semantics q1 2007 s emantics (q1,’07) week 1 jacob andersen phd student [email protected]

Semantics Q1 2007

SEMANTICS (Q1,’07)Week 1

Jacob AndersenPhD student

[email protected]

Semantics Q1 2007

Week 1 - Outline• Introduction

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

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

Semantics Q1 2007

Introduction• Semantics

– Merriam-Webster:– Example:

• “I would like a cup of coffee.”• “Colourless green ideas sleep furiously.”

• Question:Would you fly the space shuttle if you had writtenthe guidance software???

• In this course we shall explore formal ways of reasoning about the meaning / behaviour of computer programs.

Main Entry: se·man·tics Pronunciation: si-'man-tiksFunction: noun plural but singular or plural in construction1 : the study of meanings

Semantics Q1 2007

Short-film about teaching...

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Course Introduction I

SOLO 2, 3, 4

Verbs identifies skills / competences

Actor

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Course Introduction II

• No-one can force new skills into your head– You do not learn how to ride a bike by observing others (!!)

• Expected weekly time consumption:

Lecture 3

Reading 1

Exercise class 3

Hand-in 2

Sum 13

Fitting the pieces together

Lecture 3

Reading 1

Reflecting / Solving exercises

4

Exercise class 3

Hand-in 2

Sum 13

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Course Introduction III

• Course RetrospectiondSem ’05:

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Learning vs. Breaks• #Breaks, |Breaks|, ...?

Conclusion:more (shorter) breaks

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RELATIONSAND

INFERENCE SYSTEMS

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Relations I• 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:

… and as: as a short-hand for:

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

… and 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’

Semantics Q1 2007

Relations II• In general, a k-ary relation, L, over the

setsS1, S2, … , Sk

is a subsetL S1 S2 … Sk

• k is called the “arity” of L, and “L S1 S2 … Sk” is the “signature”.

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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

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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)

Question:

Why have both premises and side-conditions??

Note: The “ “ is actually just another way of writing a “” !

Semantics Q1 2007

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

Warning:Be careful !

Semantics Q1 2007

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]

Semantics Q1 2007

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"

Semantics Q1 2007

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

Semantics Q1 2007

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’)

Semantics Q1 2007

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”

Semantics Q1 2007

TRANSITION SYSTEMS

Semantics Q1 2007

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 might use the following notations:

,

’ (,’)

’ |_ ’ ’ , , …

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• 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"

Semantics Q1 2007

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

Semantics Q1 2007

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} }

Semantics Q1 2007

E.g.: NFA Transition System (1)

• Define Term. Trans. Syst. 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

Semantics Q1 2007

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> * }

Semantics Q1 2007

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, ’) ‘’

Semantics Q1 2007

• 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

Semantics Q1 2007

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

M := {(q,a,q’)|q’(q,a)}

AM :=

M = Q , , q0 , F ,

X

???

“Relaxed”v

Semantics Q1 2007

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

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

toss

heads

toss

tails

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

Semantics Q1 2007

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

Semantics Q1 2007

More Examples…• More Examples in [Plotkin, p. 22 – 26]:

– 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…

Semantics Q1 2007

THE LANGUAGE “L”

Semantics Q1 2007

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

Semantics Q1 2007

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

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• Consider program:• It can be (ambiguously) understood:

• Note:

Abstract- vs. Concrete Syntaxwhile 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”

Semantics Q1 2007

Next week

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

• …to:– describe– explain– analyse– compare

…semantics of Expressions:

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"Three minutes paper"• Please spend three minutes writing down the

most important things that you have learned today (now).

After 1 dayAfter 1 week

After 3 weeksAfter 2 weeks

Right away

Semantics Q1 2007

Exercises...

[ homepage ]

Semantics Q1 2007

See you next week…

Questions?

Semantics Q1 2007

semantics q1 2007 s emantics (q1,’07) week 1 jacob andersen phd student [email protected]

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