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Formal Semantics

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 1

Formal Semantics

 At the beginning of the book we saw formal

definitions of syntax with BNF

 And how to make a BNF that generates

correct parse trees: “where syntax meets

semantics”

 We saw how parse trees can be simplified

into abstract syntax trees (AST’s)

 Now… the rest of the story: formal

definitions of programming language

semantics

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 2

Defining Language One

 A little language of integer expressions:

– Constants

– The binary infix operators + and * , with the

usual precedence and associativity

– Parentheses for grouping

 Lexical structure: tokens are + , * , ( , ) , and

integer constants consisting of one or more

decimal digits

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 4

Syntax: Phrase Structure

 (A subset of ML expressions, Java

expressions, and Prolog terms)

 This grammar is unambiguous

 Both operators are left associative, and *

has higher precedence than +

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 5

< exp > ::= < exp > + < mulexp > | < mulexp > < mulexp > ::= < mulexp > ***** < rootexp > | < rootexp > < rootexp > ::= ( < exp > ) | < constant >

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Continuing The Definition

 That is as far as we got in Chapters 2 and 3

 One way to define the semantics of the

language is to give an interpreter for it

 We will write one in Prolog, using AST’s as

input:

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 7

+

1

2

---

3 plus(const(1),times(const(2),const(3)))

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Abstract Syntax

 Note: the set of legal AST’s can be defined

by a grammar, giving the abstract syntax of

the language

 An abstract syntax can be ambiguous, since

the order is already fixed by parsing with

the original grammar for concrete syntax

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 8

< exp > ::= plus( < exp > , < exp > ) | times( < exp > , < exp > ) | const( < constant > )

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Chapter Twenty-Three Modern Programming Languages, 2nd ed. 10

?- val1(const(1),X). X = 1.

?- val1(plus(const(1),const(2)),X). X = 3.

?- val1(plus(const(1),times(const(2),const(3))),X). X = 7.

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Problems

 What is the value of a constant?

– Interpreter says val1(const(X),X).

– This means that the value of a constant in

Language One is whatever the value of that

same constant is in Prolog

– Unfortunately, different implementations of

Prolog handle this differently

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 11

Value Of A Sum

 Some Prologs expresses sums greater than

2 31 -1 as floating-point results; others don’t

 Did we mean Language One to do this?

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 13

?- val(plus(const(2147483647),const(1)),X). X = 2.14748e+009.

?- val(plus(const(2147483647),const(1)),X). X = 2147483648.

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Defining Semantics By

Interpreter

 Our val1 is not satisfactory as a definition

of the semantics of Language One

 “Language One programs behave the way

this interpreter says they behave, running

under this implementation of Prolog on this

computer system ”

 We need something more abstract

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 14

A Rule In Natural Semantics

 Conditions above the line, conclusion below

 The same idea as our Prolog rule:

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 16

1 1 2 2

E , E v v

E v E v

→ ×

times

val1(times(X,Y),Value) :- val1(X,XValue), val1(Y,YValue), Value is XValue * YValue.

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Language One, Natural Semantics

 Of course, this still needs definitions for

+, × and eval , but at least it won’t

accidentally use Prolog’s

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 17

1 1 2 2

E , E v v

E v E v

plus

1 1 2 2

E , E v v

E v E v

→ ×

times

const ( ) n → eval ( n )

val1(plus(X,Y),Value) :- val1(X,XValue), val1(Y,YValue), Value is XValue + YValue. val1(times(X,Y),Value) :- val1(X,XValue), val1(Y,YValue), Value is XValue * YValue. val1(const(X),X).

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Outline

 Natural semantics and Prolog interpreters

– Language One

– Language Two: adding variables

– Language Three: adding functions

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 19

Defining Language Two

 That one was too easy!

 To make it a little harder, let’s add:

– Variables

– An ML-style let expression for defining them

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 20