Formalising Luck Improved Probabilistic Semantics for - - PowerPoint PPT Presentation
Context Choice-recording semantics Integrating backtracking strategies Conclusion Formalising Luck Improved Probabilistic Semantics for Property-Based Generators Diane Gallois-Wong supervised by C at alin Hrit cu, INRIA Paris
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Diane Gallois-Wong supervised by C˘ at˘ alin Hrit ¸cu, INRIA Paris September 5th 2016
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Property-Based Testing: running a program on many random inputs to test a property about it (QuickCheck) Testing insertion function for binary search trees insertBST: ∀(tr : int tree)(x : int). isBST(tr) ⇒ isBST(insertBST tr x)
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∀(tr : int tree)(x : int). isBST(tr) ⇒ isBST(insertBST tr x) Problem: isBST is sparse, most of the tests are irrelevant Solution: Property-Based Generators (PBGs) Problem: hard to implement, easily unbalanced/incomplete (e.g. generate almost only small trees) Luck: programming language dedicated to writing PBGs
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Property-Based Generator in Luck: Form of the program: predicate (boolean function) representing the property Running it: give unknowns as arguments, generate valuation V for them so that predicate evaluates to True (Can also be interpreted as a usual predicate by giving concrete values as arguments) Mixing constraint solving and instantiation / backtracking: constraint solving by default, user-controlled instantiation
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Standard lambda calculus with pairs binary sums: LT1+T2 e RT1+T2 e case e of (L x → e1) (R y → e2) recursive types Plus unknowns and special constructs to control instantiation
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fun (bst : int -> int -> int -> int tree -> bool) size low high tree = if size == 0 then tree == Empty else case tree of | 1 % Empty -> True | size % Node x l r -> ((low < x && x < high) !x) && bst (size / 2) low x l && bst (size / 2) x high r
Apply bst to given integers size, low, high to obtain a BSTs generator (predicate int tree → bool)
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Big-step operational probabilistic semantics Derivations for a generator gen produce a possible output but also the choices made to reach it, recorded in trace t gen ⇑t output Doesn’t handle backtracking: output is either an actual valuation V, or need to backtrack ∅ (error monad)
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Choice: (m, n, q) n: number of possibilities m: index of the one actually taken (0 ≤ m < n) q: probability of making this choice (q ∈ Q, 0 < q ≤ 1) Trace: sequence of choices Probability P(t) of a trace t: product of the probabilities of its choices
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( P(t): product of the probabilities of the choices in t ) πgen = [ output →
P(t) ] gen ⇑[(0,2, 1
2 ); (0,2, 1 3 ); (0,2, 2 5 )]
∅ gen ⇑[(0,2, 1
2 ); (0,2, 1 3 ); (1,2, 3 5 )]
V1 gen ⇑[(0,2, 1
2 ); (1,2, 2 3 )]
∅ gen ⇑[(1,2, 1
2 )]
V2 [V1 →
1 10; V2 → 1 2; ∅ → 1 15 + 1 3]
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Former semantics: directly derives whole probability distribution in a collecting style gen ⇑ πgen Our new choice-recording semantics is: simpler: proofs with Coq more expressive:
still able to produce πgen better handles not always terminating programs provides more detailed information, used in next part
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πgen ranges over valuations V but also need to backtrack ∅ Running a generator in practice: apply a backtracking strategy to always output a valuation Objective: determine final distribution ρbstrat
gen
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(Assume all executions terminate) gen ⇑[(0,2, 1
2 ); (0,2, 1 3 ); (0,2, 2 5 )]
∅ gen ⇑[(0,2, 1
2 ); (0,2, 1 3 ); (1,2, 3 5 )]
V1 gen ⇑[(0,2, 1
2 ); (1,2, 2 3 )]
∅ gen ⇑[(1,2, 1
2 )]
V2 V2 ∅ V1 ∅
(0, 2, 2
5 )
(1, 2, 3
5 )
(0, 2, 1
3 )
(1, 2, 2
3 )
(0, 2, 1
2 )
(1, 2, 1
2 )
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Time-homogeneous Markov chains with discrete time and finite state space Init. V2 ∅ V1 ∅
2 5 3 5 1 3 2 3 1 2 1 2
1 1 1 1 Restart-from-scratch strategy Absorbing Markov chain [V1 → 1 6; V2 → 5 6]
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Backtrack-from-leaf-to-parent strategy Init. V ∅ ∅
1 3 2 3 1 2 1 2
1 1 1 Markov chain is not absorbing: no probability distribution
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Remember hopeless nodes that lead only to ∅ leaves Backtrack to parent from ∅ leaf or node with only hopeless children Restart execution when more than a set number Bmax
Set of states: Nodes × ( P(Nodes) × {0, 1, ..., Bmax} ) ( node, ( hopeless nodes, number of ∅ leaves seen ) )
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A V2 G B ∅ F C V1 E ∅ D
2 5 3 5 1 3 2 3 1 2 1 2
(D, S, n)
1
− → (C, S ∪ {D}, n + 1) if n < Bmax (D, S, Bmax)
1
− → (A, S ∪ {D}, 0) (C, S, n)
2/5
− − → (D, S, n) (C, S, n)
3/5
− − → (E, S, n)
∈ S (C, S, n)
1
− → (E, S, n) if D ∈ S, E / ∈ S (C, S, n)
1
− → (B, S ∪ {C}, n) if D, E ∈ S (impossible, yet transition exists) (G, S, n)
1
− → (G, S, n)
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Backtracking strategy: computable function associating to a choice tree a Markov chain verifying: it is time-homogeneous with discrete time it has a finite set of states of form Nodes × M it has a single initial state of form (root, M0) absorbing states are exactly those with a non-∅ leaf it is absorbing
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Theorem Final distribution ρbstrat
gen
is computable from gen and bstrat. gen → choice tree: computable (finite) choice tree → Markov chain: bstrat computable (def) Markov chain → ρbstrat
gen :
Transition matrix of absorbing Markov chain Q R I
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A strategy is “activated only upon backtracking” if for any Markov chain in its image: for any state that is reachable from the initial state without visiting a ∅ leaf, the transitions follow the choice tree A C B
2 3 1 3
Consequence: ∀V. ρbstrat
gen (V) ≥ πgen(V)
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A strategy is a computable function from choice trees to Markov chains: very permissive Possible restriction: transitions from a state (node, M) computable only from node, M, children of node and edges to them in choice tree Examples need minor adaptation: restart-from-scratch: add memorised information which always contains root more complex strategy: add path of ancestors of the current node to memorised information
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πgen usually more intuitive for the user than ρbstrat
gen
so ideally bstrat should keep ρbstrat
gen
close to πgen Extremality result: restart-from-scratch most conservative, according to two functions evaluating closeness of distributions
DB(P, Q) = − log
i
Kullback-Leibler divergence DKL(P||Q) =
i P(i) log P(i) Q(i)
complexity term (e.g. expected number of steps in the Markov chain before being absorbed)
22/ 22 Context Choice-recording semantics Integrating backtracking strategies Conclusion
Choice-recording semantics style: simple (Coq proofs) yet more expressive than the previous one. Formalisation of backtracking strategies. Integration of them into the semantics to obtain the final probability distribution of a generator. Computability result. Future work:
identify and study narrower subclasses of strategies further explore evaluation of strategies
to characterise viable strategies.