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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

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 3

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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Parse Trees And AST’s

 The grammar generates parse trees

 The AST is a simplified form: same order

as the parse tree, but no non-terminals

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 6

< exp >

< exp > + < mulexp >

3

< mulexp > ***** < rootexp >

1 2

< mulexp >

< rootexp > < rootexp >

+

1

2

---

3

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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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Language One: Prolog Interpreter

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 9

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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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 Constant

 Some Prologs treat values greater than 2^31 -

as floating-point constants; others don’t

 Did we mean Language One to do this?

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 12

?- val1(const(2147483647),X). X = 2147483647.

?- val1(const(2147483648),X). X = 2.14748e+

?- val1(const(2147483647),X). X = 2147483647.

?- val1(const(2147483648),X). X = 2147483648.

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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

Natural Semantics

 A formal notation we can use to capture the

same basic proof rules in val

 We are trying to define the relation between

an AST and the result of evaluating it

 We will use the symbol → for this relation,

writing E → v to mean that the AST E

evaluates to the value v

 For example, our semantics should establish

times(const(2),const(3)) → 6

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 15

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 2 ) 12

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 2 ) 12

1 1 2 2

E , E v v

E v E v

→ +

→ →

plus

( 1 2 ) 12

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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Natural Semantics, Note

 There may be more than one rule for a

particular kind of AST node

 For instance, for an ML-style if-then-else

we might use something like this:

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 18

E 1 → true E 2 → v 2

if ( E 1 , E 2 , E 3 ) → v 2

E 1 → false E 3 → v 3

if ( E 1 , E 2 , E 3 ) → v 3

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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

Syntax

 (A subset of ML expressions)

 This grammar is unambiguous

 A sample Language Two expression:

let val y = 3 in y*y end

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 21

< exp > ::= < exp > + < mulexp > | < mulexp > < mulexp > ::= < mulexp > ***** < rootexp > | < rootexp > < rootexp > ::= let val < variable > = < exp > in < exp > end | ( < exp > ) | < variable > | < constant >

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Abstract Syntax

 Two more kinds of AST nodes:

– var(X) for a reference to a variable X

– let(X,Exp1,Exp2) for a let expression

that evaluates Exp2 in an environment where

the variable X is bound to the value of Exp1

 So for the Language Two program

let val y = 3 in y*y end

we have this AST:

let(y,const(3),times(var(y),var(y)))

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 22

Representing Contexts

 A representation for contexts:

– bind(Variable,Value) = the binding

from Variable to Value

– A context is a list of zero or more bind terms

 For example:

– The context in which y is bound to 3 could be

[bind(y,3)]

– The context in which both x and y are bound

to 3 could be [bind(x,3),bind(y,3)]

Chapter Twenty-Three Modern Programming Languages, 2nd ed. (^) Docsity.com 23

Looking Up A Binding

 Looks up a binding in a context

 Finds the most recent binding for a given

variable, if more than one

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 24

lookup(Variable,[bind(Variable,Value)|],Value) :- !. lookup(VarX,[|Rest],Value) :- lookup(VarX,Rest,Value).

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Language Two: Prolog Interpreter

Chapter Twenty-Three Modern Programming Languages, 2nd ed. 25

val2(plus(X,Y),Context,Value) :- val2(X,Context,XValue), val2(Y,Context,YValue), Value is XValue + YValue. val2(times(X,Y),Context,Value) :- val2(X,Context,XValue), val2(Y,Context,YValue), Value is XValue * YValue. val2(const(X),_,X). val2(var(X),Context,Value) :- lookup(X,Context,Value). val2(let(X,Exp1,Exp2),Context,Value2) :- val2(Exp1,Context,Value1), val2(Exp2,[bind(X,Value1)|Context],Value2).

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