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Proving Mutual Termination of single-threaded programs

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### Proving Mutual Terminationof single-threaded programs

Dima Elenbogen Ofer Strichman Shmuel Katz

Technion, Haifa, Israel

Notion of equivalence for this presentation

Goal: verification of the mutualterminationoftwosimilar programs.

Mutual termination

- Given equal inputs,
- P1 terminates ,P2 terminates
- Undecidable

16:32:13

Alternative: termination of a single program

- New tools have recently been developed:
- Terminator
- Mutant
- …
- Still, there are two major problems:
- Incompleteness
- Complexity

16:32:13

Mutual Termination vs. Proving Termination

- Pros:
- Computationally easier to check the mutual terminations of two programs than to prove the termination of each of them.
- Fully automated.
- It does not require finding a well-founded set.
- Program do not necessarily terminate.
- Termination check has nothing to say
- Mutual termination can still say something useful.
- Cons:
- Defines a weaker notion.

16:32:13

Goals

- Develop proof rules for mutual termination
- Present an algorithm for checking mutual termination, that
- uses the proof rules, and
- is sensitive to the magnitude of change rather than the magnitude of the programs

16:32:13

Prerequisites

- Assume:
- no loops (but there are recursive functions);
- 1-1 mapping mapbetween the functions of both sides:
- must intersect all cycles in the call graphs;
- the mapped functions have the same signature

B:

A:

2map

f1’()

f1()

2map

f5’()

f2’()

f2()

f5()

f7’()

2map

f6()

f3()

f4()

f4’()

2map

16:32:13

Mutual termination (simple case)

- Consider the call graphs:
- We want to prove that A, B are mutually terminating
- How shall we handle the recursion ?

A

B

Side 1

Side 2

16:32:14

Call-equivalence

- Definition: functions A,B are call-equivalent if…
- For equal inputs:
- For callees f,g s.t. (f,g) 2 map:
- f is called ,g is called
- f and g are called with the same arguments.

B(x, y) {

g(0,0)

if (cond2)

g(x,y)

if (cond3)

- g(x,y)
- }

A(x, y) {

if (cond1)

f(x,y)

f(0, 0)

}

The order and the number of calls do not matter

16:32:14

Preliminary inference rule (simple case)

B(w, z) {

…

if (cond2)

B(w1,z1)

else …

..

}

A(x, y) {

…

if (cond1)

A(x1,y1)

else …

…

}

A

B

call-equiv(A,B)

mutual-terminate(A, B)

(M-TERM-REC)

Side 1

Side 2

16:32:14

The premise is undecidable

How can we prove the premise?

B(w, z) {

…

if (B(w’, z’) > …)

B(w1,z1)

else …

..

}

A(x, y) {

…

if (A(x’, y’) > …)

A(x1,y1)

else …

…

}

A

B

call-equiv(A,B)

mutual-terminate(A, B)

(M-TERM-REC)

Side 1

Side 2

16:32:14

Uninterpreted functions

- Replace the recursive calls with calls to functions that
- over-approximateA, B,and
- are terminating by construction
- Natural candidates: Uninterpreted Functions
- Abstract all functionality.
- We only know they are consistent:x = y → UF(x) = UF(y)

call-equiv(A,B)

mutual-terminate(A, B)

(M-TERM-REC)

16:32:14

Replacing recursive calls (1 / 2)

- Let FUF , GUFbe F,G, after replacing the recursive call with a call to the corresponding uninterpreted functions.

F(x, y) {

…

if (cond1)

F(x1,y1)

else …

…

}

G(w, z) {

…

if (cond2)

G(w1,z1)

else …

..

}

F

G

Side 2

Side 1

16:32:14

Replacing recursive calls (2 / 2)

- Let FUF , GUFbe F,G, after replacing the recursive call with a call to the corresponding uninterpreted functions.

FUF(x, y) {

…

if (cond1)

UF(F)(x1,y1)

else …

…

}

GUF(w, z) {

…

if (cond2)

UF(G)(w1,z1)

else …

..

}

FUF

GUF

UF(G)

UF(F)

Side 2

Side 1

16:32:14

Proving mutual termination

- Let FUF , GUFbe F,G, after replacing the recursive calls with calls to uninterpreted functions.
- We can now rewrite the rule:

This premise is decidable

call-equiv(FUF,GUF)

mutual-terminate(F, G)

(M-TERM-SIMPLE)

16:32:14

General inference rule

Now we want to generalize from a single self loop to MSCCs in the call graphs:

- Definition: is called in A]

∀(F, G) ∈ map. call-equiv(FUF,GUF)

∀(F, G) ∈ map. mutual-terminate(F, G)

(M-TERM)

16:32:14

Connected MSCCs

{(g,g’),(f,f’),(h,h’)} 2 map

Connected MSCCs…

- Prove bottom-up
- Abstract mutually terminating functions
- Inline

g

g’

h

UF(h)

UF(h’)

h’

Side 2

Side 1

16:32:14

Decomposition algorithm

Legend:

Mutually terminating pair

Mutual termination undecided yet

Could not prove mutual termination

Syntactically equivalent pair

check

Unpaired function

B:

A:

check

f1()

f1’()

f2’()

f2()

U

U

f5’()

f5()

f7’()

f4()

f4’()

f3()

f6()

U

U

check

16:32:14

UF(g)

Mutual recursion{(g,g’),(f,f’)} 2 map

- Find a sub-map that intersects all cycles, e.g., {(g,g’)}
- Only when calling functions in this sub-map, replace with uninterpreted functions

g’

g

f

f ’

Side 2

Side 1

16:32:14

Decomposition with mutual recursion

Legend:

Mutually terminating pair

Mutual termination undecided yet

Could not prove mutual termination

Syntactically equivalent pair

Call-equivalent; mutual termination undecided yet

A:

B:

check

f1()

f1’()

f2()

f5()

U

U

U

U

U

U

f2’()

f5’()

U

U

U

U

U

U

16:32:14

The Regression Verification Tool (RVT)

- Given two C programs:
- loops recursive functions.
- Map functions, globals, etc.
- After that:
- Decompose to the granularity of pairs of functions
- Use a C verification engine (CBMC)

16:32:14

RVT

Version A

Version B

- Merge
- Rename identical globals
- Map functions/globals
- Decompose
- static analyses

- call-equivalence
- counterexample

RVT

C program

feedback

- enforce equality of inputs
- replace with UFs
- assert call-equivalence

CBMC

16:32:14

Improvements of completeness (1 / 2)

Partial equivalence

- Terminating executions of P1 and P2 on equal inputsresult in equal outputs.
- Taking advantage of the partial equivalence of functions:
- If we know that (f, g) ∈ mapare partially equivalent, then UF(f) = UF(g)
- We welcome additional ideas how to refine our UFs.

16:32:14

Improvements of completeness (2 / 2)

- Ignoring input arguments that do not affect the call-equivalence of a function:
- This improves mapping, as some mapped function pairs may have different prototypes.

16:32:14

Checking the termination of functions

- Suppose we know that function A terminates. Can it help us to prove the termination of A’?
- Define call-contain(A, A’) as:
- For equal inputs :
- For each pair (f, f’) 2map:

f ‘ is called in A’ with argument x

f is called in A with argument x

∀(F, F’) ∈ map. (term(F)∧call-contain(FUF ,F’UF))

∀(F, F’) ∈ map. term(F’)

(TERM)

16:32:14

Value of Mutual Termination

Full equivalence

- P1 and P2 are partially equivalent and mutually terminate.
- Introduced in:
- Luckham, Park, and M. Paterson 1970

[On formalized computer programs]

- Pratt1971 [Kernel equivalence of programs and proving kernel equivalence and correctness by test cases]
- Regression verificationof full equivalence is an important problem.
- Proving mutual termination is a crucial sub-task.

16:32:14

Questions?..

16:32:14

b)

y)

x,

Using (M-TERM-SIMPLE): example (1/2)unsigned gcd1UF

(unsigned a, unsigned b)

{ unsigned g;

if (b == 0)

g = a;

else {

a = a % b;

g = gcd1(b, a);

}

return g;

}

unsigned gcd2UF

(unsigned x, unsigned y)

{ unsigned z;

z = x;

if (y > 0)

z = gcd2(y, z % y);

}

return z;

}

?

=

UF2

term

UF1

16:32:14

Using (M-TERM-SIMPLE): example (2/2)

- Proving call-equiv(gcd1UF, gcd2UF)

Equal inputs

Equal guards

if called

then equal arguments

- Valid. gcd1,gcd2aremutually terminating.

16:32:14

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