- The paper introduces Deterministic PDGs (dPDGs) that guarantee unique execution outcomes equivalent to CFGs.
- It extends operational semantics to unstructured programs, resolving def-order dependencies and indeterminacy in execution.
- The methodology underpins formal verification of optimization transformations and opens avenues for concurrent program design.
Semantical Equivalence of the Control Flow Graph and the Program Dependence Graph
Introduction
The paper "Semantical Equivalence of the Control Flow Graph and the Program Dependence Graph" (1803.02976) addresses the operational semantics of Program Dependence Graphs (PDGs), an intermediary representation of programs that encapsulates both data and control dependencies between program statements. PDGs have been effectively utilized for optimizing compiler code due to their ability to capture dependencies without being confined by the sequential ordering found in Control Flow Graphs (CFGs). However, existing studies on PDGs have predominantly focused on well-structured programs, leaving the semantics of PDGs for more complex structures under-explored. This paper aims to extend the operational semantics to encompass broader types of PDGs.
Operational Semantics of PDGs
The paper introduces a comprehensive operational semantics for PDGs that extends the capability beyond well-structured programs. The PDG abstractly represents dependencies without imposing artificial constraints that sequential program statements might exhibit. Yet, the indeterminacy in execution order poses challenges to semantic consistency. This research defines a particular subset of PDGs, termed Deterministic PDGs (dPDGs), which yield unique execution states regardless of paths taken during execution due to their built-in structural constraints. These dPDGs ensure consistency between PDG and CFG executions, providing potential to utilize PDGs for verifying optimizations and transformations.
Deterministic PDGs
The characterization of dPDGs hinges on conditions pertaining to common ancestry and dependency edges. For dPDGs, dependencies do not allow simultaneous execution unless specific def-order dependencies are resolved, ensuring that all possible execution paths lead to consistent final results. This addresses previous limitations, establishing a robust operational framework applicable to both structured and unstructured control flow patterns.
Semantical Correspondence
A key result demonstrated in this paper is the semantic equivalence between CFGs and dPDGs for the class identified. This equivalence is crucial as it substantiates the reliability of PDGs in program optimization contexts where semantic fidelity to the original CFG is paramount. The similarity in execution sequences ensures that correctness proofs of optimizations based on PDGs can rely on well-established CFG semantics.
Implications and Future Directions
The findings have significant implications for formal verification of optimization techniques that utilize PDG transformations, especially when PDGs are converted back to CFGs following optimization. The paper calls attention to existing algorithms for such transformations which may introduce artificial constructs or duplicate nodes without formal proof of correctness. The operational semantics provided can bridge this gap, offering a robust foundation for verifying such algorithms.
Furthermore, the research suggests exploring the semantical equivalence between concurrent algorithms and PDGs. This could pave the way for new methodologies for constructing concurrent programs from PDG representations, thus broadening the applicability of PDGs in parallel computing environments.
Conclusion
The paper enriches the existing body of knowledge by extending the operational semantics of PDGs, introducing the notion of dPDGs to ensure semantical equivalence with CFGs. This groundbreaking methodology underpins a new wave of research possibilities in program optimization and concurrency, offering both theoretical insights and practical tools for advanced compiler design.
