Programming Languages and Design Lecture 3 Semantic Specifications of Programming Languages

1. Programming Languages and DesignLecture 3 Semantic Specifications of Programming Languages Instructor: Li Ma Department of Computer Science Texas Southern University, Houston January, 2008

2. Review and Preview • Last lecture • Syntax specifications of programming languages • Regular expressions and rules • Context-free grammar • Parse tree • Ambiguity • Today’s lecture • Semantic specifications of programming languages • Reference: Chapter 5 of “Foundations of Programming Languages: Design and Implementation”, S. H. Roosta • Three formal semantics: axiomatic semantics, operational semantics, denotational semantics

3. Language Description • A formal language is any set of character strings with characters chosen from a fixed, finite set of an alphabet of symbols • The strings that belong to the language are called its constructs, or phrases • Any programming language description can be classified according to its • Syntax, which deals with the formation of phrases • Semantics, which deals with the meaning of phrases • Pragmatics, which deals with the practical use of phrases

4. Semantics • Semantics referred to the study of • what words mean, the relation of words to sentences of a language, their significance, and changes in their meaning • In computer science • The investigation of it focuses on developing ways to express the meaning of programming languages • It also Involves a description of what happens during execution of a program that refers to rules specifying its meaning

5. Formal Semantics Methods • Different notational systems could be used to describe programming languages • Formal definition method is precise, but complex and abstract • Question the behavior of programming languages by mathematical method • Formal methods are important because they provide • an unambiguous language definition • standards to unify the language • a basis for correctness proof by both compilers and programs

6. Formal Semantic Methods (cont’) • Three formal methods differ in their formation • Axiomatic semantics – proof system • Operational semantics – proof system • Denotational semantics – model theory

7. Axiomatic Semantics • Associate with each syntactically correct program construct properties of variables that hold before execution starts and after the program halts • Apply mathematical logic to language constructs • The state of the computation is described by a logical statement – a predicate or an assertion

8. Principles of Axiomatic Semantics • Preconditions and postconditions are used to specify the behavior of programs at each step of execution {P} S {Q} • A logical expression • P and Q are logical assertions for precondition and postcondition, respectively • S is a construct of the subject language • If P is true before execution of S and if the execution of S terminates, then Q is true

9. Proof Rules • The goal of axiomatic semantics is to provide logical statements that capture the intended meaning of each construct in a programming language • Deduce a specification for a construct using logical statements • Also called inference rules, or deduction rules • Enable the truth of certain assertions to be deduced from the truth of certain other assertions

10. Proof Rules (cont’) R1, R2, …, Rn, Rn+1, and R are all assertions given that R1, R2, …, Rn are true, then R is true can be deduced if the truth of Rn+1 is deduced by assuming the truth of R1, R2, …, Rn, then the truth of R can be deduced

11. Program Correctness • A purely static analysis for the text of a program is used to prove the correctness of the program • Proof theory – a set of axiomatic specifications • A correct program is one that meets its specifications • Two predicates for correctness requirements of the program: • A precondition on input variables • A postcondition on input and output variables • The aim of verification is to show that if input holds before the program runs, execution terminates in a state where output holds

12. Example • Given a program fragment, prove it is correct with respect to some assertions

13. Operational Semantics • Specify a programming language by the operations through which each program construct is carried out • Hypothetical machine is an abstract machine • A collection of actions described in mathematical notation • Apply the operations to values to show the run-time behavior of a program • Individual steps should be checked to see how programs are implemented

14. Principles of Operational Semantics • Shortcomings if using actual machine • Difficult to understand actions • Computer-dependent • Correct interpreters are difficult to write • So use an abstract machine • Independent of the subject language • Define an abstract machine with primitive instructions • Simple, so no misunderstanding • States and transition between states must be explicitly defined

15. Principles of Operational Semantics (cont’) • The virtual machine interprets a program by passing through a sequence of discrete states • State transitions are defined by a set of instruction definitions written in specific notation • Three components of the abstract machine • Program • Control system • Memory system • The semantic description of the programming language specifies a translation to a metelanguage

16. VDL MetaLanguage • One of the best known metalanguage • A formal system: model the interpretation of a program • Two specification parts: • A translator: specify the translation of each program into an abstract syntax tree • Each instruction is converted into the chosen low-level language • An interpreter: specify how the program could be carried out • Interpret a program by passing through a sequence of transition states to model the operations

17. VDL MetaLanguage (cont’) • A typical VDL structure is represented as a collections of subtrees • All nonprimitive objects are modeled as syntax tree • Program, memory, input, output, environment, control mechanism, … • An initial configuration contains all the components • Control system subtree represents a single instruction • The transition functions perform the steps of the computation

18. VDL MetaLanguage (cont’) • A computation consists of • Select a terminal node of the control tree • Evaluate it according to the specific definition • Produce a new state with a modified control tree • Interpretation of the program terminates when all the nodes of the control tree have been evaluated • Evaluation of the program is concluded

19. Notations for Abstract Syntax Trees • Elementary objects • No internal structure • A terminal node of a tree • A sequence of one or more uppercase letters • For example, A, +, Ω • Or a sequence of one or more characters enclosed in single quotes • Composite objects • A set of one or more pairs of the form <s : o> • “s” for a selector, the name begins with “s-” • The null object and the associated selector have no corresponding branch in the tree diagram • A selector may be defined as a function

20. Notations for Abstract Syntax Trees (cont’) • Lists • An object may be regarded as a list of other objects • The function head yields the first element • The function tail yields a list consisting of all but first element • The function length provides the number of elements in list • Predicates • A function that takes an object as an argument and yields an elementary object as True or False • The name begin with “is-” • Equations • An equation defines a predicate for a class of objects, the selectors involved, and the kinds of objects they select

21. Example • Given a program fragment, provide the VDL syntax tree representation with respect to the syntax rules for a language

22. Denotational Semantics • Define a programming language behavior by applying mathematical functions to programs to represent their meaning • Mathematical semantics, • Programs can be translated to functions • Their properties can be proved by using the standard mathematical theory of functions • Function calculus

23. Principles of Denotational Semantics • Denotational semantics not only assigns a meaning to a complete program, but also to every phrase • Each phrase is defined in terms of the meaning of its subphrases • The meaning of each phrase – denotation • Denotational definitions are enclosed in [[ ]], to separate from syntactic definitions • If p is a syntactic phrase, meaning[[p]] is a denotational specification • For example, 2*4 and 7+1 are syntactic phrase, meaning[[2*4]]=meaning[[7+1]]=meaning[[8]]

24. Principles of Denotational Semantics (cont’) • Denotational semantics focus on the net effect of running a program, ignoring the steps by which that effect is achieved • Advantages • We can predict the behavior of a program without running it on computer • We can reason about programs • For example, to prove whether two programs are equivalent • We discuss four different aspects of denotational semantics

25. Semantic Domains • Sets of mathematical objects of a particular form • A denotational definition lists its set of elements and operations, but no properties of the operations • Domain characteristics • A primitive domain: primitive values as elements • A function domain: functions or mappings • A Cartesian production domain: A X B • A disjoint union domain: A + B • A sequence domain: homogeneous sequence

26. Semantic Functions • Semantic functions synthesize the meaning of constructs • Map syntactic objects onto objects in semantic domains • These functions are specified by giving their domains – signatures • A program can be viewed as an entity that receives input and produces output Program : Program => (Input => Output) • Each semantic domain has one semantic function

27. Semantic Equations • Specify how the semantic functions act on each construct by using the structure of the grammar rules • For example, given grammar rule: Digit ::= ‘0’ | ‘1’ | … | ‘9’ semantic function: Dig : Digit => Integer • They can be combined to form the semantic equations Dig( Digit=>’0’) = 0 Dig( Digit=>’1’) = 1 … Dig( Digit=>’9’) = 9 or Dig[[‘0’]] = 0, Dig[[‘1’]] = 1, …, Dig[[‘9’]] = 9

28. Auxiliary Functions • Predefined mathematical operations plus, times, minus, and divide in semantic functions • For example, plus: Number + Number => Number times: Number x Number => Number • Number x Number is an element of Cartesian product domain • Number is an element of primitive domain • So far, we can have a complete denotational specification of a language for some construct of that language, which should have • Production rules, semantic domain, semantic functions, auxiliary functions, and semantic equations