The Denotational Description of Programming Languages

This book explains how to formally describe programming languages using the techniques of denotational semantics. The presentation is designed primarily for computer science students rather than for (say) mathematicians. No knowledge of the theory of computation is required, but it would help to have some acquaintance with high level programming languages. The selection of material is based on an undergraduate semantics course taught at Edinburgh University for the last few years. Enough descriptive techniques are covered to handle all of ALGOL 50, PASCAL and other similar languages. Denotational semantics combines a powerful and lucid descriptive notation (due mainly to Strachey) with an elegant and rigorous theory (due to Scott). This book provides an introduction to the descriptive techniques without going into the background mathematics at all. In some ways this is very unsatisfactory; reliable reasoning about semantics (e. g. correctness proofs) cannot be done without knowing the underlying model and so learning semantic notation without its model theory could be argued to be pointless. My own feeling is that there is plenty to be gained from acquiring a purely intuitive understanding of semantic concepts together with manipulative competence in the notation. For these equip one with a powerful conceptua1 framework-a framework enabling one to visualize languages and constructs in an elegant and machine-independent way. Perhaps a good analogy is with calculus: for many practical purposes (e. g. engineering calculations) an intuitive understanding of how to differentiate and integrate is all that is needed.

1. Introduction.- 1.1. Syntax, semantics and pragmatics.- 1.2. The purposes of formal semantics.- 1.2.1. Providing precise and machine-independent concepts.- 1.2.2. Providing unambiguous specification techniques.- 1.2.3. Providing a rigorous theory to support reliable reasoning.- 1.3. Denotational semantics.- 1.4. Abstract entities and their description.- 2. A first example: the language TINY.- 2.1. Informal syntax of TINY.- 2.2. Informal semantics of TINY.- 2.2.1. Informal semantics of expressions.- 2.2.2. Informal semantics of commands.- 2.3. An example.- 2.4. Formal semantics of TINY.- 2.4.1. Syntax.- 2.4.2. States, memories, inputs, outputs and values.- 2.4.3. Semantic functions.- 2.4.3.1. Denotations of expressions.- 2.4.3.2. Denotations of commands.- 2.4.4. Semantic clauses.- 2.4.4.1. Clauses for expressions.- 2.4.4.2. Clauses for commands.- 2.4.5. Summary of the formal semantics of TINY.- 3. General concepts and notation.- 3.1. Abstract syntax.- 3.2. Sets and domains.- 3.2.1. The problem of recursively defined functions.- 3.2.2. The problem of recursively defined sets.- 3.2.3. The role of Dana Scott’s theory.- 3.2.4. The role of mathematics in this book.- 3.3. Defining domains.- 3.3.1. Standard domains.- 3.3.2. Finite domains.- 3.3.3. Domain constructors.- 3.3.3.1. Function space [D1?D2].- 3.3.3.2. Product [D1× D2×... × Dn].- 3.3.3.3. Sequences D*.- 3.3.4. Sum [D1+ D2+... + Dn].- 3.3.4. Domain equations.- 3.4. Functions.- 3.4.1. ?-notation.- 3.4.1.1. Basic idea.- 3.4.1.2. Elaborations.- 3.4.1.2.1. Explicitly indicating source and/or target.- 3.4.1.2.2. More than one argument.- 3.4.1.3. Applying ?-expressions to arguments.- 3.4.1.4. Changing bound variables.- 3.4.2. Higher order functions.- 3.4.3. Important notational conventions on precedence and association.- 3.4.4. Currying.- 3.4.5. Conditionals.- 3.4.6. Cases notation.- 3.4.7. Updating functions.- 3.4.8. Generic functions.- 3.4.9. Ways of defining functions (including recursion).- 3.4.10. Cancelling out variables.- 3.4.11. where notation.- 3.4.12. Composition and sequencing.- 3.4.12.1. Composition.- 3.4.12.2. Sequencing.- 4. Denotational description of TINY.- 4.1. Abstract syntax.- 4.1.1. Syntactic domains.- 4.1.2. Syntactic clauses.- 4.2. Semantics.- 4.2.1. Semantic domains.- 4.2.2. Auxiliary functions.- 4.2.2.1. result.- 4.2.2.2. donothing.- 4.2.2.3. checkNum.- 4.2.2.4. checkBool.- 4.2.3. Semantic functions.- 4.2.4. Semantic clauses.- 4.2.4.1. Clauses for expressions.- 4.2.4.2. Clauses for commands.- 5. Standard semantics.- 5.1. Continuations.- 5.1.1. Modelling the ‘rest of the program’.- 5.1.2. Direct and continuation semantics.- 5.1.3. Continuation semantics of TINY.- 5.1.3.1. Semantic domains and functions.- 5.1.3.2. Semantic clauses.- 5.1.4. Final answers and output.- 5.1.4.1. Final answers are not states.- 5.1.4.2. Output is not part of the state.- 5.1.4.3. Output can be infinite.- 5.2. Locations, stores and environments.- 5.2.1. Sharing.- 5.2.2. Variables and locations.- 5.2.3. Stores.- 5.2.4. Environments.- 5.3. Standard domains of values.- 5.4. Blocks, declarations and scope.- 5.5. Standard domains of continuations.- 5.5.1. Command continuations.- 5.5.2. Expression continuations.- 5.5.3. Declaration continuations.- 5.6. Standard semantic functions.- 5.7. Continuation transforming functions.- 5.7.1. cont.- 5.7.2. update.- 5.7.3. ref.- 5.7.4. deref.- 5.7.5. err.- 5.7.6. Domain checks: D?.- 5.8. Assignments and L and R values.- 5.8.1. L and R values.- 5.9. Procedures and functions.- 5.9.1. Procedures.- 5.9.2. Functions.- 5.9.3. Summary.- 5.10 Non standard semantics and concurrency.- 6. A second example: the language SMALL.- 6.1. Syntax of SMALL.- 6.1.1. Syntactic domains.- 6.1.2. Syntactic clauses.- 6.2. Semantics of SMALL.- 6.2.1. Semantic domains.- 6.2.2. Semantic functions.- 6.2.3. Semantic clauses.- 6.2.3.1. Programs.- 6.2.3.2. Expressions.- 6.2.3.3. Commands.- 6.2.3.4. Declarations.- 6.3. A worked example.- 7. Escapes and jumps.- 7.1. Escapes.- 7.1.1. Escapes from commands.- 7.1.2. Escapes from expressions.- 7.1.3. valof and resultis.- 7.2. Jumps.- 7.2.1 The semantics of jumps.- 7.2.2. Assigning label values to variables.- 8. Various kinds of procedures and functions.- 8.1. Procedures (or functions) with zero or more parameters.- 8.1.1. Zero parameters.- 8.1.2. More than one parameter.- 8.2. Recursive procedures and functions.- 8.2.1. Recursive functions in ALGOL 60 and PASCAL.- 8.3. Static and dynamic binding.- 8.3.1. Semantics of binding.- 8.3.2. Advantages and disadvantages of dynamic binding.- 8.4. Parameter passing mechanisms.- 8.4.1. Call by value.- 8.4.2. Call by reference.- 8.4.2.1. Simple call by reference.- 8.4.2.2. PASCAL call by reference.- 8.4.2.3. FORTRAN call by reference.- 8.4.3. Call by value and result.- 8.5. Procedure calling mechanisms.- 8.5.1. Call by closure (ALGOL 60 call by name).- 8.5.2. Call by text (LISP FEXPRs).- 8.5.3. Call by denotation.- 8.5.4. Quotation constructs.- 8.6. Summary of calling and passing mechanisms.- 8.7. Procedure and function denoting expressions (abstractions).- 8.8. Declaration binding mechanisms.- 9. Data structures.- 9.1. References.- 9.2. Arrays.- 9.2.1. news.- 9.2.2. newarray.- 9.2.3. subscript.- 9.3. Records.- 9.4. Data structure valued expressions.- 9.5. Files.- 10. Iteration constructs.- 10.1. repeat C until E.- 10.2. Event loops.- 10.3. For-statements.- 11. Own-variables.- 11.1. The within construct.- 11.2. Different interpretations of ALGOL 60 own-variables.- 11.3. Semantics of own-variables.- 11.3.1. Static interpretation.- 11.3.2. Intermediate interpretation.- 11.3.3. Dynamic interpretation.- 12. Types.- 12.1. Various kinds of types.- 12.2. Well-typed programs and type-checking.- 12.2.1. The denotational description of type-checking.- 12.3. The semantics of types.- Appendix: Remarks for instructors and sample exercises.- Sample exercises.- References.- Subject and Author Index.- Symbols.