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Certifying a computing host?

Formal proofs for resilience,

extensibility, security?

Need to reason about:

•  human behaviors

•  cosmic rays + natural disasters

•  hardware failure

•  software

VIEW #1: bug-free host impossible. Treat it as a biological system.

Certifying a computing host?

Formal proofs for resilience,

Need to reason about:

•  human behaviors

•  cosmic rays + natural disasters

•  hardware failure

•  software

VIEW #2: focus on software since it is a rigorous mathematical entity!

HW & Env Model

Certified software 01011010101010101111100011001110110111111101010101010101010101010111111111010101010101111000110001010101010101011111100011001001111001111111100111111100011110001010101011111101110011001110101010111111100011

110001110001110011

SOFTWARE

Formal specs &

proofs for resilience,

HW & Env Model

Find a mathematical proof showing that

if the HW/Env follows its model, the software will run according to its specification

0101101010101010111110001100111011011111110101010101010101010101011111111101010101010111100011000101010101010101111110001100100111100111111110011111110001111000101010101111110111001100111010

10101111111000111100

Application & other system SW

Certified OS kernel

Formal specs &

proofs for resilience,

HW & Env Model

Research tasks & key innovations:

•  new OS kernel that can “crash-proof” the entire system & application SW

•  new PLs for writing certified kernel plug-ins (new OCAP + DSPLs)

•  new formal methods for automating proofs & specs (VeriML)

ResourceManagers

ProtectedExecutors

DeviceDrivers

Certified Kernel

CPU CPU GPU GPU Disk NIC

NativeExecutor

TypesafeExecutor

VMMExecutor

SFIExecutor

Scheduler Scheduler

Driver Driver

FS Net

DriverDriver

LegacyX86App

Java/MSILApp

NaClApp

LegacyOS + App

Certified “hypervisor” kernel

•  Problems w. existing platforms •  Attacks: Zero-Day Kernel Vulnerabilities

(ZDKVs) & rogue driver certificates

•  leads to rogue kernels

•  leads to rogue WinCC/Step7 apps

•  leads to rogue PLC firmeware

�firmw

Rogue PC Rogue PLC WinCC

Step 7

rogue OS & its kernel

Apps �

zero-day kernel vulnerabilities

& fake/stolen driver certificates

�certified firm

Secure PC w. IFC labels

Secure PLC WinCC

Step 7

certified kernel

Apps �

COTS OS

small mechanized proof checker

New CRASH technologies

•  A small certified “hypervisor” kernel provides a reliable ZDKV-free core to fall back on, even under attacks

•  Information-Flow-Control to enforce security

•  Mechanized proof certificates are unforgeable

Protecting against Stuxnet attacks!

Process Management

Mgmt OS (Linux 1) Linux 2 App 1 Virtual

Device Certified

App 2 Applica'on*

Scheduler IPC

V-PIC V-PCI Hypercall

Trap Handling

(Interrupt, Exception, Syscall)

SVM*/*VMX*module*

Pass-Through Devices

Hardware*

Memory Management

Cer'KOS*

CertiKOS architecture

Device*Drivers* SVM&VMX*Driver* IOMMU&VT-d Driver

IOMMU/VT-d module +7@BC/:�#/167<3�#/</53;3<B�

Disk*Driver*

KBD*Driver* VGA*Driver* Timer*Driver* IOCAPIC*Driver* APIC*Driver* PCI*Driver*

Serial*Driver*

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Raw Machine / HW Spec�

kmod1.c

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abs-layer-1 spec�

kmod1.s�

high-level kernel spec�

abs-layer-k spec�

kmodk.s�

abs-layer-2 spec�

kmod2.s�

kmod2.c

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Compositonal specification & verification current target: single-core CertiKOS

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Predicate Logic

Predicate logic over integer expressions:

a language of logical assertions, for example

8x. x + 0 = x

Why discuss predicate logic?

It is an example of a simple language

It has simple denotational semantics

We will use it later in program specifications

Abstract Syntax

Describes the structure of a phrase

ignoring the details of its representation.

An abstract grammar for predicate logic over integer expressions:

intexp ::= 0 | 1 | . . .

| �intexp | intexp + intexp | intexp � intexp | . . .

| ¬assert | assert ^ assert | assert _ assert

| assert ) assert | assert , assert

| 8var . assert | 9var . assert

Resolving Notational Ambiguity

Using parentheses: (8x. ((((x) + (0)) + 0) = (x)))

Using precedence and parentheses: 8x. (x + 0) + 0 = x

arithmetic operators (⇤ / rem . . .) with the usual precedencerelational operators (= 6= < . . .)

¬^_),

The body of a quantified term extends to a delimiter.

Carriers and Constructors

Carriers: sets of abstract phrases (e.g. intexp, assert)Constructors: specify abstract grammar productions

intexp ::= 0 �! c0 2 {hi}! intexp

intexp ::= intexp + intexp �! c+ 2intexp ⇥ intexp ! intexp

Note: Independent of the concrete pattern of the production:

intexp ::= plus intexp intexp �! c+ 2intexp ⇥ intexp ! intexp

Constructors must be injective and have disjoint ranges

Carriers must be either predefined or their elements must be

constructible in finitely many constructor applications

Inductive Structure of Carrier Sets

With these properties of constructors and carriers,carriers can be defined inductively:

intexp

(0) = {}intexp

(j+1) = {c0hi, . . .} [ {c+(x0, x1) |x0, x1 2 intexp

(j)} [ . . .

assert

(0) = {}assert

(j+1) = {ctruehi, cfalsehi}[{c=(x0, x1) |x0, x1 2 intexp

(j)} [ . . .

[{c¬(x0) |x0 2 assert

(j)} [ . . .

intexp =1[

j=0intexp

assert =1[

j=0assert

Denotational Semantics of Predicate Logic

The meaning of a term e 2 intexp is [[e]]intexp

i.e. the function [[�]]intexp

maps intexp objects to their meanings.

What is the set of meanings?

The meaning [[5 + 37]]intexp

of the term 5 + 37| {z } could be the integer 42.(that is, c+(c5hi, c37hi))

However the term x + 5 contains the free variable x,

so the meaning of an intexp in general cannot be an integer. . .

Mathematical Background

Relations

Functions

Sequences

Products and Sums

membership the empty setnatural numbers

inclusion integersfinite subset

set comprehensionintersection and

is a bound variableunion ordifference and notpowersetinteger range and

Generalized Set Operations

def def

meaningless

Examples:

Relations

A relation is a set of primitive pairs .

relates and

is an identity relation

the identity on def

the domain of dom def

the range of ran def

composition of with def and

reflection of def

Relations: Properties and Examples

dom ran

Functions

A relation is a function ifand

If is a function,maps to

and are functions.If and are functions, then is a function:

is not necessarily a function:consider

is an injection if both and are functions.

Notation for Functions

Typed abstraction: def

Defined only when is defined for all(consider )

dom , if ran dom .

Placeholder: with a dash standing for the bound variable

Variation of a function :ifotherwise

dom domran ran

Sequences

— the empty function— the empty sequence— an -tuple— a (non-primitive) pair

dom == when

Products

Let be an indexed family of sets (a function with sets in its range).The Cartesian product of is

def dom dom and dom

More Products

def “ ”

Sets of Sequences

def (finite)

def (infinite)

Let ✓ be an indexed family of sets (a function with sets in its range).

The disjoint union (sum) of ✓ is

= {hi, xi | i 2 dom ✓ and x 2 ✓ i}

�x 2 T. S S1 + . . . + S

i=1“S

i2(m to n)S T ⇥ S =

n⇥ S = (0 to (n� 1))⇥ S = S + . . . + S| {z }n times

B + B =PhB, Bi = {h0, truei, h0, falsei, h1, truei, h1, falsei}

= 2⇥B

Functions of Multiple Arguments

Use tuples instead of multiple arguments:

f (a0, . . . a

n�1) ��!f ha0, . . . a

n�1i

Syntactic sugar:

�hx0 2 S0, . . . , x

n�1 2 S

n�1i. Edef

= �x 2 S0 ⇥ . . .⇥ S

n�1. (�x0 2 S0. . . . �x

n�1 2 S

n�1. E)

(x 0) . . . (x(n�1))

Use Currying:

f (a0, . . . a

n�1) ��!f a0 . . . a

=(. . . (f a0) . . .) a

where f is a Curried function �x0 2 S0. . . . �x

n�1 2 S

n�1. E.

Relations Between Sets

⇢ is a relation from S to T

() ⇢ 2 S

�!REL

() dom ⇢ ✓ S and ran ⇢ ✓ T .

Relation on S

= relation from S to S.

�!REL

⇢ 2 S

�!REL

T ) ⇢

† 2 T

�!REL

For all S and T , {} 2 S

�!REL

{} 2! S

�!REL

{} 2! {}�!REL

Total Relations

⇢ 2 S

�!REL

T is a total relation from S to T

() ⇢ 2 S

�!TREL

() 8x 2 S.9y 2 T. x ⇢ y

() dom ⇢ = S

✓ ⇢

† · ⇢

rs �

⇢ �

⇢ 2 (dom ⇢) �!TREL

T () T ◆ ran ⇢

Functions Between Sets

f is a partial function from S to T

() f 2 S

��!PFUN

() f 2 S

�!REL

T and f is a function.

“Partial”: f 2 S

�!REL

T ) dom f ✓ S

��!PFUN

T is a (total) function from S to T

() f 2 S ! T

() dom f = S.

S ! T = T

S ! T ! U = S ! (T ! U)

Surjections, Injections, Bijections

f is a surjection from S to T () ran f = T

f is a injection from S to T () f

† 2 T

��!PFUN

f is a bijection from S to T () f

† 2 T ! S

() f is an isomorphism from S to T

rs �

�++X

�++W

�**U

Back to Predicate Logic

intexp ::= 0 | 1 | . . .

| �intexp | intexp + intexp | intexp � intexp | . . .

assert ::= true | false

| intexp = intexp | intexp < intexp | intexp intexp | . . .

| ¬assert | assert ^ assert | assert _ assert

| assert ) assert | assert , assert

| 8var . assert | 9var . assert

Denotational Semantics of Predicate Logic

The meaning of term e 2 intexp is [[e]]intexp

i.e. the function [[�]]intexp

maps objects from intexp to their meanings.

What is the set of meanings?

The meaning [[5+37]]intexp

of the term 5+37 could be the integer 42.

But the term x + 5 contains the free variable x. . .

Environments

. . . hence we need an environment (variable assignment, state)

� 2 ⌃def

= var ! Z

to give meaning to free variables.

The meaning of a term is a function from the states to Z or B.

[[�]]intexp

2 intexp ! ⌃! Z

[[�]]assert

2 assert ! ⌃! B

if � = [x : 3, y : 4], then [[x+5]]intexp

� = 8

[[9z. x < z ^ z < y]] � = false

Direct Semantics Equations for Predicate Logic

v 2 var e 2 intexp p 2 assert

[[0]]intexp

� = 0

( really [[c0hi]]intexp

� = 0)

[[v]]intexp

� = �v

[[e0+e1]]intexp

� = [[e0]]intexp

� + [[e1]]intexp

[[true]]assert

� = true

[[e0=e1]]assert� = [[e0]]intexp

� = [[e1]]intexp

[[¬p]]assert

� = ¬([[p]]assert

[[p0 ^ p1]]assert� = [[p0]]assert� ^ [[p1]]assert�

[[8v. p]]assert

� = 8n 2 Z. [[p]]assert

[�|v : n]

Example: The Meaning of a Term

[[8x. x+0=x]]assert

= 8n 2 Z. [[x+0=x]]assert

[�|x : n]

= 8n 2 Z. [[x+0]]intexp

[�|x : n] = [[x]]intexp

[�|x : n]

= 8n 2 Z. [[x]]intexp

[�|x : n]+[[0]]intexp

[�|x : n]=[[x]]intexp

[�|x : n]

= 8n 2 Z. [�|x : n](x) + 0 = [�|x : n](x)

= 8n 2 Z. n + 0 = n

= true

Properties of the Semantic Equations

They are syntax-directed (homomorphic):

exactly one equation for each abstract grammar production

(constructor)

result expressed using functions (meanings)

of subterms only (arguments of constructor)

) they have exactly one solution h[[�]]intexp

, [[�]]assert

i(proof by induction on the structure of terms).

They define compositional semantic functions

(depending only on the meaning of the subterms)

) “equivalent” subterms can be substituted

Validity of Assertions

p holds/is true in � () � satisfies p () [[p]]assert

� = true

p is valid () 8� 2 ⌃. p holds in �

p is unsatisfiable () 8� 2 ⌃. [[p]]assert

� = false

() ¬p is valid

p is stronger than p

0 () 8� 2 ⌃. (p0 holds if p holds )

() (p ) p

0) is valid

p and p

0are equivalent () p is stronger than p

0is stronger than p

Inference Rules

Class Examples

` p (Axiom) ` x + 0 = x (xPlusZero)

` p(Axiom Schema) ` e1=e0 ) e0=e1

(SymmObjEq)

` p0 . . . ` pn�1` p

(Rule)` p ` p ) p0

` p0(ModusPonens)

` 8v. p(Generalization)

Formal Proofs

A set of inference rules defines a logical theory `.

A formal proof (in a logical theory):

a sequence of instances of the inference rules, wherethe premisses of each rule occur as conclusions earlier in the sequence.

1. ` x + 0 = x (xPlusZero)

2. ` x + 0 = x ) x = x + 0 (SymmObjEq)[e0 : x | e1 : x + 0]

3. ` x = x + 0 (ModusPonens, 1, 2)[p : x + 0 = x | p0 : x = x + 0]

4. ` 8x. x = x + 0 (Generalization, 3)[v : x | p : x = x + 0]

Tree Representation of Formal Proofs

` x + 0 = x ` x + 0 = x ) x = x + 0

(SymmObjEq)

` x = x + 0

` 8x. x = x + 0

Soundness of a Logical Theory

An inference rule is sound if in every instance of the rule

the conclusion is valid if all the premisses are.

A logical theory ` is sound if all inference rules in it are sound.

If ` is sound and there is a formal proof of ` p, then p is valid.

Object vs Meta implication:

` p ) 8v. p is not a sound rule, although` p

` 8v. pis.

Completeness of a Logical Theory

A logical theory ` is complete if

for every valid p there is a formal proof of ` p.

A logical theory ` is axiomatizable if

there exists a finite set of inference rules

from which can be constructed formal proofs of all assertions in `.

No first-order theory of arithmetic is complete and axiomatizable.

Variable Binding

8x. 9y. xrspq^^^^^^^^^^^

nn < y

utwv`````````````````

oo ^ 9x. xrspqOO > y

utwv_________________________________________________________________

@@⇥⇥⇥⇥⇥⇥⇥⇥

__????????

??~~~~~~~~ ^

ggOOOOOOOOOOOOOOOO

77ppppppppppppppp 9

__????????

??~~~~~~~~y

__????????

??��

__@@@@@@@@

??~~~~~~~~y

__????????

Variable Binding

8x. 9y. xrspq^^^^^^^^^^^

nn < y

utwv`````````````````

oo ^ 9x. xrspqOO > y

utwv_________________________________________________________________

pw��

��

__????????

??~~~~~~~~ ^

ggOOOOOOOOOOOOOOOO

77ppppppppppppppp 9pw⇥⇥

⇥⇥

???��

__????????

__@@@@@@@@

__????????

Bound and Free Variables

In 8v. p, v is the binding occurrence (binder) and p is its scope.

If a non-binding occurrence of v is within the scope of a binder for v,

then it is a bound occurrence; otherwise it’s a free one.

FVintexp

(0) = {} FVassert

(true) = {}FV (v) = {v} FV (e0=e1) = FV (e0) [ FV (e1)

FV (�e) = FV (e) FV (¬p) = FV (p)

FV (e0 + e1) = FV (e0) [ FV (e1) FV (p0 ^ p1) = FV (p0) [ FV (p1)

FV (8v. p) = FV (p)� {v}

Example:

FV (9y. x < y ^ 9x. x > y) = {x}

Only Assignment of Free Variables Matters

Coincidence Theorem:

If �v = �0v for all v 2 FV✓(p), then [[p]]✓� = [[p]]✓�0

(where p is a phrase of type ✓).

Proof: By structural induction.

Inductive hypothesis:

The statement of the theorem holdsfor all phrases of depth less than that of the phrase p0.

Base cases:

p0 = 0 ) [[0]]intexp

� = 0 = [[0]]intexp

p0 = v ) [[v]]intexp

� = � v = �0v = [[v]]intexp

�0, since FV (v) = {v}.

Proof of Concidence Theorem, cont’d

Coincidence Theorem:

If �v = �0v for all v 2 FV✓(p), then [[p]]✓� = [[p]]✓�0.

Inductive cases:

p0 = e0 + e1: by IH [[ei]]intexp

� = [[ei]]intexp

�0, i 2 {1,2}.[[p0]]

intexp

� = [[e0]]intexp

� + [[e1]]intexp

�= [[e0]]

intexp

�0 + [[e1]]intexp

�0 = [[p0]]intexp

p0 = 8u. q: �v = �0v, 8v 2 FV (p0) = FV (q)� {u}then [�|u : n]v = [�0|u : n]v, 8v 2 FV (q), n 2 Z

Then by IH [[q]]assert

[�|u : n]= [[q]]assert

[�0|u : n] for all n 2 Z,

hence 8n 2 Z. [[q]]assert

[�|u : n]=8n 2 Z. [[q]]assert

[�0|u : n]

[[8u. q]]assert

� =[[8u. q]]assert

Substitution

�/� 2 intexp ! intexp

�/� 2 assert ! assert

>;when � 2 var ! intexp

0/� = 0 v/� = �v

(-e)/� = -(e/�) (p0 ^ p1)/� = (p0/�) ^ (p1/�)

(e0+e1)/� = (e0/�)+(e1/�) (8v. p)/� = 8v0. (p/[�|v : v0]),

. . . where v0 /2[

u2FV (p)�{v}FV (�u)

Examples:

(x < 0 ^ 9x. x y)/[x : y+1] = y+1 < 0 ^ 9x. x y

(x < 0 ^ 9x. x y)/[y : x+1] = x < 0 ^ 9z. z x+1

Preserving Binding Structure

(x < 0 ^ 9x. x y)/[ x : y+1] = y+1 < 0 ^ 9x. x y

:B}}}}}}}

}}}}}}}

9pw⇥⇥

⇥⇥⇥

⇥⇥⇥⇥

??;C��

��

[c???????

???????

;C~~~~~~~

~~~~~~~

[c>>>>>>>>

>>>>>>>>

Zb>>>>>>>>

>>>>>>>>

Zb========

========

�! ^

:B}}}}}}}

}}}}}}}

9pw⇤⇤

⇤⇤⇤⇤

⇤⇤

??;C��

��

[c???????

???????

;C��

��

\dAAAAAAAA

AAAAAAAA

[c@@@@@@@

@@@@@@@

<D��

��

[c@@@@@@@@

@@@@@@@@

[c@@@@@@@@

@@@@@@@@

Avoiding Variable Capture

(x < 0 ^ 9x. x y)/[ y : x+1] = x < 0 ^ 9z. z x+1

:B}}}}}}}

}}}}}}}

9pw⇥⇥

⇥⇥⇥

⇥⇥⇥⇥

;C��

��

[c???????

???????

;C��

��

\dAAAAAAAA

AAAAAAAA

[c@@@@@@@

@@@@@@@

[c???????

???????

�! ^

:B}}}}}}}

}}}}}}}

9pw⇤⇤

⇤⇤⇤⇤

⇤⇤

??;C��

��

[c???????

???????

;C��

��

\dAAAAAAAA

AAAAAAAA

[c@@@@@@@

@@@@@@@

\dAAAAAAA

AAAAAAA

:B}}}}}}}}

}}}}}}}}

[c>>>>>>>

>>>>>>>

Substitution Theorems

Substitution Theorem:

If � = [[�]]intexp

�0 · � on FV (p), then ([[�]]�)p = ([[�]]�0 · (�/�))p.

Finite Substitution Theorem:

[[p/v0 ! e0, . . . vn�1 ! en�1]]� = [[p]][�|v0 : [[e0]]�, . . .].

p/v0 ! e0, . . . vn�1 ! en�1def= p/[cvar|v0 : e0| . . . |vn�1 : en�1].

Renaming:

If u /2 FV (q)� {v}, then [[8u. (q/v ! u)]]boolexp

= [[8v. q]]boolexp

( m =@;/: (3;

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