This booklet presents a reasonably self-contained theory of predicate trans former semantics. Predicate transformers were introduced by one of us (EWD) as a means for defining programming language semantics in a way that would directly support the systematic development of programs from their formal specifications. They met their original goal, but as time went on and program derivation became a more and more formal activity, their informal introduction and the fact that many of their properties had never been proved became more and more unsatisfactory. And so did the original exclusion of unbounded nondeterminacy. In 1982 we started to remedy these shortcomings. This little monograph is a result of that work. A possible -and even likely- criticism is that anyone sufficiently versed in lattice theory can easily derive all of our results himself. That criticism would be correct but somewhat beside the point. The first remark is that the average book on lattice theory is several times fatter (and probably less self contained) than this booklet. The second remark is that the predicate transformer semantics provided only one of the reasons for going through the pains of publication.

This book presents a rigorous foundation for defining Boolean categories and will appeal to graduate students and researchers in theoretical computer science.

Explores quantum computation from the perspective of the branch of theoretical computer science known as semantics.

The Formal Semantics of Programming Languages provides the basic mathematical techniques necessary for those who are beginning a study of the semantics and logics of programming languages. These techniques will allow students to invent, formalize, and justify rules with which to reason about a variety of programming languages. Although the treatment is elementary, several of the topics covered are drawn from recent research, including the vital area of concurency. The book contains many exercises ranging from simple to miniprojects.Starting with basic set theory, structural operational semantics is introduced as a way to define the meaning of programming languages along with associated proof techniques. Denotational and axiomatic semantics are illustrated on a simple language of while-programs, and fall proofs are given of the equivalence of the operational and denotational semantics and soundness and relative completeness of the axiomatic semantics. A proof of Godel's incompleteness theorem, which emphasizes the impossibility of achieving a fully complete axiomatic semantics, is included. It is supported by an appendix providing an introduction to the theory of computability based on while-programs. Following a presentation of domain theory, the semantics and methods of proof for several functional languages are treated. The simplest language is that of recursion equations with both call-by-value and call-by-name evaluation. This work is extended to lan guages with higher and recursive types, including a treatment of the eager and lazy lambda-calculi. Throughout, the relationship between denotational and operational semantics is stressed, and the proofs of the correspondence between the operation and denotational semantics are provided. The treatment of recursive types - one of the more advanced parts of the book - relies on the use of information systems to represent domains. The book concludes with a chapter on parallel programming languages, accompanied by a discussion of methods for specifying and verifying nondeterministic and parallel programs.

This volume contains the proceedings of the 8th International Conference on Mathematics of ProgramConstruction, MPC 2006,held at Kuressaare, Estonia, July 3-5, 2006, colocated with the 11th International Conference on Algebraic Methodology and Software Technology, AMAST 2006, July 5-8, 2006. TheMPCconferencesaimtopromotethedevelopmentofmathematicalpr- ciples and techniques that are demonstrably useful and usable in the process of constructing computer programs. Topics of interest range from algorithmics to support for program construction in programming languages and systems. The previous MPCs were held at Twente, The Netherlands (1989, LNCS 375), Oxford, UK (1992, LNCS 669), Kloster Irsee, Germany (1995,LNCS 947), Marstrand, Sweden (1998, LNCS 1422), Ponte de Lima, Portugal (2000, LNCS 1837), Dagstuhl, Germany (2002, LNCS 2386) and Stirling, UK (2004, LNCS 3125, colocated with AMAST 2004). MPC 2006 received 45 submissions. Each submission was reviewed by four Programme Committee members or additional referees. The committee decided to accept 22 papers. In addition, the programme included three invited talks by Robin Cockett (University of Calgary, Canada), Olivier Danvy (Aarhus Univ- sitet, Denmark) and Oege de Moor (University of Oxford, UK). The review process and compilation of the proceedings were greatly helped by Andrei Voronkov's EasyChair system that I can only recommend to every programme chair. MPC 2006 had one satellite workshop, the Workshop on Mathematically Structured Functional Programming, MSFP 2006, organized as a "small" wo- shop of the FP6 IST coordination action TYPES. This took place July 2, 2006.

This monograph extends and generalizes the UNITY methodology, introduced in the late 1980s by K. Mani Chandy and Jayadev Misra as a formalism aiding in the specification and verification of parallel programs, in several directions. This treatise further develops the ideas behind UNITY in order to explore and understand the potential and limitations of this approach: first UNITY is applied to formulate and tackle problems in parallelism such as compositionality; second, the logic and notation of UNITY is generalized in order to increase its range of applicability; finally, paradigms and abstractions useful for the design of probabilistic parallel algorithms are developed. Taken together the results presented reaffirm the promise of UNITY as a versatile medium for treating many problems of parallelism.

Much current research in computer science is concerned with two questions: is a program correct? And how can we improve a correct program preserving correctness? This latter question is known as the refinement of programs and the purpose of this book is to consider these questions in a formal setting. In fact, correctness turns out to be a special case of refinement and so the focus is on refinement. Although a reasonable background knowledge is assumed from mathematics and CS, the book is a self-contained introduction suitable for graduate students and researchers coming to this subject for the first time. There are numerous exercises provided of varying degrees of challenge.