Values

Value Typing

For the purpose of checking argument values against the parameter types of exported functions, values are classified by value types. The following auxiliary typing rules specify this typing relation relative to a store \(S\) in which possibly referenced addresses live.

Numeric Values \(t.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~c\)

\[\frac{ }{ S \href{../exec/values.html#valid-val}{\vdash} t.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~c : t }\]

Vector Values \(t.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~c\)

\[\frac{ }{ S \href{../exec/values.html#valid-val}{\vdash} t.\href{../syntax/instructions.html#syntax-instr-numeric}{\mathsf{const}}~c : t }\]

Null References \(\href{../syntax/instructions.html#syntax-instr-ref}{\mathsf{ref.null}}~t\)

  • The heap type must be valid under the empty context.

  • Then value is valid with reference type \((\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~t)\).

\[\frac{ \href{../valid/types.html#valid-heaptype}{\vdash} t \mathrel{\mbox{ok}} }{ S \href{../exec/values.html#valid-val}{\vdash} \href{../syntax/instructions.html#syntax-instr-ref}{\mathsf{ref.null}}~t : (\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-reftype}{\mathsf{null}}~t) }\]

Function References \(\href{../exec/runtime.html#syntax-ref}{\mathsf{ref{.}func}}~a\)

  • The external value \(\href{../exec/runtime.html#syntax-externval}{\mathsf{func}}~a\) must be valid with external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}}\).

  • Then the value is valid with reference type \((\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}})\).

\[\frac{ S \href{../exec/values.html#valid-externval}{\vdash} \href{../exec/runtime.html#syntax-externval}{\mathsf{func}}~a : \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}} }{ S \href{../exec/values.html#valid-val}{\vdash} \href{../exec/runtime.html#syntax-ref}{\mathsf{ref{.}func}}~a : \href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-functype}{\mathit{functype}} }\]

External References \(\href{../exec/runtime.html#syntax-ref.extern}{\mathsf{ref{.}extern}}~a\)

  • The value is valid with reference type \((\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}})\).

\[\frac{ }{ S \href{../exec/values.html#valid-val}{\vdash} \href{../exec/runtime.html#syntax-ref.extern}{\mathsf{ref{.}extern}}~a : (\href{../syntax/types.html#syntax-reftype}{\mathsf{ref}}~\href{../syntax/types.html#syntax-heaptype}{\mathsf{extern}}) }\]

Subsumption

  • The value must be valid with some value type \(t\).

  • The value type \(t\) matches another valid type \(t'\).

  • Then the value is valid with type \(t'\).

\[\frac{ S \href{../exec/values.html#valid-val}{\vdash} \href{../exec/runtime.html#syntax-val}{\mathit{val}} : t \qquad \href{../valid/types.html#valid-valtype}{\vdash} t' \mathrel{\mbox{ok}} \qquad \href{../valid/matching.html#match-valtype}{\vdash} t \href{../valid/matching.html#match-valtype}{\leq} t' }{ S \href{../exec/values.html#valid-val}{\vdash} \href{../exec/runtime.html#syntax-val}{\mathit{val}} : t' }\]

External Typing

For the purpose of checking external values against imports, such values are classified by external types. The following auxiliary typing rules specify this typing relation relative to a store \(S\) in which the referenced instances live.

\(\href{../exec/runtime.html#syntax-externval}{\mathsf{func}}~a\)

  • The store entry \(S.\href{../exec/runtime.html#syntax-store}{\mathsf{funcs}}[a]\) must exist.

  • Then \(\href{../exec/runtime.html#syntax-externval}{\mathsf{func}}~a\) is valid with external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{funcs}}[a].\href{../exec/runtime.html#syntax-funcinst}{\mathsf{type}}\).

\[\frac{ }{ S \href{../exec/values.html#valid-externval}{\vdash} \href{../exec/runtime.html#syntax-externval}{\mathsf{func}}~a : \href{../syntax/types.html#syntax-externtype}{\mathsf{func}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{funcs}}[a].\href{../exec/runtime.html#syntax-funcinst}{\mathsf{type}} }\]

\(\href{../exec/runtime.html#syntax-externval}{\mathsf{table}}~a\)

  • The store entry \(S.\href{../exec/runtime.html#syntax-store}{\mathsf{tables}}[a]\) must exist.

  • Then \(\href{../exec/runtime.html#syntax-externval}{\mathsf{table}}~a\) is valid with external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{tables}}[a].\href{../exec/runtime.html#syntax-tableinst}{\mathsf{type}}\).

\[\frac{ }{ S \href{../exec/values.html#valid-externval}{\vdash} \href{../exec/runtime.html#syntax-externval}{\mathsf{table}}~a : \href{../syntax/types.html#syntax-externtype}{\mathsf{table}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{tables}}[a].\href{../exec/runtime.html#syntax-tableinst}{\mathsf{type}} }\]

\(\href{../exec/runtime.html#syntax-externval}{\mathsf{mem}}~a\)

  • The store entry \(S.\href{../exec/runtime.html#syntax-store}{\mathsf{mems}}[a]\) must exist.

  • Then \(\href{../exec/runtime.html#syntax-externval}{\mathsf{mem}}~a\) is valid with external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{mems}}[a].\href{../exec/runtime.html#syntax-meminst}{\mathsf{type}}\).

\[\frac{ }{ S \href{../exec/values.html#valid-externval}{\vdash} \href{../exec/runtime.html#syntax-externval}{\mathsf{mem}}~a : \href{../syntax/types.html#syntax-externtype}{\mathsf{mem}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{mems}}[a].\href{../exec/runtime.html#syntax-meminst}{\mathsf{type}} }\]

\(\href{../exec/runtime.html#syntax-externval}{\mathsf{global}}~a\)

  • The store entry \(S.\href{../exec/runtime.html#syntax-store}{\mathsf{globals}}[a]\) must exist.

  • Then \(\href{../exec/runtime.html#syntax-externval}{\mathsf{global}}~a\) is valid with external type \(\href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{globals}}[a].\href{../exec/runtime.html#syntax-globalinst}{\mathsf{type}}\).

\[\frac{ }{ S \href{../exec/values.html#valid-externval}{\vdash} \href{../exec/runtime.html#syntax-externval}{\mathsf{global}}~a : \href{../syntax/types.html#syntax-externtype}{\mathsf{global}}~S.\href{../exec/runtime.html#syntax-store}{\mathsf{globals}}[a].\href{../exec/runtime.html#syntax-globalinst}{\mathsf{type}} }\]