Principles of Programming Languages


Syntax and Semantics

  • Describing Syntax and Semantics
    The General Problem of Describing Syntax
    Formal Methods of Describing Syntax
    Attribute Grammars
    Describing the Meanings of Programs: Dynamic Semantics
    2.1 Introduction
    Syntax: the form or structure of the expressions, statements, and program units
    Semantics: the meaning of the expressions, statements, and program units
    Syntax and semantics provide a language‘s definition
    * Users of a language definition
    * Other language designers
    * Implementers
    * Programmers (the users of the language)
    2.2 The General Problem of Describing Syntax: Terminology
    * A sentence is a string of characters over some alphabet
    * A language is a set of sentences
    * A lexeme is the lowest level syntactic unit of a language (e.g., *, sum, begin)
    * A token is a category of lexemes (e.g., identifier)
    Formal Definition of Languages

    * A recognition device reads input strings of the language and decides whether the input strings belong to the language
    * Example: syntax analysis part of a compiler
    * Detailed discussion in Chapter 4
    * A device that generates sentences of a language
    * One can determine if the syntax of a particular sentence is correct by comparing it to the structure of the generator
    2.3 Formal Methods of Describing Syntax
    Backus-Naur Form and Context-Free Grammars
    * Most widely known method for describing programming language syntax
    Extended BNF
    * Improves readability and writability of BNF
    Grammars and Recognizers
    BNF and Context-Free Grammars
    * Context-Free Grammars
    * Developed by Noam Chomsky in the mid-1950s
    * Language generators, meant to describe the syntax of natural languages
    * Define a class of languages called context-free languages
    Backus-Naur Form (BNF)
    Backus-Naur Form (1959)
    * Invented by John Backus to describe Algol 58
    * BNF is equivalent to context-free grammars
    * BNF is ametalanguage used to describe another language
    * In BNF, abstractions are used to represent classes of syntactic structures--they act like syntactic variables (also called nonterminal symbols)
    BNF Fundamentals
    Non-terminals: BNF abstractions
    Terminals: lexemes and tokens
    Grammar: a collection of rules
    * Examples of BNF rules:
    <ident_list> identifier | identifer, <ident_list>
    <if_stmt> if <logic_expr> then <stmt>
    BNF Rules
    A rule has a left-hand side (LHS) and a right-hand side (RHS), and consists of terminal and nonterminal symbols
    A grammar is a finite nonempty set of rules
    An abstraction (or nonterminal symbol) can have more than one RHS
    <stmt> <single_stmt>
    | begin <stmt_list> end
    Describing Lists
    Syntactic lists are described using recursion
    <ident_list> ident
    | ident, <ident_list>
    A derivation is a repeated application of rules, starting with the start symbol and ending with a sentence (all terminal symbols)
    An Example Grammar
    <program> <stmts>
    <stmts> <stmt> | <stmt> ; <stmts>
    <stmt> <var> = <expr>
    <var> a | b | c | d
    <expr> <term> + <term> | <term> - <term>
    <term> <var> | const
    An example derivation
    <program> => <stmts> => <stmt>
    * <var> = <expr> => a =<expr>
    * a = <term> + <term>
    * a = <var> + <term>
    * a = b + <term>
    * a = b + const
    Every string of symbols in the derivation is a sentential form
    A sentence is a sentential form that has only terminal symbols
    A leftmost derivation is one in which the leftmost nonterminal in each sentential form is the one that is expanded
    A derivation may be neither leftmost nor rightmost
    Parse Tree
    A hierarchical representation of a derivation

    Ambiguity in Grammars
    A grammar is ambiguous iff it generates a sentential form that has two or more distinct parse trees
    An Ambiguous Expression Grammar
    * Ex: Two distinct parse trees for the same sentence, A = B + A * C

    An Unambiguous Expression Grammar
    If we use the parse tree to indicate precedence levels of the operators, we cannot have ambiguity
    Associativity of Operators
    Operator associativity can also be indicated by a grammar
    <expr> -> <expr> + <expr> | const (ambiguous)
    <expr> -> <expr> + const | const (unambiguous)
    Extended BNF
    Optional parts are placed in brackets ([ ])
    <proc_call> -> ident [(<expr_list>)]
    Alternative parts of RHSs are placed inside parentheses and separated via vertical bars
    <term> <term> (+|-) const
    Repetitions (0 or more) are placed inside braces ({ })
    <ident> letter {letter|digit}
    BNF and EBNF
    <expr> <expr> + <term>
    | <expr> - <term>
    | <term>
    <term> <term> * <factor>
    | <term> / <factor>
    | <factor>
    <expr> <term> {(+ | -) <term>}
    <term> <factor> {(* | /) <factor>}
    2.4 Attribute Grammars
    Context-free grammars (CFGs) cannot describe all of the syntax of programming languages Additions to CFGs to carry some semantic info along parse trees
    Primary value of attribute grammars (AGs):
    * Static semantics specification
    * Compiler design (static semantics checking)
    Attribute Grammars: Definition
    An attribute grammar is a context-free grammar G = (S, N, T, P) with the following additions:
    * For each grammar symbol x there is a set A(x) of attribute values
    * Each rule has a set of functions that define certain attributes of the nonterminals in the rule
    * Each rule has a (possibly empty) set of predicates to check for attribute consistency
    * Let X0 X1 ... Xn be a rule
    * Functions of the form S(X0) = f(A(X1), ... , A(Xn)) define synthesized attributes
    * Functions of the form I(Xj) = f(A(X0), ... , A(Xn)), for i <= j <= n, define inherited
    * Initially, there are intrinsic attributes on the leaves
    Attribute Grammars: Example
    <assign> -> <var> = <expr>
    <expr> -> <var> + <var> | <var>
    <var> A | B | C
    actual_type: synthesized for <var> and <expr>
    expected_type: inherited for <expr>
    Syntax rule: <expr> <var>[1] + <var>[2]
    Semantic rules:
    <expr>.actual_type <var>[1].actual_type
    <var>[1].actual_type == <var>[2].actual_type
    <expr>.expected_type == <expr>.actual_type
    Syntax rule: <var> id
    Semantic rule:
    <var>.actual_type lookup (<var>.string)
    How are attribute values computed?
    * If all attributes were inherited, the tree could be decorated in top-down order.
    * If all attributes were synthesized, the tree could be decorated in bottom-up order.
    * In many cases, both kinds of attributes are used, and it is some combination of top-down and bottom-up that must be used.
    <expr>.expected_type inherited from parent
    <var>[1].actual_type lookup (A)
    <var>[2].actual_type lookup (B)
    <var>[1].actual_type =? <var>[2].actual_type
    <expr>.actual_type <var>[1].actual_type
    <expr>.actual_type =? <expr>.expected_type

    2.5 Semantics
    There is no single widely acceptable notation or formalism for describing semantics Operational Semantics
    * Describe the meaning of a program by executing its statements on a machine, either simulated or actual. The change in the state of the machine (memory, registers, etc.) defines the meaning of the statement
    To use operational semantics for a high-level language, a virtual machine is needed A hard ware pure interpreter would be too expensive
    A software pure interpreter also has problems:
    * The detailed characteristics of the particular computer would make actions difficult to understand
    * Such a semantic definition would be machine- dependent
    Operational Semantics
    A better alternative: A complete computer simulation
    The process:
    * Build a translator (translates source code to the machine code of an idealized computer)
    * Build a simulator for the idealized computer
    Evaluation of operational semantics:
    * Good if used informally (language manuals, etc.)
    * Extremely complex if used formally (e.g., VDL), it was used for describing semantics of PL/I.
    Axiomatic Semantics
    * Based on formal logic (predicate calculus)
    * Original purpose: formal program verification
    * Approach: Define axioms or inference rules for each statement type in the language (to allow transformations of expressions to other expressions)
    * The expressions are called assertions
    Axiomatic Semantics
    An assertion before a statement (a precondition) states the relationships and constraints among variables that are true at that point in execution
    An assertion following a statement is a postcondition
    A weakest precondition is the least restrictive precondition that will guarantee the postcondition
    Pre-post form: {P} statement {Q}
    An example: a = b + 1 {a > 1}
    One possible precondition: {b > 10}
    Weakest precondition: {b > 0}
    Program proof process: The postcondition for the whole program is the desired result. Work back through the program to the first statement. If the precondition on the first statement is the same as the program spec, the program is correct.
    An axiom for assignment statements (x = E):
    {Qx->E} x = E {Q}
    An inference rule for sequences
    o For a sequence S1;S2:
    o {P1} S1 {P2}
    o {P2} S2 {P3}
    An inference rule for logical pretest loops
    For the loop construct:
    {P} while B do S end {Q}
    Characteristics of the loop invariant
    I must meet the following conditions:
    * P => I (the loop invariant must be true initially)
    * {I} B {I} (evaluation of the Boolean must not change the validity of I)
    * {I and B} S {I} (I is not changed by executing the body of the loop)
    * (I and (not B)) => Q (if I is true and B is false, Q is implied)
    * The loop terminates (this can be difficult to prove)
    The loop invariant I is a weakened version of the loop postcondition, and it is also a precondition. I must be weak enough to be satisfied prior to the beginning of the loop, but when combined with the loop exit condition, it must be strong enough to force the truth of the postcondition
    Evaluation of axiomatic semantics:
    * Developing axioms or inference rules for all of the statements in a language is difficult
    * It is a good tool for correctness proofs, and an excellent framework for reasoning about programs, but it is not as useful for language users and compiler writers
    * Its usefulness in describing the meaning of a programming language is limited for language users or compiler writers
    Denotational Semantics
    * Based on recursive function theory
    * The most abstract semantics description method
    * Originally developed by Scott and Strachey (1970)
    * The process of building a denotational spec for a language (not necessarily easy):
    * Define a mathematical object for each language entity
    * Define a function that maps instances of the language entities onto instances of the corresponding mathematical objects
    * The meaning of language constructs are defined by only the values of the program's variables
    * The difference between denotational and operational semantics: In operational semantics, the state changes are defined by coded algorithms; in denotational semantics, they are defined by rigorous mathematical functions
    * The state of a program is the values of all its current variables
    s = {<i1, v1>, <i2, v2>, …, <in, vn>}
    * Let VARMAP be a function that, when given a variable name and a state, returns the current value of the variable
    VARMAP(ij, s) = vj
    Decimal Numbers
    * The following denotational semantics description maps decimal numbers as strings of symbols into numeric values
    <dec_num> 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
    · <dec_num> (0 | 1 | 2 | 3 | 4 |
    5 | 6 | 7 | 8 | 9)
    Mdec('0') = 0, Mdec ('1') = 1, …, Mdec ('9') = 9
    Mdec (<dec_num> '0') = 10 * Mdec (<dec_num>)
    Mdec (<dec_num> '1’) = 10 * Mdec (<dec_num>) + 1

    Mdec (<dec_num> '9') = 10 * Mdec (<dec_num>) + 9

    Map expressions onto Z {error}
    We assume expressions are decimal numbers, variables, or binary expressions having one arithmetic operator and two operands, each of which can be an expression
    Me(<expr>, s) =
    case <expr> of
    <dec_num> => Mdec(<dec_num>, s)
    <var> =>
    if VARMAP(<var>, s) == undef
    then error
    else VARMAP(<var>, s)
    <binary_expr> =>
    if (Me(<binary_expr>.<left_expr>, s) == undef
    OR Me(<binary_expr>.<right_expr>, s) =
    * undef)
    then error
    if (<binary_expr>.<operator> == ‗+‘ then
    Me(<binary_expr>.<left_expr>, s) +
    Me(<binary_expr>.<right_expr>, s)
    else Me(<binary_expr>.<left_expr>, s) *
    Me(<binary_expr>.<right_expr>, s)
    Assignment Statements
    o Maps state sets to state sets
    Ma(x := E, s) =
    if Me(E, s) == error
    then error
    else s‘ = {<i1‘,v1‘>,<i2‘,v2‘>,...,<in‘,vn‘>},
    where for j = 1, 2, ..., n,
    vj‘ = VARMAP(ij, s) if ij <> x
    = Me(E, s) if ij == x
    Logical Pretest Loops
    o Maps state sets to state sets
    Ml(while B do L, s) =
    if Mb(B, s) == undef
    then error
    else if Mb(B, s) == false
    then s
    else if Msl(L, s) == error
    then error
    else Ml(while B do L, Msl(L, s))
    The meaning of the loop is the value of the program variables after the statements in the loop have been executed the prescribed number of times, assuming there have been no errors
    In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions
    Recursion, when compared to iteration, is easier to describe with mathematical rigor Evaluation of denotational semantics
    * Can be used to prove the correctness of programs
    * Provides a rigorous way to think about programs
    * Can be an aid to language design
    * Has been used in compiler generation systems
    * Because of its complexity, they are of little use to language users
    BNF and context-free grammars are equivalent meta-languages
    * Well-suited for describing the syntax of programming languages
    An attribute grammar is a descriptive formalism that can describe both the syntax and the semantics of a language
    Three primary methods of semantics description
    * Operation, axiomatic, denotational
    1) Write short notes on lexemes, language recognizers and language generators.
    2) Distinguish between 2 mathematical models of a language description.
    3) A concise and understandable description of a programming language is essential to language’s success. Comment on this.
    4) What is BNF notation? Explain with examples.
    5) Give BNF notation for identifier , for loop, while loop in C. Give the corresponding syntax graph.
    6) What is grammar? What is derivation? Give program of a simple grammar? Give the
    derivation of the above program?
    7) Explain with example how operator associativity can be incorporated in grammars.
    8) What purpose do predicates serve in an attribute grammar?
    9) What is the use of attribute grammar? Explain.
    10) What are the differences between synthesized and inherited attributes? Explain.
    11) What are the difficulties in using an attribute grammar to describe all of the syntax and static semantics of a contemporary language?
    12) What do you mean by static semantics? Give examples of static rules that are difficult and impossible to describe with BNF.
    13) Describe the basic concepts of denotational semantics.
    14) In what way do operational semantics differ from denotational semantics.
    15) Define axiomatic semantics. Comment on its applicability.
    16) What is the significance of mathematical logic in the context of axiomatic semantics. Explain with examples.
    17) What is the significance of denotational semantics in describing the meaning of programs. Explain with examples.

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