Semantic reasoning about the sea of nodes

Abstract

The Sea of Nodes intermediate representation was introduced by Cliff Click in the mid 90s as an enhanced Static Single Assignment (SSA) form. It improves on the initial SSA form by relaxing the total order on instructions in basic blocks into explicit data and control dependencies. This makes programs more flexible to optimize. This graph-based representation is now used in many industrial-strength compilers, such as HotSpot or Graal. While the SSA form is now well understood from a semantic perspective-even formally verified optimizing compilers use it in their middle-end-very few semantic studies have been conducted about the Sea of Nodes. This paper presents a simple but rigorous formal semantics for a Sea of Nodes form. It comprises a denotational component to express data computation, and an operational component to express control flow. We then prove a fundamental, dominance-based semantic property on Sea of Nodes programs which determines the regions of the graph where the values of nodes are preserved. Finally, we apply our results to prove the semantic correctness of a redundant zero-check elimination optimization. All the necessary semantic properties have been mechanically verified in the Coq proof assistant.

Verified Lustre Normalization with Node Subsampling

“…For example, the Coq model used to encode an intermediate representation for LLVM [34] is based on successive updates to an environment mapping variables to values, with a control state that successively transitions between and within blocks of instruction sequences. Even in formalizations where the semantics within a block does not depend on instruction ordering [15], the overall model remains operational.…”

A Formal Semantics of the GraalVM Intermediate Representation

“…1 gives the Isabelle representation of the graph nodes. 4 ConstantNode corresponds to a Java constant, so has a value constant as its only field, with no input or successor edges. Similarly, ParameterNode has a single natural number field that is an index into the list of parameter values of the current method.…”

“…Our approach to handling φ nodes is similar to that used by Demange et al for their formalization of reasoning about the sea of nodes in Coq [4]. End nodes (5.3) represent the end of a basic block in SSA terminology.…”

“…The closest research to that presented here is the work of Demange et al [4] who provide the semantics of an abstract sea-of-nodes representation in Coq, which focuses on the semantics of φ nodes and regions. The semantics is used to prove a semantic property and a simple optimization transformation.…”

Visualizing JIT Compiler Graphs

“…Intermediate Representation (IR): a type of graph also known as sea-ofnodes [11,14,18]. Unlike other graphs used in program analysis, such as controlflow or data-flow graphs which have specific types of nodes, nodes in the seaof-nodes graph represent different types: from scalar values and arithmetic op- erators to variables and control-flow nodes and function entry nodes.…”