1. Complete Program Synthesis for Linear Arithmetics
Mikaël Mayer
Monday, January 25, 2009
LARA Supervisors:
Viktor Kuncak
Ruzica Piskac

2. Program Verification Our goal is to verify functional programs.
Verification in the usual sense: given some specifications, you either prove that the program conforms to them, or find a bug (in the form of a counterexample)

3. Program Synthesis Our goal is to verify functional programs.
Verification in the usual sense: given some specifications, you either prove that the program conforms to them, or find a bug (in the form of a counterexample)

4. Program Synthesis Challenges
Define high-level language input.
Reuse existing knowledge
Guarantee quality of synthesized programs
Human
« Give me a program to generate pictures representing complex functions using this formalism(… )»
But not:
If not:
For each program in possible programs:
If program meets specifications:
Return program
Non efficient
Undecidable
Program Verification
Efficiency
Correctness-by-construction

5. Outline Examples Linear Integer Arithmetic synthesis algorithm
Parametrized LIA synthesis algorithm
Future of program synthesis
Contribution : Decision procedure extension.
Contribution: Synthesis extension.

6. Linear Integer Arithmetic Synthesis
> 0
= B
< y
5x-13y+5z
3x+4y-2A
3x+1
A,B   provided at run-time
Find x, y, z  

7. Examples 27 9
Weights 3 1 T

8. Examples 27 9 3 1
Weights 19

9. Examples
Weights 3 27 19 9 1

10. Examples 3 27 T 9 1  , 1 3 9 27 Î £ - = + w T Weights =0 Right Left
Outside 

11. Complicated OK Examples  , 1 3 9 27 Î £ - = + w T Weights T =19
40+T >= 0 && 40-T >= 0
def getWeights(T : Int):(Int, Int, Int, Int) = {
val ya = Math.min(1, ((13+T) - (27 + (13+T)%27)%27)/27)
val yc = Math.min(1, ((4+T-27*ya) - (9 + (4+T-27*ya)%9)%9)/9)
val yb = Math.min(1, ((1+T-27*ya-9*yc) - (3 + (1+T-27*ya-9*yc)%3)%3)/3)
val w9 = yc
val w3 = yb
val w27 = ya
val w1 = T-27*ya-3*yb-9*yc
(w1, w27, w3, w9) }
Complicated
T =19
ya = 1
yc = -1
yb = 0
w9 = -1 // On the right
w3 = 0 // Not taken
w27 = 1 // On the left
w1 = 1 // On the left OK 

12. Linear Integer Arithmetic Synthesis
Two steps
Equalities
Inequalities

13. Linear Integer Arithmetic Synthesis
Equalities – Process 20 30 12 ) 3 .( , = + $ z y x b a 10 15 6 . , = + $ z y x c ) , ( 3 6
var 2 5
... / z y x v u c b a - + = 2 ) 20 , 3 (1
gcd =
Bézout’s Theorem b a 3 2 +
Condition
Program fragment

14. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions?
Find one solution
Find all homogenous solutions ( ) c z y x ÷ ø ö ç è æ - = + 1
var 10 15 6 ÷ ø ö ç è æ = + v u z y x ?
var 10 15 6
Extended Euclidian algorithm
gcd(15,10)=5
Replace x by 5.

15. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions?
Find all homogenous solutions ÷ ø ö ç è æ = + v u z y x ? 5
var 10 15 30
Simplify
Find a witness for y and z

16. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions?
Find all homogenous solutions ÷ ø ö ç è æ - = + v u z y x ? 6 5
var 2 3
Solve homogeneous

17. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions?
Find all homogenous solutions ÷ ø ö ç è æ - = + v u z y x ? 6 5
var 2 3
Solve homogeneous

18. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions?
Find all homogenous solutions ÷ ø ö ç è æ - = + v u c z y x 3 6 1 2 5
var 10 15 ÷ ø ö ç è æ - = + v u z y x 3 2 6 5
var
2 variables
Complete solution
Result

19. Linear Integer Arithmetic Synthesis
Equalities – Describe all solutions ) ,
gcd(
var
... .. 1 2 3 n i k g u x L M O = ÷ ø ö ç è æ + -
Lower triangular
Find a witness
General solution

20. Linear Integer Arithmetic Synthesis6 )% 3 12 ( 2 4 a b y - +
Linear Integer Arithmetic Synthesis
Inequalities
Removed a variable x.
Added a new variable u.
Added an equality. a b y x 3 12 6 2 4 . , - + $ b y a x + - $ 4 2 3 . ,
Elimination of x ú û ê ë - + ù é 6 3 12 2 4 a b y { } ë û ) , ( 6 / 3 12
var
... ] 5 [
for y x a b k - + = Î a b y - + 5 8 6 )% 3 12 (
Program fragment k u a b y + = - Ù Î $ 6 3 12 5 8 ]. , [
New equality, new variable u
Condition

21. Linear Integer Arithmetic Synthesis
Complexity
E = Equalities
N = Inequalities
V = Output variables ÷ ø ö ç è æ + = - ) ,
min( 2
Ops( 1 E V N O

22. Parametrized Linear Integer Synthesis
> 0
= B
< y
5x-13y+(C-A)z
3x+(AB)y-2A
Cx+1
A,B,C   provided at run-time
Find x, y, z  
V. Weispfenning 1997 : Uniform Presburger Arithmetic

23. Example y j x i + × - = ) 1 ( j i ³ j y x i < £ + × = i j + × = j i
Quotient computation in 1990 by Z. Manna, R. Waldinger y j x i + × - = ) 1 ( j i j y x i
< + × = i j + × = j i
<
def quotientRemainder(i : Int, j: Int):(Int, Int) = {
if(i >= j) {
val (x, y) = quotientRemainder(i-j, j)
(x+1, y)
} else {
(0, i) }
Can we do better ?

24. Example < j False 1 ) ( - £ × i j x j y x i < £ + × = j x i <
Quotient computation in 2010 by M. Mayer
< j
False 1 ) ( - × i j x j y x i
< + × = j x i
< × - = j
False i + 1
> j i j x × + - 1 ë û ) , (
var / y x j i × - =
> j
Condition
Program fragment

25. Example ë û ) , ( var / y x j i × - =
Quotient computation in 2010 by M. Mayer
Comparison
def quotientRemainder(i : Int, j: Int):(Int, Int) = {
if(i >= j) {
val (x, y) = quotientRemainder(i-j, j)
(x+1, y)
} else {
(0, i) } ë û ) , (
var / y x j i × - =
Condition
Recursive
Comfusy

26. Parametrized Linear Integer Synthesis
Inequalities:
Test each coefficient sign >0, =0, <0
Generate if-then-elses if needed.
Add the coefficient sign to the precondition.

27. Test if all coefficients are zero
Parametrized Linear Integer Synthesis
Equalities ) ( . , = + × $ dz cy bx f e a z y x { } ï þ ý ü î í ì + - = × v k u z y x g d c b f e a ij 22 21 20 12 11 10 02 01
var
... ) ,
Bézout( (
gcd(
else
&
if( 2 ) 20 , 3 (1
gcd =
Test if all coefficients are zero
Bézout’s Theorem b a 3 2 + ) ,
gcd( ( = + × Ù Ú f e a d c b
Condition
Program fragment

28. Future of Synthesis Program optimization Other types
Recursive Abstract data types
Strings …
Efficient high-order specification languages

29. Thank you

30. Related work
A deductive approach to program synthesis, Z. Manna and R. Waldinger,
Complexity and uniformity of elimination in presburger airthmetic, V. Weispfenning, 1997.
Combinatorial sketching for finite programs, A. Solar-Lezama & al., 2006.
From Program Verification to Synthesis, S. Sivastava & al., 2010.
A practical algorithm for extract array dependence analysis, W. Pugh, 1992.
Partial Evaluation and Automatic Program Generation, 1993.
This presentation is based on the following publications:
On Complete Functional Synthesis. (Mayer, Suter, Piskac, Kuncak 2009)
Complete Program Synthesis for Linear Arithmetics (Mayer 2010)

31. Extra
Happy New Year 2010

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Happy New Year 2010
Quatrains
Until the end

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Constraints

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Constraints

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Expected output

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Expected output

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Specifications

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Result

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Happy New Year 2010
Result