[Paper Review] An Algebraic Framework for Compositional Program Analysis

Azadeh Farzan, Zachary Kincaid|arXiv (Cornell University)|Oct 13, 2013

Logic, programming, and type systems23 references3 citations

TL;DR

This paper introduces an algebraic framework based on Pre-Kleene algebras to design, implement, and formally verify compositional program analyses. It enables efficient, scalable analysis via path-expression algorithms and provides a unified foundation for computing loop invariants and procedure summaries through algebraic operations like sequencing, choice, and iteration.

ABSTRACT

The purpose of a program analysis is to compute an abstract meaning for a program which approximates its dynamic behaviour. A compositional program analysis accomplishes this task with a divide-and-conquer strategy: the meaning of a program is computed by dividing it into sub-programs, computing their meaning, and then combining the results. Compositional program analyses are desirable because they can yield scalable (and easily parallelizable) program analyses. This paper presents algebraic framework for designing, implementing, and proving the correctness of compositional program analyses. A program analysis in our framework defined by an algebraic structure equipped with sequencing, choice, and iteration operations. From the analysis design perspective, a particularly interesting consequence of this is that the meaning of a loop is computed by applying the iteration operator to the loop body. This style of compositional loop analysis can yield interesting ways of computing loop invariants that cannot be defined iteratively. We identify a class of algorithms, the so-called path-expression algorithms [Tarjan1981,Scholz2007], which can be used to efficiently implement analyses in our framework. Lastly, we develop a theory for proving the correctness of an analysis by establishing an approximation relationship between an algebra defining a concrete semantics and an algebra defining an analysis.

Motivation & Objective

• To provide a generic, algebraic foundation for designing compositional program analyses that support efficient, parallelizable analysis.
• To enable novel loop invariant computation by treating loop semantics as an algebraic iteration operation rather than iterative approximation.
• To decouple the analysis specification from the algorithmic implementation via a clear interface between abstract algebras and path-expression algorithms.
• To establish a formal correctness proof method using a soundness relation between concrete and abstract algebras, replacing ad-hoc analysis-specific arguments.

Proposed method

• Define program analyses using a Pre-Kleene algebra structure with three core operations: sequencing, choice, and iteration.
• Represent program paths as regular expressions and reinterpret them using the semantic algebra's operations to compute abstract program meanings.
• Use path-expression algorithms—such as those by Tarjan (1981a) and Scholz & Blieberger (2007)—to efficiently compute the result of the analysis via algebraic evaluation.
• Establish correctness by proving a soundness relation between a concrete semantic algebra and the abstract analysis algebra, analogous to a Galois connection in the iterative framework.
• Support interprocedural analysis by modeling procedure summaries as algebraic elements and combining them via algebraic operations.
• Handle local variables and procedure frames using an algebraic treatment of quantification, similar to but more abstract than Cousot and Cousot’s footprint/frame model.

Experimental results

Research questions

• RQ1 How can compositional program analyses be systematically designed and verified using a unified algebraic framework?
• RQ2 Can loop invariants be computed more effectively using an algebraic iteration operator rather than iterative approximation?
• RQ3 What is the relationship between the concrete semantics of a program and its abstract analysis in an algebraic setting?
• RQ4 How can path-expression algorithms be used to compute analysis results efficiently and correctly within this framework?
• RQ5 To what extent can this framework support advanced features like interprocedural analysis and local variable handling in a modular way?

Key findings

• The framework enables the design of compositional program analyses using a minimal algebraic structure—Pre-Kleene algebras—equipped with sequencing, choice, and iteration.
• Loop invariants can be computed non-iteratively by applying the algebraic iteration operator to the loop body, enabling new invariant discovery techniques.
• Path-expression algorithms such as those by Tarjan (1981a) and Scholz & Blieberger (2007) can be used to compute analysis results efficiently by evaluating regular expressions over the abstract algebra.
• Correctness of an analysis is established via a soundness relation between the concrete and abstract algebras, providing a general proof method independent of analysis-specific details.
• The framework supports interprocedural analysis by modeling procedure summaries as algebraic elements and combining them via algebraic operations, enabling scalable and modular analysis.
• The approach generalizes and unifies existing compositional analyses, including those based on WPDS and Kleene algebra, while enabling new constructions such as non-iterative loop abstractions.