Control-flow graph

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Control-flow graph (CFG) is a graphical representation of the flow of control within a program. It's a directed graph where nodes represent basic blocks of code, and edges represent control flow between these blocks. Each basic block typically contains a sequence of instructions that execute sequentially without any branching or looping within the block itself.

Some CFG examples:(a) an if-then-else (b) a while loop (c) a natural loop with two exits, e.g. while with an if...break in the middle; non-structured but reducible (d) an irreducible CFG: a loop with two entry points, e.g. goto into a while or for loop

In a CFG, control flow edges indicate the possible transitions between basic blocks, such as conditional branches, loops, function calls, and returns. Analyzing the control flow graph can help in understanding program behavior, optimizing code, and detecting potential security vulnerabilities or bugs.

Technical details

Each node in a control-flow graph, represents a basic block, a sequential segment of code without jumps or jump targets within. Jump targets initiate a block, while jumps end it. Directed edges represent jumps in the control flow. Typically, two blocks hold special significance: the entry block, marks the entry point of control into the graph, the exit block, serves as the point from which all control flow departs. Because of its construction procedure, in a CFG, every edge A→B has the property that:

outdegree(A) > 1 or indegree(B) > 1 (or both).

The construction of a Control Flow Graph (CFG) can be theoretically derived from a program's full flow graph, where each node represents a single instruction. This derivation process involves a technique known as edge contraction. Edges that do not satisfy a predetermined predicate are contracted, essentially merging them. This predicate typically specifies properties like having a single exit node for the originating edge and a single entry node for the target. While conceptually interesting, this edge-contraction algorithm holds limited practical value. Its primary utility lies in providing a visual aid for understanding CFG construction.

In practice, a more efficient method for CFG construction is employed. This method directly analyzes the program's structure to identify its constituent basic blocks. Basic blocks are essentially sequences of instructions that exhibit sequential execution without jumps or conditional branches. By directly identifying these basic blocks, the CFG can be constructed in a more efficient and practical manner.


Consider the following fragment of code:

0: (A) t0 = read_num
1: (A) if t0 mod 2 == 0
2: (B)     print t0 + " is even."
3: (B)     goto 5
4: (C) print t0 + " is odd."
5: (D) end program

In the given code snippet, there are four basic blocks identified: A spanning from line 0 to line 1, B from line 2 to line 3, C at line 4, and D at line 5. Notably, A serves as the "entry block," D as the "exit block," and lines 4 and 5 serve as jump targets. Consequently, a graph representing this fragment exhibits edges connecting A to B, A to C, B to D, and C to D.

Reachability in Control Flow Graphs

Reachability in Control Flow Graphs (CFGs):

Within the domain of compiler optimization, the concept of reachability plays a crucial role in analyzing CFGs. It pertains to a specific property of graphs that proves valuable for optimization techniques.

Unreachable Code and Removal

When a subgraph within a CFG is not reachable from the subgraph containing the program's entry block, it signifies that this subgraph cannot be executed during any possible program execution path. Consequently, such code is deemed unreachable. Under normal circumstances, this unreachable code can be safely removed from the program without affecting its functionality. This removal contributes to program size reduction and potentially improves execution efficiency.

Infinite Loops and the Halting Problem

The concept of reachability is also instrumental in detecting potential infinite loops. If the CFG's exit block is unreachable from the entry block, it suggests the program might be trapped in an infinite loop, continually executing instructions without ever reaching the designated termination point. However, it is essential to acknowledge the limitations of this approach. The Halting Problem establishes the theoretical impossibility of definitively determining whether an arbitrary program will halt or run indefinitely. Therefore, not all infinite loops can be reliably identified using reachability analysis alone.

Reachability and Optimizations

It's noteworthy that unreachable code and infinite loops can arise even in the absence of explicit coding by the programmer. Certain optimization techniques, such as constant propagation, constant folding, and subsequent jump threading, can inadvertently create these scenarios. These optimizations may merge multiple basic blocks into a single one, eliminate edges from the CFG, or otherwise restructure the control flow graph. As a consequence, previously reachable sections of the program might become disconnected and unreachable due to these optimizations.

This passage adopts a more formal and objective tone, using vocabulary suitable for an encyclopedia. It clarifies the concept of reachability within CFGs and its significance for optimization. It also acknowledges the limitations of reachability analysis in detecting infinite loops.

Domination Relationships in Control Flow Graphs (CFGs)

Within the realm of compiler optimization and program analysis, the concept of domination plays a pivotal role in understanding a program's control flow. Domination establishes a relationship between basic blocks within a CFG, signifying the influence one block exerts over another in terms of execution paths.

Dominators and Paths

A block denoted as M is said to dominate another block N if every conceivable execution path originating from the program's entry block and reaching block N must necessarily traverse through block M. Intuitively, M acts as a control point that all paths leading to N must pass through. By definition, the entry block dominates all blocks within the CFG, as all execution paths begin there.

Postdominators and Exit

The concept of postdomination operates in the opposite direction. A block M is said to postdominate block N if every potential execution path starting from block N and culminating at the program's exit block must invariably pass through block M. In simpler terms, M exerts control over all paths exiting from N and leading to program termination. Conversely, the exit block postdominates all blocks within the CFG, as all execution paths eventually reach the exit.

Immediate Domination and Uniqueness

A more granular relationship exists within domination, known as immediate domination. Block M is said to immediately dominate block N if two conditions hold true:

  • M dominates N (as defined earlier).
  • There exists no intermediary block P such that M dominates P and P dominates N. Essentially, M represents the closest dominator on all paths leading from the entry block to N. Notably, each block possesses a unique immediate dominator.

Postdominators and Analogy

Similar to immediate domination, the concept of immediate postdominator is established. This relationship mirrors immediate domination but operates in the context of postdominance, considering paths leading to the exit block.

Dominator Tree: Structure and Calculation

The dominator tree serves as a supplementary data structure that visually depicts the domination relationships within a CFG. This tree structure features an arc connecting block M to block N if M acts as the immediate dominator of N. A crucial property of the dominator tree is its tree-like structure. This arises from the fact that each block has a unique immediate dominator. Additionally, the dominator tree is rooted at the program's entry block, reflecting its dominance over all other blocks. The Lengauer–Tarjan algorithm offers an efficient method for calculating the dominator tree of a CFG.

Postdominator Tree

Analogous to the dominator tree, a postdominator tree can be constructed. This tree structure mirrors the dominator tree but operates in the context of postdominance, with the exit block serving as the root.

This revised passage employs a more formal and encyclopedic tone, avoiding colloquialisms and focusing on clear definitions and explanations. It emphasizes the significance of domination relationships in CFG analysis and presents the dominator tree as a valuable tool for visualizing these relationships.

Edges in Control Flow Graphs (CFGs)

In the domain of compiler optimization and program analysis, control flow graphs (CFGs) represent the execution flow of a program. Edges within a CFG depict the potential transitions between basic blocks, the fundamental building blocks of the graph. Here, we explore various types of edges encountered in CFGs:

Back Edges

A back edge signifies an edge that directs execution flow from a block to a preceding block that has already been encountered during a depth-first search (DFS) traversal of the CFG. These edges are a characteristic feature of loops within the program. When processing a loop, the execution path cycles back to a previously visited block, creating a back edge.

Critical Edges

A critical edge is defined as an edge that does not fall into two specific categories:

The only edge leaving its source block (i.e., the block from which the edge originates).

The only edge entering its destination block (i.e., the block where the edge leads). In simpler terms, a critical edge is not the sole path out of its source block nor the sole path into its destination block. These edges pose a challenge for certain optimizations. To facilitate the insertion of computations along a critical edge without impacting other execution paths, the edge needs to be split. This splitting process involves creating a new basic block in the middle of the critical edge, enabling the introduction of desired computations at that point.

Abnormal Edges

An abnormal edge deviates from the typical structure of CFG edges. It refers to an edge whose destination block remains unknown during the construction of the CFG. This situation often arises due to exception handling constructs within the program. The presence of abnormal edges can hinder optimization efforts, as the unknown destination makes it difficult to analyze potential execution paths.

Impossible Edges (Fake Edges)

An impossible edge (also known as a fake edge) is a special type of edge introduced solely to uphold a specific property within the CFG. This property dictates that the exit block (the block signifying program termination) must postdominate all other blocks in the graph. Postdominance implies that every conceivable execution path leading to the exit block must necessarily pass through the postdominator block. In certain scenarios, ensuring this property might necessitate the addition of an impossible edge. However, it's crucial to understand that this edge cannot be traversed during actual program execution. It serves a purely structural purpose within the CFG to maintain the postdomination relationship.

This revised passage utilizes a more encyclopedic tone, providing clear definitions and explanations for various edge types in CFGs. It emphasizes the role these edges play in program analysis and optimization.

Loop management

Within the context of control flow graphs (CFGs), a crucial concept emerges: the loop header. It plays a pivotal role in identifying and analyzing loops within a program.

Definition and Properties

A loop header, sometimes referred to as the loop's entry point, is a block that satisfies two key criteria: It acts as a dominator. This signifies that any execution path originating from the program's entry block and reaching a block within the loop body must necessarily traverse through the loop header. It serves as the target of a loop-forming back edge. A back edge, as the name suggests, directs execution flow from a block back to a previously encountered block. In the context of loops, this back edge points to the loop header, creating a cyclical execution pattern.

Domination and Loop Body

A significant property of a loop header is that it dominates all blocks constituting the loop body. This implies that every conceivable execution path within the loop must pass through the loop header. This dominance relationship is essential for understanding the control flow within the loop.

Multiple Loop Headers and Entry Points

A single block can function as the loop header for multiple loops. This scenario arises when multiple loops share a common entry point. Conversely, a loop can possess multiple entry points. In such cases, the concept of a loop header becomes inapplicable. There's no single dominant block that all execution paths must traverse to enter the loop.

Loop Header Transformation (Optimization)

In specific optimization techniques, it can be advantageous to split a loop header block (denoted as M) into two distinct blocks:

  • Mpre (Loop Pre-header): Initially empty, this block can be populated with loop-invariant code (code that remains constant throughout loop iterations). Techniques like loop-invariant code motion can identify and place such code in the pre-header.
  • Mloop (Loop Header): This block inherits the contents of the original loop header M and all loop-forming back edges. Edges not related to the loop are redirected to point to Mpre. Additionally, a new edge is inserted from Mpre to Mloop, establishing Mpre as the immediate dominator of Mloop.

This transformation aims to improve the efficiency of certain optimizations by separating loop-invariant code from the core loop logic within Mloop.

This revised passage adopts a more encyclopedic tone, focusing on clear definitions and properties of loop headers in CFGs. It explains their role in loop identification and the potential optimization technique of loop header splitting.

Reducible and Irreducible Control Flow Graphs (CFGs)

In the realm of compiler design and optimization, control flow graphs (CFGs) represent a program's execution flow. These graphs depict the potential transitions between basic blocks, the fundamental building blocks of the program. Here, we delve into the concepts of reducible and irreducible CFGs:

Reducible CFGs

A reducible CFG possesses a specific characteristic regarding its edges. These edges can be systematically partitioned into two distinct, non-overlapping sets:

Forward Edges: These edges depict a sequential flow of execution. They form a directed acyclic graph (DAG), meaning there are no cycles within this set of edges. Additionally, all nodes within the CFG must be reachable from the entry node (the starting point of program execution) by traversing these forward edges.

Back Edges: Back edges represent loops within the program. They point execution flow from a block back to a previously encountered block, creating a cyclical pattern.

Properties of Reducible CFGs

A crucial property of reducible CFGs is the domination relationship between nodes connected by back edges. For any back edge from node A to node B, node B must dominate node A. Domination signifies that every conceivable execution path originating from the entry block and reaching node A must necessarily traverse through node B. This property facilitates analysis and optimization techniques that rely on dominance relationships.

Structured Programming and Reducible CFGs

Structured programming languages, with their emphasis on well-defined control flow constructs (like if, for, while, break, and continue), are often designed to produce reducible CFGs. These constructs inherently lead to control flow patterns that adhere to the properties of reducible CFGs.

Irreducible CFGs

In contrast to reducible CFGs, irreducible CFGs violate the strict partitioning of edges. These graphs may contain edges that cannot be neatly classified as either forward edges or back edges, making them unsuitable for certain optimization techniques.

Causes of Irreducible CFGs

The use of the goto statement, a relic of unstructured programming, can introduce irreducible control flow. Goto statements allow for non-sequential jumps within the program, potentially leading to a CFG structure that defies the properties of reducible graphs.

Certain compiler optimizations, in their pursuit of efficiency, might inadvertently transform a reducible CFG into an irreducible one. This can occur when optimizations introduce control flow patterns that deviate from the characteristics of reducible CFGs.

This explanation utilizes a more encyclopedic tone, providing clear definitions and properties of reducible and irreducible CFGs. It highlights the connection between structured programming and reducible CFGs, and explores the factors that can lead to irreducible CFGs.

Loop Connectedness in Control Flow Graphs (CFGs)

Within the domain of compiler optimization, loop connectedness emerges as a concept that aids in analyzing the complexity of data-flow analysis on control flow graphs (CFGs).

Depth-First Search Tree (DFST)

The concept of loop connectedness hinges on a specific type of tree structure: a depth-first search tree (DFST). This tree is constructed by applying a depth-first search algorithm on the CFG. Here are some key points regarding the DFST used in loop connectedness:

  • Root and Coverage: The DFST must be rooted at the start node of the CFG, signifying the program's entry point.
  • Completeness: The DFST must encompass every single node within the CFG. No node should remain unexplored during the depth-first search process.

Back Edges

Within the CFG, edges that originate from a node and point towards one of its DFST ancestors (including the node itself) are classified as back edges. These edges essentially represent loops within the program's control flow. The cyclical nature of loops manifests as edges that traverse back up the DFST, connecting a node to an earlier point in the search process.

Loop Connectedness Definition

The loop connectedness of a CFG is a numerical value calculated based on the back edges present within the graph. It represents the highest number of back edges encountered along any cycle-free path in the CFG. Intuitively, loop connectedness provides a measure of the loop nesting depth within the program. A higher loop connectedness suggests the presence of more deeply nested loops.

Reducible CFGs and Consistency

An important property of loop connectedness is its independence of the chosen DFST in reducible CFGs. Reducible CFGs, as explained earlier, possess a specific structure where edges can be categorized as forward edges or back edges. This well-defined structure ensures that the loop connectedness value remains consistent regardless of the specific DFST used for calculation.

Applications: Data-Flow Analysis

The concept of loop connectedness plays a role in reasoning about the time complexity of data-flow analysis on CFGs. Data-flow analysis is a technique employed by compilers to analyze the flow of data values within a program. Loop connectedness provides insights into the potential complexity of performing data-flow analysis, particularly in the presence of nested loops.

This revised passage adopts a more encyclopedic tone, clearly defining loop connectedness and its relationship with DFSTs and back edges. It emphasizes the significance of loop connectedness in reducible CFGs and its connection to data-flow analysis complexity.

Inter-Procedural Control Flow Graphs (ICFGs)

Beyond the analysis of individual procedures, compilers often require a broader view of a program's execution flow. This is where inter-procedural control flow graphs (ICFGs) come into play.

Understanding Control Flow Graphs (CFGs)

As a prerequisite, it's helpful to understand the concept of a control flow graph (CFG). A CFG depicts the execution flow within a single procedure or function. It represents basic blocks (sequences of instructions without jumps or branches) as nodes and the potential transitions between them as edges.

Extending Beyond Procedures: ICFG Construction

An ICFG builds upon the foundation of CFGs. It aims to capture the control flow across the entire program, encompassing not just individual procedures but also the interactions between them. Here's a simplified explanation of ICFG construction:

  • Start by creating a CFG for each procedure within the program.
  • Identify call sites (points where one procedure invokes another).
  • From each call site, add an edge to the entry node of the called procedure's CFG.
  • Every return statement within a procedure's CFG has a corresponding edge pointing back to the instruction following the call site that invoked the procedure.

ICFG for Program Analysis

By incorporating these inter-procedural edges, the ICFG provides a comprehensive view of the program's execution flow, considering not only the internal logic of each procedure but also the potential call sequences between them. This broader view is instrumental in various program analysis techniques:

  • Inter-procedural data-flow analysis: This analysis tracks the flow of data values across procedure boundaries, which is crucial for tasks like optimizing variable usage and dead code elimination.
  • Alias analysis: This analysis determines when different memory locations might refer to the same data, enabling optimizations like memory disambiguation.
  • Static program slicing: Identifying the parts of a program that can potentially affect a specific variable or statement benefits debugging and verification efforts.

Challenges and Limitations

While powerful, ICFG construction can be computationally expensive, especially for large programs with numerous procedures and complex call hierarchies. Additionally, certain programming features like recursion can introduce challenges in ICFG construction, as the number of potential call paths can become unbounded. In essence, ICFGs serve as a bridge between the control flow within individual procedures and the overall execution flow of a program. They offer valuable insights for various program analysis techniques.

See also

Data-flow graph