Automatic Complexity Analysis of Simple Imperative Programs

Zachi Mann

Consider the following “Core” Language:

•

• Non-deterministic branchingBounded non-deterministic loops

Variables hold only positive integer values

Problem Statement

• The polynomial-bound analysis problem is to find, for any given command, which output variables are bounded by a polynomial in the input variables.

• A variant: The linear-bound problem identifies linearly-bounded output values instead.

• The problem can be easily extended to identify the overall runtime upper bound.

A Short Example

may grow exponentially!

output value bounded by a polynomial in it’s input ?

Ben-Amram, Jones & Kristiansen (2008)

• Sound & complete compositional analysis algorithm running in polynomial time.

• Algorithm assigns a certificate(DFR) to each command: DFR

Ben-Amram, Jones & Kristiansen (2008)• DFR left part represents the variables dependencies.

type:

DFR

Ben-Amram, Jones & Kristiansen (2008)

• Example:

• DFR right part is used to identify computations that double an input value by adding two copies of it:

DFR

Ben-Amram (2010)

• Core language is extended with the constant zero.

• The analysis problem is shown to be PSPACE-complete.

Back to the Example

may grow exponentially is polynomially bounded

Ben-Amram (2010)

• Problem solved by applying context sensitive analysis.– Dependence Judgments:

– Inference Rules:

– Example (assume program has 3 variables):

Ben-Amram (2010)

• Inference rules (cont.)Another example - multiplication:

Formally:

Ben-Amram (2010)

• The link between inference rules and DFR:– The inference rules generate the variable

dependences (arcs) within the DFR.

Implementation• Bottom Up– For every command the set of all true judgments are

generated.• Top Down– An abstract interpreter generates for every command

the dependence judgments according to the inference rules and the reset contexts that were computed so far from previous commands.

• Output Extension– Algorithm output is extended to include a description

of a computation path which results with each computed variable dependence.

Loops

A command is analysed in 3 steps:1. Consider case where loop body is skipped

Similar to analysis of the skip command.2. Consider any iterations

Least Fixed Point.3. Loop Correction

Performance Comparison

Related Work

• What happens if features are added/dropped from the core language? – Deterministic loops, the constant 1,

weak assignment, …• Is the analysis problem decidable? With what

run time complexity?– Some answers in paper by

Amir Ben Amram & Lars Kristiansen (2011)