c laus b rabrand © s emantics (q1,’06) a ug 31, 2006 c laus b rabrand © 2005–2006, university...

CLAUS BRABRAND © SEMANTICS (Q1,’06) AUG 31, 2006

CLAUS BRABRAND

© 2005–2006 , University of Aarhus

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

SEMANTICS (Q1,’06)

WEEK 1

CLAUS BRABRAND SEMANTICS (Q1,’06)[ 2 ]

AUG 31, 2006

Week 1 - Outline

Introduction Welcome Short-film about teaching (20')... Course Presentation [ homepage ]

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


AUG 31, 2006

Short-film about teaching...


AUG 31, 2006

Course Introduction

[ homepage ]


AUG 31, 2006

Learning vs. Breaks

#Breaks, |Breaks|, ...?

Conclusion:more (shorter) breaks


INFERENCE SYSTEMS

Keywords:

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


AUG 31, 2006

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’


AUG 31, 2006

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


AUG 31, 2006

Terminology

Meaning: Inductive:

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

“m is even, if n is even (provided m = n+2)”

|_even n

|_even m

m = n+2

premise(s)

conclusion

side-condition(s)


AUG 31, 2006

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

Even so, this is what we mean


AUG 31, 2006

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]


AUG 31, 2006

Activation Exercise 1

Activation Exercise: 1. Specify the signature of the relation: '<<'

x << y "y is-double-that-of x"

2. Specify the relation via an inference system i.e. axioms and rules

3. Prove that indeed: 3 << 6 "6 is-double-that-of 3"

When you are done, take a break (see you 13:15)


AUG 31, 2006

Activation Exercise 2

Activation Exercise: 1. Specify the signature of the relation: '//'

x // y "x is-half-that-of y"

2. Specify the relation via an inference system i.e. axioms and rules

3. Prove that indeed: 3 // 6 "3 is-half-that-of 6"

Syntactically differentSemantically the same relation


AUG 31, 2006

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!

|_R 0|_R n

|_R n+2

[rule1][axiom1]

|_R Z

(0 ‘|_R’) ( n ‘|_R’ n+2 ‘|_R’)


AUG 31, 2006

Inductive Interpretation

A inference system:

…induces a function:

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

|_R 0|_R n

|_R n+2

[rule1][axiom1]

FR: P(Z) P(Z)

|_R Z

FR(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(FR) = FRn(Ø)n

|_R P(Z)

From rel. to rel.

2N =

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


AUG 31, 2006

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”


TRANSITION SYSTEMS

Keywords:

Configuration, Final Configuration, Transition, Action / label


AUG 31, 2006

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:

,

’ (,’)

’ |_ ’ ’ …


AUG 31, 2006

The transition may be illustrated as:

In this course, we will be (mostly) using:

For instance:

A Transition

’

’

= system configuration = < program , data >

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

...and have "" describe "one step of computation"


AUG 31, 2006

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 T : : t


AUG 31, 2006

E.g.: Finite Automaton

A Non-det Finite Automaton (NFA) is a 5tuple:

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} }


AUG 31, 2006

E.g.: NFA 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:

M = Q , , q0 , F , Given


AUG 31, 2006

E.g.: NFA Trans. Sys. (Cont’d)

Define behavior of M:

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> * }


AUG 31, 2006

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, ’) ‘’


AUG 31, 2006

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


AUG 31, 2006

E.g.: NFA 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 ,


AUG 31, 2006

E.g.: NFA Trans. Sys. (Cont’d)

Behavior:

Define ”*” as the reflexive 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’’


AUG 31, 2006

NFA: 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


AUG 31, 2006

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


AUG 31, 2006

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


AUG 31, 2006

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


AUG 31, 2006

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”


NEXT WEEK…

Semantics of…:

“Simple Commands and Expressions”


AUG 31, 2006

Next week

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

…to: describe explain analyse compare

…semantics of Expressions:


AUG 31, 2006

"Three minutes paper"

Please spend three minutes writing down the most important things that you have learned today (now).

Efter en dagEfter en uge

Efter 3 ugerEfter 2 uger

Med det samme


Exercises...

[ homepage ]


See you next week…

Questions?

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