in finitary regular languages, trees, synchronization trees, 2-categories, and others.

This two volume set LNCS 9234 and 9235 constitutes the refereed conference proceedings of the 40th International Symposium on Mathematical Foundations of Computer Science, MFCS 2015, held in Milan, Italy, in August 2015. The 82 revised full papers presented together with 5 invited talks were carefully selected from 201 submissions. The papers feature high-quality research in all branches of theoretical computer science. They have been organized in the following topical main sections: logic, semantics, automata, and theory of programming (volume 1) and algorithms, complexity, and games (volume 2).

This book constitutes the refereed proceedings of the 12th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2009, held in York, UK, in March 2009, as part of ETAPS 2009, the European Joint Conferences on Theory and Practice of Software. The 30 revised full papers presented together with two invited talks were carefully reviewed and selected from 102 full paper submissions. The topics addressed are semantics, logics and automata, algebras, automata theory, processes and models, security, probabilistic and quantitative models, synthesis, and program analysis and semantics.

This book constitutes the refereed proceedings of the First International Symposium on Dependable Software Engineering: Theories, Tools, and Applications, SETTA 2015, held in Nanjing, China, in November 2015. The 20 full papers presented together with 3 invited talks were carefully reviewed and selected from 60 submissions.The papers are organized on topical sections on probabilistic systems; hybrid and cyber-physical systems; testing, simulation and inference; bisimulation and correctness; design and implementation; symbolic execution and invariants; and verification and case studies.

ETAPS 2004 was the seventh instance of the European Joint Conferences on Theory and Practice of Software. ETAPS is an annual federated conference that was established in 1998 by combining a number of existing and new conferences. This year it comprised ?ve conferences (FOSSACS, FASE, ESOP, CC, TACAS), 23 satellite workshops, 1 tutorial, and 7 invited lectures (not including those that are speci?c to the satellite events). The events that comprise ETAPS address various aspects of the system - velopment process, including speci?cation, design, implementation, analysis and improvement. The languages, methodologies and tools that support these - tivities are all well within its scope. Di?erent blends of theory and practice are represented, with an inclination towards theory with a practical motivation on the one hand and soundly based practice on the other. Many of the issues inv- ved in software design apply to systems in general, including hardware systems, and the emphasis on software is not intended to be exclusive.