C LAUS B RABRAND © S EMANTICS (Q1,’05) S EP 1, 2005 C LAUS B RABRAND © 2005, University of...

CLAUS BRABRAND © SEMANTICS (Q1,’05) SEP 1, 2005

CLAUS BRABRAND

© 2005, University of Aarhus

[ [email protected] ] [ http://www.daimi.au.dk/~brabrand/ ]

SEMANTICS (Q1,’05)

WEEK 1

CLAUS BRABRAND SEMANTICS (Q1,’05)[ 2 ]

SEP 1, 2005

Week 1 - Outline

Introduction Welcome Course Presentation [ homepage ]

Prerequisitional (discrete) Mathematics Relations Inference Systems Transition Systems The Language “L” Virtual Machine Semantics


INFERENCE SYSTEMS

Keywords:

relations, axioms, rules, (co-)inductive interpretation, fixed-points


SEP 1, 2005

Relations

Example1: “even” relation: Written as: as a short-hand for:

… and as: as a short-hand for:

Example2: “equals” relation: Written as: as a short-hand for:


Example3: “DFA transition” relation: Written as: as a short-hand for:


|_even Z

|_even 4

|_even 5

4 |_even

5 |_even

2 3 (2,3) ‘=’

‘=’ Z Z

(2,2) ‘=’2 = 2

‘’ Q Q

q q’ (q, , q’) ‘’

(p, , p’) ‘’p p’


SEP 1, 2005

Inference System

Inference System: Inductive (recursive) specification of relations Consists of axioms and rules

Example: Axiom:

“0 (zero) is even”!

Rule: “If n is even, then m is even (where m = n+2)”

|_even 0

|_even n

|_even m

m = n+2

|_even Z


SEP 1, 2005

Terminology

Meaning: “If n is even, then m is even (provided m = n+2)”; or “m is even, if n is even (provided m = n+2)”

|_even n

|_even m

m = n+2

premise(s)

conclusion

side-condition(s)


SEP 1, 2005

Abbreviation

Often, rules are abbreviated:

Rule: “If n is even, then m is even (provided m = n+2)”; or “m is even, if n is even (provided m = n+2)”

Abbreviated rule: “If n is even, then n+2 is even”; or “n+2 is even, if n is even”

|_even n

|_even n+2

|_even n

|_even m

m = n+2


SEP 1, 2005

Relation Definition (Interpretation)

Actually, an inference system:

…is a demand specification for a relation:

The three relations: R = {0, 2, 4, 6, …} (aka., 2N) R’ = {0, 2, 4, 5, 6, 7, 8, …} R’’ = {…, -2, -1, 0, 1, 2, …} (aka., Z)

…all satisfy the (above) specification!

|_even 0|_even n

|_even n+2

[rule1][axiom1]

|_even Z

(0 ‘|_even’) ( n ‘|_even’ n+2 ‘|_even’)


SEP 1, 2005

Inductive Interpretation

A relation:

…induces a function:

Definition: ‘lfp’ (least fixed point) ~ least solution:

|_even 0|_even n

|_even n+2

[rule1][axiom1]

F: P(Z) P(Z)

|_even Z

F(R) = {0} { n+2 | n R }

F(Ø) = {0} F2(Ø) = F({0}) = {0,2} F3(Ø) = F2({0}) = F({0,2}) = {0,2,4} …

|_even := lfp(F) = Fn(Ø)n

|_even P(Z)

From rel. to rel.

2N =

Fn(Ø) ~ “Anything that can be proved in ‘n’ steps”


SEP 1, 2005

Co-inductive Interpretation ()

A relation:

…induces a function:

Definition: ‘gfp’ (greatest fixed point) ~ greatest solution:

|_even 0|_even n

|_even n+2

[rule1][axiom1]

F: P(Z) P(Z)

|_even Z

F(R) = {0} { n+2 | n R }

F(Z) = Z

|_even := gfp(F) = Fn(Z)n

|_even P(Z)

From rel. to rel.

F2(Z) = F(Z) = Z F3(Z) = F2(Z) = F(Z) = Z … Z =

Fn(Z) ~ “Anything that cannot be disproved in ‘n’ steps”


SEP 1, 2005

Example: “less-than-or-equal-to”

Relation:

Is ”1 2” ?!? Yes, because there exists an inference tree:

In fact, it has two inference trees:

0 0 n mn m+1

[rule1][axiom1]

‘’ N N

n mn+1 m+1

[rule2]

0 00 11 2

[rule2]

[rule1][axiom1] 0 0

1 11 2

[rule1]

[rule2][axiom1]


TRANSITION SYSTEMS

Keywords:

Configuration, Final Configuration, Transition, Action / label


SEP 1, 2005

Definition: Transition System

A Transition System is (just) a structure:

is a set of configurations is a binary relation

(called the transition relation)

We will write instead of

Other times we use the following notation:

,

’ (,’)

’ |_ ’ ’ …


SEP 1, 2005

The transition may be illustrated as:

We will (often) be using:

For instance:

A Transition

’

’

= system configuration = < program , data >

< “x:=y ; y:=0” , [x=3,y=7] > < “y:=0” , [x=7,y=7] >


SEP 1, 2005

Def: Terminal Transition System

A Terminal Transition System is a structure:

is the set of configurations is the transition relation T is a set of final configurations

…satisfying:

i.e. “all configurations in ‘T’ really are terminal”.

, , T

T : ’ : ’


SEP 1, 2005

Example: Finite Automaton

A Finite Automaton (FA) is a quintuplet:

Q finite set of states finite set of input symbols q0 Q initial state F Q set of acceptance states : Q P(Q) state transition relation

M = Q , , q0 , F ,

toss

toss

heads

tails

0 1 2

Q = { 0,1,2 } = { = { toss,heads,tails } (0,toss) {1,2} ,q0 = 0 (1,heads) {0} ,F = { 0 } (2,tails) {0} }


SEP 1, 2005

Example: FA Transition System (1)

Define (FA) Trans. Sys. by:

Configurations:

Transition relation:

i.e., we have whenever

Final Configurations:

M := Q *

M , M , TM

TM := { <q,> | q F }

M := { ( <q,aw> , <q’,w> ) | q,q’Q, a, w*, q’(q,a) }

“State component”“Data component”

q’ (q,a)<q,aw> M <q’,w>

Recall:


SEP 1, 2005

Example: FA Trans. Sys. (Cont’d)

Behavior:

toss

toss

heads

tails

0 1 2

<0, toss;heads;toss;tails> M <1, heads;toss;tails> M

<0, toss;tails> M

<2, tails> M

<0, > TM

L(M) := { w * | T : <q0,w> * }


SEP 1, 2005

Def: Labelled Transition System

A Labelled Transition System is a structure:

is the set of configurations A is the set of actions (=

labels) A is the transition relation

Note: we will write instead of

, A ,

’a (, a, ’) ‘’


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The labelled transition may be illustrated as:

The labels(/actions) add the opportunity of describing transitions:

Internally (e.g., information about what went on internally) Externally (e.g., information about communication /w env) …or both.

A Labelled Transition

’

’a

a


SEP 1, 2005

Example: FA Transition System (2)

Given Finite Automaton:

Define Labelled Terminal Transition System:

Configurations:

Labels:

Transition relation:

Final configurations:

M := Q

M , AM , M , TM

TM := F

q M q’ := q’(q,a)

AM := a

M = Q , , q0 , F ,


SEP 1, 2005

Example: FA Trans. Sys. (Cont’d)

Behavior:

Define ”*” as the transitive closure of ””on sequences of labels:

toss

toss

heads

tails

0 1 2

0 M 1 M 0 M 2 M 0 TM

L(M) := { w A* | qT : q0 * qw

toss

heads

toss

tails

L(M) := { a a’ … a’’ A* | qT : q0 q’ … qa a’ a’’


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FA: Machine 1 vs. Machine 2

The two transition systems are very different:

Machine 1: ”I transitioned from <q,aw> to <q’,w>”

Implicit: ”…by consuming part of (internal) data component”

Machine 2: ”I transitioned from q to q’

…by inputing an ’a’ symbol from the (external) environment!”

<q,aw> M <q’,w>

q M q’a


SEP 1, 2005

More Examples…

More Examples in [Plotkin, p. 6 – 13]: Three Counter Machine (***) Context-Free Grammars (**) Petri Nets (*)

They illustrate expressive power of transition systems no new points

...except formalizing input/output behavior (also later here…)

Read these yourself…


THE LANGUAGE “L”

Keywords:

Expressions, Boolean Expressions, Commands, Abstract Syntax, Concrete Syntax


SEP 1, 2005

The Language ”L” Basic Syntactic Sets:

Truthvalues: Set ranged over by: t , t’, t0, …

Numbers: Set ranged over by: m, n, …

Variables: Set ranged over by: v, v’,

…

T = {tt, ff}

N = {0, 1, 2, …}

VAR = {a, b, c, …, z}

Meta-variables


SEP 1, 2005

The Language ”L” Derived Syntactic Sets:

Arithmetic Expressions (e Exp):

Boolean Expressions (b BExp):

Commands (c Com):

e ::= n | v | e + e’ | e – e’ | e e’

b ::= t | e = e’ | b or b’ | ~ b

c ::= nil | v := e | c ; c’ | if b then c else c’ | while b do c


SEP 1, 2005

Consider program: It can be (ambiguously) understood:

Note:

Abstract- vs. Concrete Syntax

while b do c ; c’

while b do c ; c’ while b do c ; c’

…either as: …or as:

“Concrete syntax”

while

b c

;

c’

while

b

c

;

c’

“Abstract syntax”

Parsing: “Concrete syntax” “Abstract syntax”


VIRTUAL MACHINE SEMANTICS

Keywords:

The “SMC” Machine, Virtual Machine Semantics, Machine Code Semantics


SEP 1, 2005

Value Stack: Set ranged over by: S

Memories: Functions ranged over by: M

Control Stack: Set ranged over by: C

SMC Machine (Trans. Sys. Semantics): Configurations:

Virtual Machines (SMC Expressions)

= Value Stack Memories Control Stack

( T N )*

VAR N

( Exp { +, –, } )*

Initially Commands, and later bits of commands

For accumulating partial results


SEP 1, 2005

Configurations:

Notation:

i.e., “Memory Update” More common syntax for this:

Memory Update

= ( T N VAR )* VAR N ( Exp { +, –, } )*

m , if v = v’M’(v’) = M(v’) , otherwise

M[m/v]

M[m/v] = M’ where

M[mv]


SEP 1, 2005

Virtual Machine Transitions

Configurations:

Transitions: Defined by case according to top of control stack:

= ( T N VAR )* VAR N ( Exp { +, –, } )*

[case n]: < S , M , n C > < n S , M , C >

[case v]: < S , M , v C > < M(v) S , M , C >

[case ee’]: < S , M , e e’ C > < S , M , e e’ C >

[case +/-/]: < m’ m S , M , C > < n S , M , C > …where n = m m’

Syntactic ‘+’

Semantic ‘+’


SEP 1, 2005

Virtual Machine: Example

Given program: and memory:(((x + 1) – y) 7)

< , M, (((x + 1) – y) 7) >

< , M, (x + 1) ; y ; – ; 7 ; >

< , M, ((x + 1) – y) ; 7 ; >

M = [x=9,y=4]

< , M, x ; 1 ; + ; y ; – ; 7 ; >

< , M, ((x + 1) – y) ; 7 ; > < , M, (x + 1) ; y ; – ; 7 ; >

< , M, x ; 1 ; + ; y ; – ; 7 ; >

< 9, M, 1 ; + ; y ; – ; 7 ; >

< 9, M, 1 ; + ; y ; – ; 7 ; > < 1 ; 9, M, + ; y ; – ; 7 ; >

< 9 ; 1, M, + ; y ; – ; 7 ; > < 10, M, y ; – ; 7 ; >

< 10, M, y ; – ; 7 ; > < 4 ; 10, M, – ; 7 ; >

< 4 ; 10, M, – ; 7 ; > < 6, M, 7 ; >

< 6, M, 7 ; > < 7 ; 6, M, >

< 7 ; 6, M, > < 42, M, >


SEP 1, 2005

VM Semantics: Major Drawbacks!

Advantage: Easy to implement (and efficient)

Drawbacks:

Non-intuitive Too concrete (e.g., stack) Indirect semantics (not syntax directed) Computational step?

“High-level language understood in terms of low-level machine code”

“Many other machine along these lines […]. They all have a tendency to pull the syntax into pieces or at any rate to wander around the syntax creating various complex symbolic structures which do not seem particularly forced by the demands of the language itself”

- Gordon Plotkin, ‘81


NEXT WEEK…

Semantics of…:

“Simple Commands and Expressions”


SEP 1, 2005

Next week

Use everything…: Inference Systems, Transition Systems, Syntax, …

…to: describe explain analyse compare

…semantics of Expressions:


See you next week…

Questions?

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