Swarat ChaudhuriRoberto Lublinerman

Pennsylvania State University

Sumit Gulwani

Microsoft Research

CAUCHY

Continuity analysis of programs

Uncertainty and robustness

• Trends: cyber-physical systems, integration of computation and science, …

• Uncertainty: stale satellite data, erroneous sensor measurements, …

• Does your program handle uncertainty robustly? – Correctness in settings without

uncertainty does not imply correctness in uncertain environments.

Robustness analysis of programs

• Robustness: small perturbations to a program’s operating conditions do not change its behavior significantly.– Continuity:

Infinitesimal changes to inputs only cause infinitesimal changes to outputs.

– Discrete continuity:Similar, except for non-infinitesimalchanges to discrete numbers.

– Derivative has low complexity. – Asymptotic stability

• Verify these!

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First step: Continuity analysis of programs

• Continuity of mathematical functions:– definition.– Equivalently, infinitesimal changes in

inputs only cause infinitesimal changes in outputs.

• Continuity of programs – Associate metric spaces with types, lift

it into a metric over states. – Same question. Do infinitesimal

changes in program inputs only cause infinitesimal changes to outputs?

This paper: structural analysis of continuity.

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Example: an implementation of Dijkstra’s algorithm

• Small change to real array: each element changes at most by a small amount.

• Small change to graph with real edge weight: each edge weight changes at most by a small amount.

• Is this program continuous?

procedure Dijkstra (G: graph, src: node):

for each node v in G: { d[v] := Infinity }

d[src] := 0; Worklist := set of all nodes in G;

while Worklist is not empty {

Remove node w from Worklist s.t. d[w] is minimal;

for each neighbor v of w: {

z := d[w] + G[w,v];

if (z < d[v]) { d[v] := z; prev[v] := w; } } }

Example: Dijkstra’s algorithm

• Small change to real array: each element changes at most by a small amount.

• Small change to graph with real edge weight: each edge weight changes at most by a small amount.

• Is this program continuous?

procedure Dijkstra (G: graph, src: node):

for each node v in G: { d[v] := Infinity }

d[src] := 0; Worklist := set of all nodes in G;

while Worklist is not empty {

Remove node w from Worklist s.t. d[w] is minimal;

for each neighbor v of w: {

z := d[w] + G[w,v];

if (z < d[v]) { d[v] := z; prev[v] := w; } } }

• Depends on what is observable.• At point of output, d is a continuous function of G, but prev is not.

• Sorting algorithms are continuous

… but only if output = array of keys.• Minimum spanning tree algorithms are continuous

… but only if the output is the weight of the tree.• Integer knapsack is continuous in values of items but not

in their weights.• Fractional knapsack is continuous in values and weights.

Continuity at work!

2.0 3.0 4.0 3.0

2.0 3.0 3.0 4.0

2.0 3.0 4.0 2.99

2.0 2.99 3.0 4.0

• Key Idea: Prove branch-equivalence at the zeroes of b—i.e., conditions under which guard can flip on small changes.

• Example: d[v] is continuous after if (d[v] < z) d[v] := z.

The guard (d[v] < z) flips (under small changes) only when d[v] = z. Then, d[v] has similar values on both branches.

• Automate using an SMT-solver. (cf. Translation validation)

Challenge #1: Control flow

if b

P1 P2

1. 2. P1 and P2 are continuous. P is continuous

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Challenge #2: Noninductiveness

while Worklist is not empty {

Remove node w from Worklist s.t. d[w] is minimal;

for each neighbor v of w: {

z := d[w] + G[w,v];

if (z < d[v]) { d[v] := z; prev[v] := w; } } }

• Small change to d at iteration-entry can completely change the value of d at the end.

• Thus, continuity is not inductive.

u1

d[u1] = 2.00u2

d[u2] = 2.00u3

d[u2] = 4.00

u2

d[u2] = 2.00u1

d[u1] = 2.00

Key idea: Induction over epochswhile Worklist is not empty {

Remove node w from Worklist s.t. d[w] is minimal;

• To be reordered, iterations must be approximately tied on selection criterion. Epoch = cluster of such iterations.

• Prove that iterations within epochs are commutative. • Proof can be discharged using an SMT-solver.

u3

d[u2] = 4.00

u1

d[u2] = 2.01u2

d[u1] = 2.00

u3

d[u2] = 4.02

u2

d[u2] = 2.00u1

d[u1] = 2.00

Key idea: Induction over epochs

while Worklist is not empty {

Remove node w from Worklist s.t. d[w] is minimal;

Now do induction over epochs.

u3

d[u2] = 4.00

u1

d[u2] = 2.01u2

d[u1] = 2.00u3

d[u2] = 4.02

Original

Perturbed

But often, simple induction is enough

for k := 1 to N

for i, j := 1 to N:

if G[i, j] > G[i, k] + G [k, j]

G[i, j] := G[i, k] + G[k. j];

Floyd-Warshall shortest path algorithm

Challenge #3: Early or late termination

• Key Idea: Prove idempotence under conditions when guard can flip.

• Example: while (z > 0) { x := x + z; z := z * w; }

If z = 0, then loop body is idempotent.

while b

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Original

Perturbed

• Soundness with respect to definition. (Tricky!) • Proof rules discharged using Z3 SMT solver.• Able to prove 11 of the 13 continuous algorithms

targeted:– Sorting (Merge sort, Bubble sort, Insertion sort, Selection sort).– Minimum Spanning Tree (Prim’s and Kruskal’s)– Shortest Paths (Floyd-Warshall, Bellman-Ford, Dijkstra)– Knapsack (Fractional and integer)

• Epoch induction needed in 5/13 cases. • Early termination check needed in 3/13 cases.• Current work exploring “real” applications (embedded

medical devices and GPS apps).

Results

Ongoing work: discrete derivatives of programs

• Instead of infinitesimal changes to real variables, consider unit changes to finite-precision variables.

• More natural in the quantitative setting. • Changes the game somewhat:

– E.g., Addition is not continuous. • But most of the rules/insights still apply. • Goal: Mechanically generate discrete

derivatives of programs:– E.g., Discrete derivative of Dijkstra’s

algorithm in O(n).

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The Cauchy challenge

Cauchy

Develop an analytical calculus of computation

Limits of programs

Hybrid representations

Continuity analysis

Analytic approximations ofprograms

(Discrete) derivatives of programs

Applications in cyber-physical systems, approximate computation. Also, pedagogical value.

Fourier analysis of programs

Conclusion

• Robustness is an important correctness property for programs operating under uncertainty.

• Continuity is one, but by no means the only, robustness property.

• This paper offers one, but by no means the only, continuity analysis.

• First step towards an analytical calculus of computation.