Conventions

WebAssembly code is executed when instantiating a module or invoking an exported function on the resulting module instance.

Execution behavior is defined in terms of an abstract machine that models the program state. It includes a stack, which records operand values and control constructs, and an abstract store containing global state.

For each instruction, there is a rule that specifies the effect of its execution on the program state. Furthermore, there are rules describing the instantiation of a module. As with validation, all rules are given in two equivalent forms:

  1. In prose, describing the execution in intuitive form.

  2. In formal notation, describing the rule in mathematical form. 1

Note

As with validation, the prose and formal rules are equivalent, so that understanding of the formal notation is not required to read this specification. The formalism offers a more concise description in notation that is used widely in programming languages semantics and is readily amenable to mathematical proof.

Prose Notation

Execution is specified by stylised, step-wise rules for each instruction of the abstract syntax. The following conventions are adopted in stating these rules.

  • The execution rules implicitly assume a given store \(S\).

  • The execution rules also assume the presence of an implicit stack that is modified by pushing or popping values, labels, and frames.

  • Certain rules require the stack to contain at least one frame. The most recent frame is referred to as the current frame.

  • Both the store and the current frame are mutated by replacing some of their components. Such replacement is assumed to apply globally.

  • The execution of an instruction may trap, in which case the entire computation is aborted and no further modifications to the store are performed by it. (Other computations can still be initiated afterwards.)

  • The execution of an instruction may also end in a jump to a designated target, which defines the next instruction to execute.

  • Execution can enter and exit instruction sequences that form blocks.

  • Instruction sequences are implicitly executed in order, unless a trap or jump occurs.

  • In various places the rules contain assertions expressing crucial invariants about the program state.

Formal Notation

Note

This section gives a brief explanation of the notation for specifying execution formally. For the interested reader, a more thorough introduction can be found in respective text books. 2

The formal execution rules use a standard approach for specifying operational semantics, rendering them into reduction rules. Every rule has the following general form:

\[\mathit{configuration} \quad\href{../exec/conventions.html#formal-notation}{\hookrightarrow}\quad \mathit{configuration}\]

A configuration is a syntactic description of a program state. Each rule specifies one step of execution. As long as there is at most one reduction rule applicable to a given configuration, reduction – and thereby execution – is deterministic. WebAssembly has only very few exceptions to this, which are noted explicitly in this specification.

For WebAssembly, a configuration typically is a tuple \((S; F; \href{../syntax/instructions.html#syntax-instr}{\mathit{instr}}^\ast)\) consisting of the current store \(S\), the call frame \(F\) of the current function, and the sequence of instructions that is to be executed. (A more precise definition is given later.)

To avoid unnecessary clutter, the store \(S\) and the frame \(F\) are omitted from reduction rules that do not touch them.

There is no separate representation of the stack. Instead, it is conveniently represented as part of the configuration’s instruction sequence. In particular, values are defined to coincide with \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}\) instructions, and a sequence of \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}\) instructions can be interpreted as an operand “stack” that grows to the right.

Note

For example, the reduction rule for the \(\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}}\) instruction can be given as follows:

\[(\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~n_1)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~n_2)~\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}} \quad\href{../exec/conventions.html#formal-notation}{\hookrightarrow}\quad (\href{../syntax/types.html#syntax-valtype}{\mathsf{i32}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~(n_1 + n_2) \mathbin{\mathrm{mod}} 2^{32})\]

Per this rule, two \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}\) instructions and the \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}}\) instruction itself are removed from the instruction stream and replaced with one new \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}\) instruction. This can be interpreted as popping two values off the stack and pushing the result.

When no result is produced, an instruction reduces to the empty sequence:

\[\href{../syntax/instructions.html#syntax-instr-control}{\mathsf{nop}} \quad\href{../exec/conventions.html#formal-notation}{\hookrightarrow}\quad \epsilon\]

Labels and frames are similarly defined to be part of an instruction sequence.

The order of reduction is determined by the definition of an appropriate evaluation context.

Reduction terminates when no more reduction rules are applicable. Soundness of the WebAssembly type system guarantees that this is only the case when the original instruction sequence has either been reduced to a sequence of \(\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}\) instructions, which can be interpreted as the values of the resulting operand stack, or if a trap occurred.

Note

For example, the following instruction sequence,

\[(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_1)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_2)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{neg}}~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_3)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}}~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{mul}}\]

terminates after three steps:

\[\begin{split}\begin{array}{ll} & (\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_1)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_2)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{neg}}~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_3)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}}~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{mul}} \\ \href{../exec/conventions.html#formal-notation}{\hookrightarrow} & (\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_1)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_4)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_3)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{add}}~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{mul}} \\ \href{../exec/conventions.html#formal-notation}{\hookrightarrow} & (\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_1)~(\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_5)~\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{mul}} \\ \href{../exec/conventions.html#formal-notation}{\hookrightarrow} & (\href{../syntax/types.html#syntax-valtype}{\mathsf{f64}}.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~x_6) \\ \end{array}\end{split}\]

where \(x_4 = -x_2\) and \(x_5 = -x_2 + x_3\) and \(x_6 = x_1 \cdot (-x_2 + x_3)\).

1

The semantics is derived from the following article: Andreas Haas, Andreas Rossberg, Derek Schuff, Ben Titzer, Dan Gohman, Luke Wagner, Alon Zakai, JF Bastien, Michael Holman. Bringing the Web up to Speed with WebAssembly. Proceedings of the 38th ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 2017). ACM 2017.

2

For example: Benjamin Pierce. Types and Programming Languages. The MIT Press 2002