School of Computing and Mathematics, University of Huddersfield

CHA2545: WEEK 4

LECTURE:

DENOTIONAL SEMANTICS OF A SIMPLE LANGUAGE

TUTORIAL:

Do exercises given out

DURING THE WEEK:

Go through ARJ’s supporting WEB notes (from main web page).

School of Computing and Mathematics, University of Huddersfield

The Four Parts of a Semantic Definition:

Syntactic domains Abstract Syntax Semantic Domains Meaning Functions

Example syntax domains (see handout)

Nml, Ide, Exp, Cmd

School of Computing and Mathematics, University of Huddersfield

DENOTATIONS

The semantics of the various syntactic structures of a language is defined via meaning functions which take the syntactic structure as a parameter and returns its denotation.

A “denotation” is a mathematical representation of the meaning of a construct - its an object belonging to a SEMANTIC DOMAIN.

School of Computing and Mathematics, University of Huddersfield

SEMANTIC DOMAINS - Examples

Simple Semantic Domains -

T - truth values

N - natural numbers

Z - integers

Complex Domains

are made up by combining Simple Domains with operators such as +, x, ->

School of Computing and Mathematics, University of Huddersfield

COMPLEX DOMAINS - Examples Disjoint Union Domain of all integers and truth

values

Z + T Cross Product Domain of all pairs of integers

Z x Z Function Domain consisting of all functions

from integers to truth values

Z -> Z

and pairs of integers to integers

(Z x Z) -> Z

School of Computing and Mathematics, University of Huddersfield

COMPLEX DOMAINS - Another Example

All “expressible” values in a simple programming language with truth values and natural numbers:

E = N + T + {error}

School of Computing and Mathematics, University of Huddersfield

The STORE

The simplest way to represent the STATE of program execution is by a STORE - representing the values stored at any point in the execution of a program.

The store is represented by a function domain

S = [ Ide -> E ] + {error}

We usually let ‘s’ be a typical store or state.

School of Computing and Mathematics, University of Huddersfield

Commands

Commands change the program state

The domain of commands is therefore the functions between states, i.e.

S -> S

School of Computing and Mathematics, University of Huddersfield

Semantic Function: Emeans

Emeans takes a syntactically well formed expression, and gives it a denotation in terms of a

function between stores and expressible values

so

Emeans : Exp -> S -> E

The definition of Emeans depends on the syntactic subclass to which it is applied.

School of Computing and Mathematics, University of Huddersfield

Semantic Function: Cmeans

Cmeans takes a syntactically well formed command, and gives it a denotation in terms of a

function between stores and stores

so

Cmeans : Cmd -> S -> S

The definition of Cmeans depends on the syntactic subclass to which it is applied

School of Computing and Mathematics, University of Huddersfield

The Problem with Recursive Functions - what do they mean?

Cmeans[[while B do C]] =

s.(Emeans[[B]]s ->

Cmeans[[while B do C]](Cmeans[[C]]s), s)

OR:

Cmeans[[while B do C]] = f where

f = s.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)

School of Computing and Mathematics, University of Huddersfield

Recursive Functions - The Fixpoint Equation

Given:

f = s.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)

Then:

H = f.s.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)

Hf = (f.s.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)) f

= s.(Emeans[[B]]s -> f(Cmeans[[C]]s), s)

= f

SO WE HAVE Hf = f - the fixpoint equation!!