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Probabilistic Correctness Properties
P(x) := 0 ≤ x ≤ 0.2 P(x) := 0 ≤ x ≤ 0.8
1 ] 0.2 1 ] 0.8
x ~ Uniform([0, 1])
1
post := Pr[P(x) = 1] > ⅓
✗ ✓
Efficient Synthesis with Probabilistic Constraints Samuel Drews, - - PowerPoint PPT Presentation
Efficient Synthesis with Probabilistic Constraints Samuel Drews, - - PowerPoint PPT Presentation
Efficient Synthesis with Probabilistic Constraints Samuel Drews, Aws Albarghouthi, Loris DAntoni Probabilistic Correctness Properties x ~ Uniform([0, 1]) post := Pr[ P ( x ) = 1] > 0 1 P ( x ) := 0 x 0.2 ] 0 0.2 1 P (
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Probabilistic Program Synthesis
Progs post Q P
Pr[Q(x) ≠ P(x)]
1 ⅓
Values of a: Decreasing “Error” Pr[Q(x) ≠ P(x)] Satisfy post Progs = {P(x) := 0 ≤ x ≤ a | a ∈ R} inputs ~ Uniform([0, 1]) post := Pr[P(x) = 1] > ⅓ Q(x) := constant-0 function
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DIGITS (CAV17)
Sample Set S {0.4, 0.6} Program Correct? Error [0, 0.3] ✗ 0.3 1 unsat
- 1
[0, 0.5] ✓ 0.5 1 1 [0, 1] ✓ 1.0 Constraints f : S→{0,1} For each f : S → {0,1}, query a synthesis oracle Oracles compute post and Error Sample m inputs Progs = {P(x) := 0 ≤ x ≤ a | a ∈ R} inputs ~ Uniform([0, 1]) post := Pr[P(x) = 1] > ⅓ Q(x) := constant-0 function
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Progs post
DIGITS Convergence (CAV17)
Theorem (Convergence of DIGITS) (Under certain assumptions,) with high probability, DIGITS enumerates some correct, (near-)near-optimal program P.
𝜁 P Q P*
Pr[Q(x) ≠ P*(x)]
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Which Begs the Questions:
Do we really need to make exponentially many queries? How high is the “high probability” of optimal convergence? No! Depends on the number of samples.
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DIGITS Trie Implementation (CAV17)
unsat
Sample Set S {0.4, 0.6} Program Correct? Error [0, 0.3] ✗ 0.3 1 unsat
- 1
[0, 0.5] ✓ 0.5 1 1 [0, 1] ✓ 1.0 Constraints f : S→{0,1}
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Efficiency of the Trie “Heuristic”
5050 <<< 2100 Polynomial Trendline: f(x) = ½ x2 + ½ x R2 = 1
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Polynomial?
Two equivalent ways of thinking:
- The trie avoids unsatisfiable parts of the search space
- The trie stays close to satisfiable parts of the search space
# Queries @ next depth = O(# Satisfiable @ current depth)
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For [0, a] intervals and m points, # Satisfiable Queries (m) = m + 1
1
3 samples
] ] ] ]
4 choices of right endpoint [0, a] Search Trie, depth m: O(m+1) m O(m2)
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Progs post
Finite VC Dimension → Polynomial Bound
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Progs post
Finite VC Dimension → Polynomial Bound
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Progs post
Finite VC Dimension → Polynomial Bound
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Progs post
Finite VC Dimension → Polynomial Bound
unsat
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Progs post
Finite VC Dimension → Polynomial Bound
unsat unsat unsat
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Progs post
Finite VC Dimension → Polynomial Bound
unsat unsat unsat
“Sauer-Shelah Lemma”: Only O(md) are satisfiable!
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Finite VC Dimension → Polynomial Bound
Theorem: If DIGITS runs on Progs using m samples, it performs O(md+1) synthesis queries. Progs Search Trie, |S| = m: O(md) m O(md+1)
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Which Begs the Questions:
Do we really need to make exponentially many queries? How high is the “high probability” of optimal convergence? No!---In fact, we only make O(md+1) Depends on the number of samples.
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𝜐-DIGITS
Use the constraint set to estimate the error Pr[Q(x) ≠ P(x)]: Fix threshold 𝜐 ∈ (0,1] and skip any f : S → {0,1} such that |{x ∈ S : Q(x) ≠ f(x)}| > 𝜐|S|
𝜐 Q P4 P2 P3 Progs post Q P4 P2 P3 P1 Progs post
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Progs post Q
Choosing 𝜐
𝜐 Progs post Q 𝜐
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Progs post Q
Adaptive 𝜐-DIGITS
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Progs post Q
Adaptive 𝜐-DIGITS
P1
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𝜐 Progs post Q
Adaptive 𝜐-DIGITS
P1
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𝜐 Progs post Q
Adaptive 𝜐-DIGITS
P1 P2
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𝜐 Progs post Q
Adaptive 𝜐-DIGITS
P1 P2
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Adaptive 𝜐-DIGITS Efficacy (Fairness Benchmarks)
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Which Begs the Questions:
Do we really need to make exponentially many queries? How high is the “high probability” of optimal convergence? No!---in fact, we only make O(md+1) Depends on the number of samples.
- --and we can do better with 𝜐-DIGITS
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Thanks!
Progs Search Trie, |S| = m: O(md) m O(md+1)
𝜐 Q P4 P2 P3 Progs post Q P4 P2 P3 P1 Progs post