Programming Languages


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1 Programming Languages Chapter 3 Describing Syntax and Semantics Instructor: ChihYi Chiu 邱志義
2 Outline Introduction The General Problem of Describing Syntax Formal Methods of Describing Syntax Attribute Grammars Describing the Meanings of Programs: Dynamic Semantics 2
3 Introduction Syntax: the form or structure of the expressions, statements, and program units Semantics: the meaning of the expressions, statements, and program units For example, the syntax of a Java while statement is while (<boolean_expr>) <statement> the semantics is when the current value of the Boolean expression is true, the statement is executed; otherwise control continues after the while construct 3
4 The General Problem of Describing Syntax Terminology A sentence (statement) 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, punctuation) 4
5 The General Problem of Describing Syntax Consider the statement: index = 2 * count + 17; Lexemes index Tokens identifier = equal_sign 2 int_literal * mult_op count identifier + plus_op 17 int_literal ; semicolon 5
6 Language Recognizers A recognition device reads input strings of the language and decides whether the input strings belong to the language Example: the syntax analysis part of a compiler Detailed discussion in Chapter 4 6
7 Language Generators 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 7
8 Formal Methods of Describing Syntax BackusNaur Form (BNF) and Context Free Grammars Most widely known method for describing programming language syntax Extended BNF Improves readability and writability of BNF Grammars and Recognizers 8
9 BNF and ContextFree Grammars Contextfree grammars One of grammar classes developed by Noam Chomsky in the mid1950s The forms of the tokens of programming language can be described by regular grammars The syntax of whole programming language can be described by contextfree grammars 9
10 Origins of BackusNaur Form BackusNaur Form (1959) Invented by John Backus to describe ALGOL 58; slightly modified by Peter Naur to describe ALGOL 60 BNF is equivalent to contextfree grammars In this chapter, the contextfree grammar is referred as grammar or BNF 10
11 BNF Fundamentals BNF is a metalanguage 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) 11
12 BNF Fundamentals Nonterminals: BNF abstractions Terminals: lexemes and tokens Grammar: a collection of rules 12
13 BNF Rules Examples <assign> <var> = <expression> <if_stmt> if <logic_expr> then <stmt> <ident_list> identifier identifier, <ident_list> 13
14 BNF Rules Lefthand side (LHS): the abstraction being defined Righthand side (RHS): the definition of the LHS, consisting of mixture of tokens, lexemes, and references to other abstractions A rule has a lefthand side (LHS) and a righthand 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 14
15 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) 15
16 An Example Grammar 16
17 Derivation <program> => begin <stmt_list> end => begin <stmt> end => begin <var> = <expression> end => begin A = <expression> end => begin A = <var> + <var> end => begin A = B + <var> end => begin A = B + C end 17
18 Derivation 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 18
19 Another Example How to generate the statement by the leftmost derivation? A = B * (A + C) 19
20 Parse Tree A hierarchical representation of a derivation Try A = A + (B * (C)) 20
21 Ambiguity in Grammars A grammar is ambiguous if and only if it generates a sentential form that has two or more distinct parse trees 21
22 Ambiguity in Grammars 22
23 23
24 Ambiguity in Grammars It is mathematically not possible to determine whether an arbitrary grammar is ambiguous In many cases, an ambiguous grammar can be rewritten to be unambiguous but still generate the desired language 24
25 Operator Precedence When an expression includes two different operators, for example, x + y * z, a semantic issue is the order of evaluation of the two operators 25
26 Unambiguous Grammar Write down the derivation of the sentence A = B + C * A in leftmost and rightmost 26
27 27
28 Associativity of Operators Definition: when an expression includes two operators that have the same precedence, for example A / B * C, a semantic rule is required to specify which should have precedence 28
29 Left associative 29
30 Associativity of Operators In mathematics, addition is associative, i.e., (A + B) + C = A + (B + C) Floatingpoint addition is not necessarily associative 30
31 Associativity of Operators Operator associativity can also be indicated by a grammar <expr> > <expr> + <expr> const (ambiguous) <expr> > <expr> + const const (unambiguous) <expr> <expr> + const <expr> + const const 31
32 Associativity of Operators When a grammar rule has its LHS also appearing at the beginning of its RHS, the rule is said to be left recursive, which specifies left associativity Unfortunately, left recursion disallows the use of some syntax analysis algorithms Fortunately, left associativity can be enforced by the compiler 32
33 Associativity of Operators In most languages, the exponentiation operator is right associative To indicate right associativity, right recursion can be used A grammar rule is right recursive if the LHS appears at the right end of the RHS <factor> > <exp> ** <factor> <exp> <exp> > (<expr>) id For example, a**b**c 33
34 An Unambiguous Grammar for ifthenelse Consider the grammar <if_stmt> if <logic_expr> then <stmt> if <logic_expr> then <stmt> else <stmt> <stmt> <if_stmt> This grammar is ambiguous! Write down the possible parse trees of this sentential form if <logic_expr> then if <logic_expr> then <stmt> else <stmt> 34
35 35
36 An Unambiguous Grammar for ifthenelse <stmt> > <matched> <unmatched> <matched> > if <logic_expr> then <matched> else <matched> any nonif statement <unmatched> > if <logic_expr> then <stmt> if <logic_expr> then <matched> else <unmatched> Write down the parse tree of this sentential form using the above grammar if <logic_expr> then if <logic_expr> then <stmt> else <stmt> 36
37 Extended BNF Optional parts are placed in brackets [ ] <selection> > if (<expression>) <statement> [else <statement>] 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_list> <identifier> {, <identifier>} 37
38 Extended BNF The brackets, braces, and parentheses are metasymbols, which means they are notational tools and not terminal symbols in the syntactic entities 38
39 39
40 Grammars and Recognizers Given a contextfree grammar, a recognizer for the language generated by the grammar can be algorithmically constructed A number of software systems have been developed that perform this construction One of the first is yacc 40
41 Attribute Grammars Contextfree grammars (CFGs) cannot describe all of the syntax of programming languages An attribute grammar is an extension to CFGs to carry some semantic info along parse trees Primary value of attribute grammars (AGs) Static semantics specification Compiler design (static semantics checking) 41
42 Static Semantics Consider type compatibility In Java, a floatingpoint value cannot be assigned to an integer type variable, although the opposite is legal Specifying this restriction in BNF requires additional nonterminal symbols and rules If all of the typing rules of Java were specified in BNF, the grammar would become too large to be useful 42
43 Static Semantics Consider variable declaration If all variables must be declared before they are referenced, it has been proven this rule cannot be specified in BNF 43
44 Static Semantics Many static semantic rules of a language state its type constraints Static semantics is so named because the analysis required to check these specifications can be done at compile time 44
45 Static Semantics Due to the problems of describing static semantic with BNF, attribute grammars was designed by Knuth to describe both syntax and the static semantics of programs Attribute grammars are a formal approach to both describing and checking the correctness of the static semantics rules of a program 45
46 Basic Concepts of Attribute Grammas Attribute grammas are contextfree grammars Attributes are associated grammar symbols; similar to variables Attribute computation functions are associated with grammar rules Predicate functions state the static semantic rules of the language 46
47 Attribute Grammars Defined Let X 0 X 1... X n be a rule, and A(X) be the attribute set of symbol X Functions of the form S(X 0 ) = f(a(x 1 ),..., A(X n )) define synthesized attributes, which pass semantic information up a parse tree Functions of the form I(X j ) = f(a(x 0 ),..., A(X n )), for i j n, define inherited attributes, which pass semantic information down and across a tree 47
48 Attribute Grammars Defined A predicate function has the form of a Boolean expression on the union of the attribute set {(A(X 0 ),..., A(X n )} and a set of literal attribute values If all the attribute values in a parse tree have been computed, the tree is said to be fully attributed 48
49 Intrinsic Attributes Intrinsic attributes are synthesized attributes of leaf nodes whose values are determined outside the parse tree (e.g., symbol table) Initially, in an unattributed parse tree, the only attributes with values are the intrinsic attributes of the leaf nodes Next, the semantic functions can be used to compute the remaining attribute values 49
50 Examples of Attribute Grammars Consider the name on the end of an Ada procedure must match the procedure s name Syntax rule: <proc_def> procedure <proc_name>[1] <proc_body> end <proc_name>[2]; Predicate: <proc_name>[1].string == <proc_name>[2].string 50
51 Examples of Attribute Grammars Let s check the type rules of a simple assignment statement The only variable names are A, B, and C The variables can be one of two types: int or real The type of the expression when the operand types are not the same is always real; when they are the same, the expression type is that of the operands The types of operands in the right side can be mixed, but the assignment is valid only if the LHS and the value resulting from evaluating the RHS have the same type 51
52 Syntax <assign> <var> = <expr> <expr> <var> + <var> <var> <var> A B C actual_type: a synthesized attribute for <var> and <expr>; it is used to store the actual type, int or real, of a variable or expression For variable, the actual type is intrinsic For expression, it is determined from the actual types of the child node expected_type: an inherited attribute for <expr>; it is used to store the type, either int or real, that is expected for the expression, as determined by the type of the variable on the left side of the assignment statement 52
53 53
54 Decorating the Parse Tree 54
55 Decorating the Parse Tree How are attribute values computed? If all attributes were inherited, the tree could be decorated in topdown order If all attributes were synthesized, the tree could be decorated in bottomup order In many cases, both kinds of attributes are used, and it is some combination of topdown and bottomup that must be used 55
56 1. <var>.actual_type lookup(a) (Rule 4) 2. <expr>.expected_type <var>.actual_type (Rule 1) 3. <var>[2].actual_type lookup(a) (Rule 4) <var>[3].actual_type lookup(b) (Rule 4) 4. <expr>.actual_type either int or real (Rule 2) 5. <expr>.actual_type == <expr>.expected_type is either TRUE or FALSE (Rule 2) 56
57 A Fully Attributed Parse Tree 57
58 Evaluation Checking the static semantic rules of a language is an essential part of all compilers The main difficult in using an attribute grammar to describe all of the syntax and static semantics is its size and complexity 58
59 Dynamic Semantics About the meaning of expressions, statements, and program units There is no single widely acceptable notation or formalism for describing semantics Operational semantics Axiomatic semantics Denotational semantics 59
60 Operational Semantics Describe the meaning of a statement or program by translating it into a more easily understood language; this language is at the intermediate level Natural operational semantics (the highest level) The final result of the execution of a complete program Structural operational semantics (the lowest level) The precise meaning of a single statement, often by examination of the translated version of the statement 60
61 The Basic Process C statement for (expr1; expr2; expr3;) { } Operational Semantics expr1; loop: if expr2 == 0 goto out expr3; goto loop out: 61
62 Operational Semantics A list of statements in the lowlevel language ident = var ident = ident + 1 ident = ident 1 goto label if var relop var goto label // relop = {=, <>, >, < >=, <=} ident = var bin_op var ident = un_op var // un_op = unary operator These statements are all simple and therefore easy to understand and implement // bin_op = binary arithmetic operator 62
63 Evaluation Operational semantics provides an effective means of describing semantics for language users and language implementors The statements of one programming language are described in terms of the statements of a lowerlevel programming language This approach can lead to circularities, where concepts are indirectly defined in terms of themselves The other two approaches are based on logic and mathematics, not programming language 63
64 Axiomatic Semantics Based on formal logic (predicate calculus) Original purpose: formal program verification The logic expressions are called assertions The semantic of a statement is defined by the statement s effect on assertions about the data being processed by the statement 64
65 Assertions Precondition: an assertion immediately preceding a program statement describes the constraints on the program variables at that point Postcondition: an assertion immediately following a statement describes the new constraints on those variables after execution of the statement Weakest precondition: the least restrictive precondition that will guarantee the postcondition 65
66 Assertions Pre, post form: {P} statement {Q} An example a = b + 1 {a > 1} One possible precondition: {b > 10} Weakest precondition: {b > 0} 66
67 Program Verification 1) A program proof is begun by using the desired results of the program s execution as the postcondition of the last statement of the program 2) The postcondition is used to compute the weakest precondition for the last statement 3) This precondition is then used as the postcondition for the second last statement 4) The process continues until the beginning of the program is reached 5) The precondition of the first statement states the conditions under which the program will compute the desired results 6) If these conditions are implied by the input specification, the program is verified to be correct 67
68 Inference Rule A method of inferring the truth of one assertion on the basis of the values of other assertions General form: S 1, S2,..., Sn S If S1, S2,, and Sn (antecedent) are true, then the truth of S (consequent) can be inferred 68
69 Axiom An axiom is a logical statement that is assumed to be true An axiom is an inference rule without an antecedent 69
70 Assignment Statement Let x = E be a general assignment statement and Q be its postcondition Its precondition P is defined by the axiom P = Q x E It means P is computed as Q with all instances of x replaced by E 70
71 Assignment Statement Example: a = b / 2 1 {a < 10} The weakest precondition is computed as: b / 21 < 10 b < 22 Thus, the weakest precondition for the given assignment and postcondition is {b < 22} The axiom for the assignment statement (x = E): {Q x>e } x = E {Q} 71
72 Assignment Statement A given assignment statement with a precondition and a postcondition can be considered a theorem If the assignment axiom produces the given precondition, the theorem is proved 72
73 Assignment Statement Example {x > 3} x = x 3 {x > 0} Using the assignment axiom on x = x 3 { x > 0} produces {x > 3}, which is the given precondition Therefore, we have proven the example logical statement 73
74 Assignment Statement How to prove this statement: {x > 5} x = x 3 {x > 0} We need an inference rule, named the rule of consequence 74
75 Rule of Consequence The general form {P} S{Q},P' P, Q {P'}S{Q'} Q' means implies A postcondition can always be weakened and a precondition can always be strengthened 75
76 Rule of Consequence How to prove this statement: {x > 5} x = x 3 {x > 0} Use the rule of consequence {P} S{Q},P' P, Q {P'}S{Q'} Q' we prove the consequent is true: {x 3}x x  3{x 0},{x 5} {x {x 5}x x  3{x 3},{x 0} 0} {x 0} 76
77 Sequences Suppose we have the following adjacent statements {P1} S1 {P2} {P2} S2 {P3} An inference rule for sequences {P1}S1{P2},{P2}S2 {P3} {P1}S1; S2 {P3} 77
78 Sequences If S1 and S2 are assignment statements S1: x1 = E1 S2: x2 = E2 Then we have {P3 x2 E2 } x2 = E2 {P3} {(P3 x2 E2 ) x1 E1 } x1 = E1 {P3 x2 E2 } The weakest precondition for the sequence with postcondition P3 is {(P3 x2 E2 ) x1 E1 } 78
79 Sequences Example: y = 3 * x + 1; x = y + 3; {x < 10} The precondition for the last assignment statement is {y < 7}, which is used as the postcondition for the first statement. The first statement s precondition is {x < 2}. 79
80 Selection The general form of selection statements is if B then S1 else S2 The inference rule is {Band P}S1{Q},{(not B) and P}S2 {Q} {P}if B then S1 else S2 {Q} It indicates selection statements must be proven for both conditions 80
81 Selection Example if (x > 0) y = y 1 else y = y + 1 {y > 0} The precondition for the then clause is {y > 1}, while the precondition for the else clause is {y > 1}. Because {y > 1} {y >1}, the rule of consequence allow us to use {y > 1} for precondition of the selection statement. 81
82 Logical Pretest Loops An inference rule for logical pretest loops {P} while B do S end {Q} {I and B}S{I} {I}while B do S{I and (not B)} where I is the loop invariant (the inductive hypothesis). The complexity lies in finding an appropriate loop invariant. 82
83 Logical Pretest Loops The axiomatic description of a while loop is written as {P} while B do S end {Q} 83
84 Logical Pretest Loops The complete axiomatic description of a while construct requires all of the following to be true (the selection of I must meet the following conditions): P => I  the loop invariant must be true initially {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  the loop must be terminated 84
85 Logical Pretest Loops 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 85
86 Logical Pretest Loops Example 1: while y x do y = y + 1 end {y = x} For zero iteration, the weakest precondition is {y = x} For one iteration, it is wp(y = y + 1, {y = x}) = {y + 1 = x} or {y = x 1} where wp(statement, postcondition) = weak precondition 86
87 Logical Pretest Loops For two iterations, it is wp(y = y + 1, {y = x  1}) = {y + 1 = x 1}, or {y = x 2} For three iterations, it is wp(y = y + 1, {y = x  2}) = {y + 1 = x 2}, or {y = x 3} It is obvious {y < x} will suffice for one or more iterations; combining with {y = x} for the zero iteration, we get {y x}, which can be used for the loop invariant. 87
88 Logical Pretest Loops To make sure the four criteria to be true Because P = I, we have P I {I and B} S {I}. Proof: {y x and y x} y = y + 1 {y x} {I and (not B)} Q. Proof: {y x and y = x} {y = x} Loop termination. The question is whether the loop terminates: {y x} while y x do y = y + 1 end {y = x} 88
89 Logical Pretest Loops Example 2 while s > 1 do s = s / 2 end {s = 1} For zero iteration, the weakest precondition is {s = 1} For one iteration, it is wp(s = s / 2, {s = 1}) = {s / 2 = 1}, or {s = 2} For two iterations, it is wp(s = s / 2, {s = 2}) = {s / 2 = 2}, or {s = 4} 89
90 Logical Pretest Loops We can see the invariant is {s is a nonnegative power of 2} However, this is not a weakest precondition Consider using the precondition {s > 1}. Can the following logical statement be proven? {s > 1} while s > 1 do s = s / 2 end {s = 1} 90
91 Logical Pretest Loops It can not be proven using the precondition {s > 1} Violate rules 2 & 3 What do you think the invariant? 91
92 Logical Pretest Loops Finding loop invariants is not always easy It is often ignore the requirement of loop termination condition because its difficulty If loop termination can be shown, the axiomatic description of the loop is called total correctness If the other conditions can be met but termination is not guaranteed, it is called partial correctness 92
93 Program Proof Example: {x = A AND y = B} t = x; x = y; y = t; {x = B AND y = A} Use the assignment axiom and the inference rule for sequences for proof. 93
94 Program Proof Example: 1. {n 0} 2. count = n; 3. fact = 1; 4. while count 0 do 5. fact = fact * count; 6. count = count 1; 7. end 8. {fact = n!} 94
95 Let I = (fact = (count+1)* * n) & (count 0) (1) P => I (2) {I and B} S {I} {I and B} is ((fact = (count+1)* * n) & (count 0)) & (count 0) = ((fact = (count+1)* * n) & (count>0)) In line 6, {P} count = count1 {I} {P} is {(fact = count*(count+1)* *n) & (count >= 1)} In line 5, {P} fact = fact*count {(fact = count*(count+1)* *n) & (count >= 1)} {P} is {(fact = (count+1)* *n) & (count >= 1)} same to {I and B} 95
96 (3) I & (NOT B) => Q (fact = (count+1)* *n) & (count>=0)) & (count=0)) fact = n! We have proven the choice of I meets the requirements for a loop invariant! In line 3, {P} fact = 1 {fact = (count+1)* *n) & (count >= 0)} {P} is (1 = (count+1)* *n) & (count >= 0)) In line 2, {P} count = n {(1=(count+1)* *n) & (count >= 0)) {P} to be {(n+1)* *n = 1) & (n >=0)}?? 96
97 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 97
98 Denotational Semantics Denotational semantics is the most rigorous, widely known method for describing the meaning of programs Based on recursive function theory The most abstract semantics description method Originally developed by Scott and Strachey (1970) 98
99 Denotational Semantics The process of building a denotational specification for a language 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 99
100 Denotation Semantics vs. Operational Semantics In operational semantics, the state changes are defined by coded algorithms In denotational semantics, the state changes are defined by rigorous mathematical functions 100
101 Denotational Semantics Example I The syntax of binary numbers <bin_num> 0 1 <bin_num> 0 <bin_num> 1 101
102 Denotational Semantics Example I The semantic function, named M bin, maps the syntactic objects (described in grammar rules) to the objects in N M bin ( 0 ) = 0 M bin ( 1 ) = 1 M bin (<bin_num> 0 ) = 2 * M bin (<bin_num>) M bin (<bin_num> 1 ) = 2 * M bin (<bin_num>)
103 Denotational Semantics Example I The meanings, or denoted objects, can be attached to the nodes of the parse tree 103
104 Denotational Semantics Example II The syntax of decimal literals <dec_num> <dec_num> ( ) The denotational mappings are M dec ( 0 ) = 0, M dec ( 1 ) = 1,, M dec ( 9 ) = 9 M dec (<dec_num> 0 ) = 10 * M dec (<dec_num>) M dec (<dec_num> 1 ) = 10 * M dec (<dec_num>) + 1 M dec (<dec_num> 9 ) = 10 * M dec (<dec_num>)
105 The State of a Program The state s of a program is the values of all its current variables s = {<i 1, v 1 >, <i 2, v 2 >,, <i n, v n >} Let VARMAP be a function that, when given a variable name and a state, returns the current value of the variable VARMAP(i j, s) = v j 105
106 Expressions We assume expressions are decimal numbers, variables, or binary expressions having one arithmetic operator and two operands, each of which can be an expression BNF syntax <expr> <dec_num> <var> <binary_expr> <binary_expr> <left_expr> <operator> <right_expr> <left_expr> <dec_num> <var> <right_expr> <dec_num> <var> <operator> + * 106
107 M e (<expr>, s) = case <expr> of <dec_num> => M dec (<dec_num>, s) <var> => if VARMAP(<var>, s) == undef error else VARMAP(<var>, s) <binary_expr> => if (M e (<binary_expr>.<left_expr>, s) == undef OR M e (<binary_expr>.<right_expr>, s) = undef) then error else if (<binary_expr>.<operator> == + M e (<binary_expr>.<left_expr>, s) + M e (<binary_expr>.<right_expr>, s) else M e (<binary_expr>.<left_expr>, s) * M e (<binary_expr>.<right_expr>, s) 107
108 Assignment Statements An assignment statement is an expression evaluation plus the setting of the leftside variable to the expression s value In this case, the meaning function maps a state to a state 108
109 M a (x = E, s) = if M e (E, s) == error error else s = {<i 1,v 1 >,<i 2,v 2 >,...,<i n,v n >}, where for j = 1, 2,..., n, if i j == x v j = M e (E, s) else v j = VARMAP(i j, s) 109
110 Logical Pretest Loops Using two other mapping functions M sl : maps statement lists to states M b : maps Boolean expressions to Boolean values (or error) 110
111 M l (while B do L, s) = if M b (B, s) == undef error else if M b (B, s) == false s else if M sl (L, s) == error error else M l (while B do L, M sl (L, s)) 111
112 Loop Meaning In essence, the loop has been converted from iteration to recursion, where the recursive control is mathematically defined by other recursive state mapping functions Compared to iteration, recursion is easier to describe with mathematical rigor 112
113 Evaluation 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 113
114 Summary BNF and contextfree grammars are equivalent metalanguages Wellsuited 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 114
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