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Recursive Program Synthesis
Aws Albarghouthi (UToronto), Sumit Gulwani (MSR), and Zachary Kincaid (UToronto) CAV 2013 Saint Petersburg, Russia
Program Synthesis
2
Magic
Specification Program
Recursive Program Synthesis Aws Albarghouthi (UToronto), Sumit - - PowerPoint PPT Presentation
Recursive Program Synthesis Aws Albarghouthi (UToronto), Sumit - - PowerPoint PPT Presentation
Recursive Program Synthesis Aws Albarghouthi (UToronto), Sumit Gulwani (MSR), and Zachary Kincaid (UToronto) CAV 2013 Saint Petersburg, Russia Program Synthesis Magic Specification Program 2 Program Synthesis Magic Specification
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Program Synthesis
2
Magic
Specification Program
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What are your Dreams?
3
Magic
Specification Program
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What are your Dreams?
3
Magic
Specification Program First-order Logic
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What are your Dreams?
3
Magic
Specification Program First-order Logic Temporal Logic
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What are your Dreams?
3
Magic
Program First-order Logic Temporal Logic I/O Examples
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Why Synthesis from I/O?
I/O examples are easy to specify
4
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Why Synthesis from I/O?
I/O examples are easy to specify Non-expert users can specify I/O behaviour
- See, e.g., FlashFill for Excel, Smartphone scripts, etc.
- Desired programs are usually simple
4
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Contribution
5
Escher
I/O Examples Recursive Program
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Contribution
5
Escher
I/O Examples Recursive Program User
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Contribution
5
Escher
I/O Examples Recursive Program Parameterized by a set of building blocks (operations) integers, lists, trees, etc. User
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Contribution
5
Escher
I/O Examples Recursive Program Parameterized by a set of building blocks (operations) Novel search-based synthesis technique integers, lists, trees, etc. User
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Contribution
5
Escher
I/O Examples Recursive Program Parameterized by a set of building blocks (operations) Novel search-based synthesis technique integers, lists, trees, etc. User No templates required
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High Level View
6
Forward Search
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High Level View
6
Forward Search
I1 In . . .
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High Level View
6
Forward Search
I1 In . . .
- O1. . .On
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High Level View
6
Forward Search
I1 In . . .
- O1. . .On
P1 Pn . . .
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High Level View
6
Forward Search Conditional Inference
I1 In . . .
- O1. . .On
P1 Pn . . . I1 O1
+
I2 O2
P1 P2
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High Level View
6
Forward Search Conditional Inference
I1 In . . .
- O1. . .On
P1 Pn . . . I1 O1
+
I2 O2
P1 P2
= if ( ) else
C
P1
P2
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High Level View
6
Forward Search Conditional Inference
I1 In . . .
- O1. . .On
P1 Pn . . . I1 O1
+
I2 O2
P1 P2
= if ( ) else
C
P1
P2
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High Level View
6
Forward Search Conditional Inference
I1 In . . .
- O1. . .On
P1 Pn . . . I1 O1
+
I2 O2
P1 P2
= if ( ) else
C
P1
P2
Recursive call synthesis Reuse I/O as recursive call specification or query user
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Example
Synthesize list length from examples:
7
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Example
Synthesize list length from examples:
7
→ [],[2],[1,2]
Inputs:
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Example
Synthesize list length from examples:
7
→ [],[2],[1,2]
Inputs:
0,1,2
Outputs:
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Ex: Forward Search
8
Components: inc, isEmpty, tail, zero
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Ex: Forward Search
8
Components: inc, isEmpty, tail, zero Programs of size 1:
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Ex: Forward Search
8
Components: inc, isEmpty, tail, zero Programs of size 1:
1 P1 = i → [],[2],[1,2]
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Ex: Forward Search
8
Components: inc, isEmpty, tail, zero Programs of size 1:
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
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Ex: Forward Search
8
Components: inc, isEmpty, tail, zero Programs of size 1:
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
matches output for first input Recall outputs
0,1,2
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Ex: Conditional Inference
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
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Ex: Conditional Inference
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
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Ex: Conditional Inference
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
Goal Graph (GG):
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
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Ex: Conditional Inference
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
Goal Graph (GG):
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
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Ex: Conditional Inference
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
AND node Goal Graph (GG):
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
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Ex: Conditional Inference
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
AND node Goal Graph (GG):
1 → P2 = zero → 0,0,0 0,1,2
Outputs:
“Backward” search
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Ex: Forward Search 2
10
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2:
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Ex: Forward Search 2
10
→
- 2
P3 = tail(P1) → err,[],[2] P4 = inc(P2) → 1,1,1 P5 = isEmpty(P1) → T,F,F
1 P1 = i → [],[2],[1,2]
1 → P2 = zero → 0,0,0
Programs of size 1: Programs of size 2: Satisfies one of our goals
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Ex: Conditional Inference 2
11
4 2 →
- P5 = isEmpty(P1) → T,F,F
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
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Ex: Conditional Inference 2
11
4 2 →
- P5 = isEmpty(P1) → T,F,F
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
- P5
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Ex: Conditional Inference 2
11
4 2 →
- P5 = isEmpty(P1) → T,F,F
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
- P5
if ( ) else
- P5
- P2
...
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Ex: Forward Search 3
12
Programs of size 2 →
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
→
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
Only allow calls satisfying well-founded relation
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Ex: Forward Search 3
12
Programs of size 2 Programs of size 3
→
- 3 P6 = length(P3) → err,0,1
Use I/O values to simulate recursive call
→
- P3 = tail(P1) → err,[],[2]
...
Only allow calls satisfying well-founded relation
But what if there isn’t such an I/O pair? Query the user for new examples!
(too long for 15 min talk)
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Ex: Conditional Inference
13
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
One iteration later:
- P5
inc(length(tail(i))) P8
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Ex: Conditional Inference
13
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
One iteration later:
- P5
inc(length(tail(i))) P8
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Ex: Conditional Inference
13
Pr 0,1,2
T,F,F ?,1,2 0,?,?
cond then else goal
P2
One iteration later:
- P5
inc(length(tail(i))) P8
if isEmpty(i) then 0 else inc(length(tail(i)))
P5 P8 P2
Synthesized program:
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Escher: Recap
Forward Search
- Apply component to all synthesized programs
- Heuristic-based search
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Escher: Recap
Forward Search
- Apply component to all synthesized programs
- Heuristic-based search
Conditional Inference
- Uses goal graph to synthesize conditionals
- Conditionals synthesized on demand
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Escher: Recap
Forward Search
- Apply component to all synthesized programs
- Heuristic-based search
Conditional Inference
- Uses goal graph to synthesize conditionals
- Conditionals synthesized on demand
Recursive call synthesis
- Use I/O specification
- Query user for more output values if needed
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Implementation
Prototype implementation of Escher
- Pluggable components
Experimented with:
- integer, list, and tree manipulating programs
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Experiments
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Escher w/o
ObsEquiv w/o GoalGraph
w/o OE+GG
collect_leaves
0.04 0.09 68.9 81.8
count_leaves
0.06 0.2 9.3 12.3
hbal_tree
1.5 MEM TIME MEM
nodes_at_level 10.74 MEM
TIME MEM
(tree manipulating programs) Time in seconds
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Experiments
17
(#components sensitivity: mult)
Escher: 0.7s
with all components
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Experiments
17
(#components sensitivity: mult)
10 20 30 40 1 2 3 4 5Sketch sensitivty
SAT-based synthesis [ASPLOS’06]
NUMBER OF EXTRA COMPONENTS T I M E ( S )
Escher: 0.7s
with all components
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Experiments
17
(#components sensitivity: mult)
10 20 30 40 1 2 3 4 5Sketch sensitivty
SAT-based synthesis [ASPLOS’06]
NUMBER OF EXTRA COMPONENTS T I M E ( S )
Had to supply Sketch with high-level conditional
Escher: 0.7s
with all components
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Conclusion
18
Escher
I/O Examples Recursive Program User
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Conclusion
18
Escher
I/O Examples Recursive Program User Interactive
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Conclusion
18
Escher
I/O Examples Recursive Program User Interactive Highly-customizable
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Conclusion
18
Escher
I/O Examples Recursive Program User Interactive Highly-customizable Novel synthesis techniques
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Questions?
How can we synthesize loops?
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Questions?
How can we synthesize loops? Integration with SMT
- based search
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Questions?
How can we synthesize loops? Integration with SMT
- based search
Applications of Escher in eduction
- e.g., for interacting with students
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Questions?
How can we synthesize loops? Integration with SMT
- based search
Applications of Escher in eduction
- e.g., for interacting with students
Theoretical under-pinnings
- e.g., see Madhusudan [CSL’11]
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