NICTA Advanced Course Theorem Proving Principles...

NICTA Advanced Course

Theorem Proving

Principles, Techniques, Applications

{P} . . . {Q}

CONTENT

➜ Intro & motivation, getting started with Isabelle

➜ Foundations & Principles

• Lambda Calculus

• Higher Order Logic, natural deduction

• Term rewriting

➜ Proof & Specification Techniques

• Inductively defined sets, rule induction

• Datatypes, recursion, induction

• More recursion, Calculational reasoning

• Hoare logic, proofs about programs

• Locales, Presentation

CONTENT 2

LAST TIME

➜ Recdef

➜ More induction

➜ Well founded orders

➜ Well founded recursion

➜ Calculations: also/finally

➜ [trans]-rules

LAST TIME 3

A CRASH COURSE IN SEMANTICS

IMP - A SMALL IMPERATIVE LANGUAGE

Commands:

datatype com = SKIP| Assign loc aexp ( := )

| Semi com com ( ; )

| Cond bexp com com (IF THEN ELSE )| While bexp com (WHILE DO OD)

types loc = stringtypes state = loc ⇒ nat

types aexp = state ⇒ nattypes bexp = state ⇒ bool

IMP - A SMALL IMPERATIVE LANGUAGE 5

Commands:

IMP - A SMALL IMPERATIVE LANGUAGE 5-A

Commands:

IMP - A SMALL IMPERATIVE LANGUAGE 5-B

EXAMPLE PROGRAM

Usual syntax:B := 1;

WHILE A 6= 0 DOB := B ∗ A;

A := A − 1

Expressions are functions from state to bool or nat:

B := (λσ. 1);

WHILE (λσ. σ A 6= 0) DOB := (λσ. σ B ∗ σ A);

A := (λσ. σ A − 1)

EXAMPLE PROGRAM 6

EXAMPLE PROGRAM

Usual syntax:B := 1;

WHILE A 6= 0 DOB := B ∗ A;

A := A − 1

Expressions are functions from state to bool or nat:

B := (λσ. 1);

WHILE (λσ. σ A 6= 0) DOB := (λσ. σ B ∗ σ A);

A := (λσ. σ A − 1)

EXAMPLE PROGRAM 6-A

WHAT DOES IT DO?

So far we have defined:

➜ Syntax of commands and expressions

➜ State of programs (function from variables to values)

Now we need: the meaning (semantics) of programs

How to define execution of a program?

➜ A wide field of its own (visit a semantics course!)

➜ Some choices:

• Operational (inductive relations, big step, small step)• Denotational (programs as functions on states, state transformers)• Axiomatic (pre-/post conditions, Hoare logic)

WHAT DOES IT DO? 7

WHAT DOES IT DO?

➜ Some choices:

WHAT DOES IT DO? 7-A

WHAT DOES IT DO?

Now we need:

the meaning (semantics) of programs

➜ Some choices:

WHAT DOES IT DO? 7-B

WHAT DOES IT DO?

➜ Some choices:

WHAT DOES IT DO? 7-C

WHAT DOES IT DO?

➜ Some choices:

WHAT DOES IT DO? 7-E

WHAT DOES IT DO?

➜ Some choices:

WHAT DOES IT DO? 7-F

STRUCTURAL OPERATIONAL SEMANTICS

〈SKIP, σ〉 −→ σ

e σ = v

〈x := e, σ〉 −→ σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True 〈c1, σ〉 −→ σ′

〈IF b THEN c1 ELSE c2, σ〉 −→ σ′

b σ = False 〈c2, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8

e σ = v

〈x := e, σ〉 −→

σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True 〈c1, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8-A

e σ = v

〈x := e, σ〉 −→ σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True 〈c1, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8-B

e σ = v

〈x := e, σ〉 −→ σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True

〈c1, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8-C

e σ = v

〈x := e, σ〉 −→ σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True 〈c1, σ〉 −→ σ′

b σ = False

〈c2, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8-D

e σ = v

〈x := e, σ〉 −→ σ[x 7→ v]

〈c1, σ〉 −→ σ′ 〈c2, σ′〉 −→ σ′′

〈c1; c2, σ〉 −→ σ′′

b σ = True 〈c1, σ〉 −→ σ′

STRUCTURAL OPERATIONAL SEMANTICS 8-E

b σ = False

〈WHILE b DO c OD, σ〉 −→

b σ = True 〈c, σ〉 −→ σ′ 〈WHILE b DO c OD, σ′〉 −→ σ′′

〈WHILE b DO c OD, σ〉 −→ σ′′

STRUCTURAL OPERATIONAL SEMANTICS 9

b σ = False

〈WHILE b DO c OD, σ〉 −→ σ

STRUCTURAL OPERATIONAL SEMANTICS 9-A

b σ = False

b σ = True

〈c, σ〉 −→ σ′ 〈WHILE b DO c OD, σ′〉 −→ σ′′

σ′′

STRUCTURAL OPERATIONAL SEMANTICS 9-B

b σ = False

b σ = True 〈c, σ〉 −→ σ′

〈WHILE b DO c OD, σ′〉 −→ σ′′

σ′′

STRUCTURAL OPERATIONAL SEMANTICS 9-C

b σ = False

STRUCTURAL OPERATIONAL SEMANTICS 9-D

DEMO: THE DEFINITIONS IN ISABELLE

PROOFS ABOUT PROGRAMS

Now we know:

➜ What programs are: Syntax

➜ On what they work: State

➜ How they work: Semantics

So we can prove properties about programs

Example:Show that example program from slide 6 implements the factorial.

lemma 〈factorial, σ〉 −→ σ′ =⇒ σ′B = fac (σA)

(where fac 0 = 0, fac (Suc n) = (Suc n) ∗ fac n)

PROOFS ABOUT PROGRAMS 11

Now we know:

PROOFS ABOUT PROGRAMS 11-A

Now we know:

PROOFS ABOUT PROGRAMS 11-B

DEMO: EXAMPLE PROOF

TOO TEDIOUS

Induction needed for each loop

Is there something easier?

TOO TEDIOUS 13

TOO TEDIOUS

Induction needed for each loop

Is there something easier?

TOO TEDIOUS 13-A

FLOYD/HOARE

Idea: describe meaning of program by pre/post conditions

Examples:

{True} x := 2 {x = 2}

{y = 2} x := 21 ∗ y {x = 42}

{x = n} IF y < 0 THEN x := x + y ELSE x := x− y {x = n− |y|}

{A = n} factorial {B = fac n}

Proofs: have rules that directly work on such triples

FLOYD/HOARE 14

FLOYD/HOARE

Examples:{True} x := 2 {x = 2}

{y = 2} x := 21 ∗ y {x = 42}

FLOYD/HOARE 14-A

FLOYD/HOARE

{y = 2} x := 21 ∗ y {x = 42}

FLOYD/HOARE 14-B

FLOYD/HOARE

{y = 2} x := 21 ∗ y {x = 42}

FLOYD/HOARE 14-C

FLOYD/HOARE

{y = 2} x := 21 ∗ y {x = 42}

FLOYD/HOARE 14-D

FLOYD/HOARE

{y = 2} x := 21 ∗ y {x = 42}

FLOYD/HOARE 14-E

MEANING OF A HOARE-TRIPLE

{P} c {Q}

What are the assertions P and Q?

➜ Here: again functions from state to bool

(shallow embedding of assertions)

➜ Other choice: syntax and semantics for assertions (deep embedding)

What does {P} c {Q} mean?

Partial Correctness:|= {P} c {Q} ≡ (∀σ σ′. P σ ∧ 〈c, σ〉 −→ σ′ =⇒ Q σ′)

Total Correctness:|= {P} c {Q} ≡ (∀σ. P σ =⇒ ∃σ′. 〈c, σ〉 −→ σ′ ∧ Q σ′)

This lecture: partial correctness only (easier)

MEANING OF A HOARE-TRIPLE 15

{P} c {Q}

MEANING OF A HOARE-TRIPLE 15-A

{P} c {Q}

MEANING OF A HOARE-TRIPLE 15-B

{P} c {Q}

MEANING OF A HOARE-TRIPLE 15-C

{P} c {Q}

MEANING OF A HOARE-TRIPLE 15-D

{P} c {Q}

MEANING OF A HOARE-TRIPLE 15-E

HOARE RULES

{P} SKIP {P}

{P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P} IF b THEN c1 ELSE c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

{P} WHILE b DO c OD {Q}

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-A

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-B

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-C

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q}

{P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-D

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-E

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-F

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′

{P ′} c {Q′}

Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-G

HOARE RULES

{P} SKIP {P} {P [x 7→ e]} x := e {P}

{P} c1 {R} {R} c2 {Q}

{P} c1; c2 {Q}

{P ∧ b} c1 {Q} {P ∧ ¬b} c2 {Q}

{P ∧ b} c {P} P ∧ ¬b =⇒ Q

P =⇒ P ′ {P ′} c {Q′} Q′ =⇒ Q

{P} c {Q}

HOARE RULES 16-H

HOARE RULES

` {P} SKIP {P} ` {λσ. P (σ(x := e σ))} x := e {P}

` {P} c1 {R} ` {R} c2 {Q}

` {P} c1; c2 {Q}

` {λσ. P σ ∧ b σ} c1 {R} ` {λσ. P σ ∧ ¬b σ} c2 {Q}

` {P} IF b THEN c1 ELSE c2 {Q}

` {λσ. P σ ∧ b σ} c {P}∧

σ. P σ ∧ ¬b σ =⇒ Q σ

` {P} WHILE b DO c OD {Q}

∧σ. P σ =⇒ P ′ σ ` {P ′} c {Q′}

∧σ. Q′ σ =⇒ Qσ

` {P} c {Q}

HOARE RULES 17

ARE THE RULES CORRECT?

Soundness: ` {P} c {Q} =⇒|= {P} c {Q}

Proof: by rule induction on ` {P} c {Q}

Demo: Hoare Logic in Isabelle

ARE THE RULES CORRECT? 18

ARE THE RULES CORRECT? 18-A

ARE THE RULES CORRECT? 18-B

NICER, BUT STILL KIND OF TEDIOUS

Hoare rule application seems boring & mechanical.

Automation?

Problem: While – need creativity to find right (invariant) P

Solution:

➜ annotate program with invariants

➜ then, Hoare rules can be applied automatically

Example:{M = 0 ∧ N = 0}

WHILE M 6= a INV {N = M ∗ b} DO N := N + b; M := M + 1 OD{N = a ∗ b}

NICER, BUT STILL KIND OF TEDIOUS 19

Automation?

Solution:

Example:{M = 0 ∧ N = 0}

NICER, BUT STILL KIND OF TEDIOUS 19-A

Automation?

Solution:

Example:{M = 0 ∧ N = 0}

NICER, BUT STILL KIND OF TEDIOUS 19-B

Automation?

Solution:

Example:{M = 0 ∧ N = 0}

NICER, BUT STILL KIND OF TEDIOUS 19-C

Automation?

Solution:

Example:{M = 0 ∧ N = 0}

NICER, BUT STILL KIND OF TEDIOUS 19-D

WEAKEST PRECONDITIONS

pre c Q = weakest P such that {P} c {Q}

With annotated invariants, easy to get:

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

pre (c1; c2) Q = pre c1 (pre c2 Q)

pre (IF b THEN c1 ELSE c2) Q = λσ. (b −→ pre c1 Q σ) ∧

(¬b −→ pre c2 Q σ)

pre (WHILE b INV I DO c OD) Q = I

WEAKEST PRECONDITIONS 20

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

(¬b −→ pre c2 Q σ)

WEAKEST PRECONDITIONS 20-A

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

(¬b −→ pre c2 Q σ)

WEAKEST PRECONDITIONS 20-B

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

(¬b −→ pre c2 Q σ)

WEAKEST PRECONDITIONS 20-C

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

(¬b −→ pre c2 Q σ)

WEAKEST PRECONDITIONS 20-D

pre SKIP Q = Q

pre (x := a) Q = λσ. Q(σ(x := aσ))

(¬b −→ pre c2 Q σ)

WEAKEST PRECONDITIONS 20-E

VERIFICATION CONDITIONS

{pre c Q} c {Q} only true under certain conditions

These are called verification conditions vc c Q:

vc SKIP Q = True

vc (x := a) Q = True

vc (c1; c2) Q = vc c2 Q ∧ (vc c1 (pre c2 Q))

vc (IF b THEN c1 ELSE c2) Q = vc c1 Q ∧ vc c2 Q

vc (WHILE b INV I DO c OD) Q = (∀σ. Iσ ∧ bσ −→ pre c I σ)∧

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-A

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-B

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-C

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-D

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-E

vc SKIP Q = True

(∀σ. Iσ ∧ ¬bσ −→ Q σ)∧

vc c I

vc c Q ∧ (pre c Q =⇒ P ) =⇒ {P} c {Q}

VERIFICATION CONDITIONS 21-F

SYNTAX TRICKS

➜ x := λσ. 1 instead of x := 1 sucks

➜ {λσ. σ x = n} instead of {x = n} sucks as well

Problem: program variables are functions, not values

Solution: distinguish program variables syntactically

Choices:➜ declare program variables with each Hoare triple

• nice, usual syntax• works well if you state full program and only use vcg

➜ separate program variables from Hoare triple (use extensible records),indicate usage as function syntactically

• more syntactic overhead• program pieces compose nicely

SYNTAX TRICKS 22

SYNTAX TRICKS

SYNTAX TRICKS 22-A

SYNTAX TRICKS

SYNTAX TRICKS 22-B

SYNTAX TRICKS

SYNTAX TRICKS 22-C

SYNTAX TRICKS

SYNTAX TRICKS 22-D

SYNTAX TRICKS

SYNTAX TRICKS 22-E

SYNTAX TRICKS

SYNTAX TRICKS 22-F

RECORDS IN ISABELLE

Records are a tuples with named components

Example:

record A = a :: natb :: int

➜ Selectors: a :: A ⇒ nat, b :: A ⇒ int, a r = Suc 0

➜ Constructors: (| a = Suc 0, b = −1 |)

➜ Update: r(| a := Suc 0 |)

Records are extensible:record B = A +

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-A

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-B

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-C

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-D

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-E

RECORDS IN ISABELLE

Example:

c :: nat list

(| a = Suc 0, b = −1, c = [0, 0] |)

RECORDS IN ISABELLE 23-F

Available now in Isablle:

➜ procedures

➜ with blocks and local variables

➜ and (mutual) recursion

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

➜ object orientation

MORE 25

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-A

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-B

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-C

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-D

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-E

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-F

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-G

➜ procedures

➜ exceptions

➜ arrays

➜ pointers

We’re working at:

➜ nondeterminsm

➜ probability

MORE 25-H

WE HAVE SEEN TODAY ...

➜ Syntax and semantics of IMP

➜ Hoare logic rules

➜ Soundness of Hoare logic

➜ Verification conditions

➜ Example program proofs

WE HAVE SEEN TODAY ... 26

EXERCISES

➜ Write a program in IMP that calculates quotient and reminder ofx ∈ IN and y ∈ IN

➜ Find the right invariant for its while loop.

➜ Show its correctness in Isabelle:

` {True} program { Q ∗ y + R = x ∧ R < y }

➜ Write an IMP program that sorts arrays (lists) by insertion sort.

➜ Formulate and show its correctness in Isabelle.

EXERCISES 27

NICTA Advanced Course Theorem Proving Principles...

Documents

Medientag Digitalisierung@Talanx/media/Files/T/Talanx/pdfcontent/newsroom/2017/tx... · Gamification. Travel Insurance App. ... und der Gewinnanteil ist ... Höhere Akzeptanz und

Contentagr.asu.edu.eg/uploads/agr/Lecture_3_201503174901.pdfContent: • Structure ofStructure of Peptide ... bonds in a sequential order. dipeptide, tripeptide, oligopeptide, polypeptide

Content-Based and Sheltered Instructionk12.wa.us/MigrantBilingual/Webinar/CBIShelteredInstruction6-5-15.pdfContent-Based and Sheltered Instruction: Components for Success . ... instruction

ComunidadGUIDE of the Best Fruits and Vegetables ... › pdf_UK09 › 096_119.pdfcontent, all these products have great health benefits and are recommended by specialists. That’s

Content Validity in Psychological Assessment: A …personal.kent.edu/~dfresco/CRM_Readings/Haynes_1995.pdfContent Validity in Psychological Assessment: A Functional Approach to Concepts

AGENDA MANYALLALUK LOCAL AUTHORITY MEETING …ropergulf.nt.gov.au/wp-content/uploads/2018/10/MAN_22102018_AGN_AT_SAVED.pdf10.5 SCHOOL NUTRITION PROGRAM Ben Kleing provided an update

Content and Hierarchy in Pixel-Based Methods for Reverse ...homes.cs.washington.edu/~jfogarty/publications/chi2011.pdfContent and Hierarchy in Pixel-Based Methods for Reverse Engineering

La problemática de la lectura en el bachilleratocmas.siu.buap.mx/portal_pprd/work/sites/filosofia/resources/PDFContent/... · Este artículo plantea una reflexión sobre esta situación

CONTENT ANALYSIS OF TELEVISION TALK SHOWSgallup.com.pk/wp-content/uploads/2016/06/MonthlyReportAugust20141.pdfContent Analysis of Television Talk Shows [ 1 ] CONTENT ANALYSIS OF TELEVISION

seL4: Formal Verification of an OS Kernelisabelle.in.tum.de/~kleing/papers/sosp09.pdf · models with a high degree of abstraction from hardware. As a compromise that blends both views,

Tolerancias excluyentes y excluidas. Un apunte genealógicocmas.siu.buap.mx/portal_pprd/work/sites/filosofia/resources/PDFContent/... · 11 Números 16-17 enero-junio – julio-diciembre

[MS-CDEPLOY]: Content Deployment Remote Import Web …MS-CDEPLOY].pdfContent Deployment Remote Import Web Service Protocol site collection of

Content Delivery Networks: Protection or Threat?haddock.case.edu/files/documents/cdnSecurity-esorics09.pdfContent Delivery Networks: Protection or Threat? ... Content Delivery Network

Focus on the People - Universiti Sains Malaysiapippa.usm.my/images/PDFContent/2019/Performance...Organizational performance is a Human Resource challenge. Ultimately, it is the people

Developing the Therapeutic Alliance as a Bridge to …ctndisseminationlibrary.org/PDF/284.pdfContent and Process “Ingredients” 8 Essential Therapist Behavior 10 Reflective Listening

Content Oriented Surveillance System Based on Information ...icnsra.nz.comm.waseda.ac.jp/wp-content/uploads/2016/12/s1-3.pdfContent Oriented Surveillance System Based on Information-Centric

A Survey on Content-based Image Retrievalfaculty.utrgv.edu/zhixiang.chen/cs6174/papers/Quweider_paper3.pdfContent-Based Image Retrieval (CBIR) emerged as a promising substitute to

ENERGY SECTOR OF MONGOLIA, COUNTRY REPORT › data › 8016.pdfCONTENT • General information • Reserves of Energy Mineral Resources • Current Energy Policy and Measures • Current

Gerardo Rivas. Curriculum Vitae largocmas.siu.buap.mx/.../PDFContent/34/CV_Gerardo_Rivas.pdf · 2014. 12. 11. · gerardo rivas y ana marÍa martÍnez de la escalera (coords.), el

Flash - Purdue Universityjshan/publications/2011/HowPublishingPaperPERS2011.pdfcontent in figures and tables and the messages you are trying to communi- cate through these figures