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llbracket

Represents a specialized double square bracket used primarily in mathematical semantics and formal logic notation.

Overview

Serves as a semantic delimiter in advanced mathematics, particularly in denotational semantics, category theory, and formal logic.

  • Often paired with \rrbracket to enclose mathematical expressions or semantic interpretations
  • Common in computer science literature when describing program semantics
  • Frequently appears in theoretical mathematics to denote special mappings or interpretations
  • Distinguished from regular square brackets by its more formal and specialized meaning in mathematical contexts

Examples

Denoting a semantic interpretation function in formal semantics.

⟦ϕ⟧={x:x satisfies ϕ}\llbracket \phi \rrbracket = \{x : x \text{ satisfies } \phi\} 
\llbracket \phi \rrbracket = \{x : x \text{ satisfies } \phi\}

Representing an integer interval in number theory.

⟦1,n⟧={k∈N:1≤k≤n}\llbracket 1,n \rrbracket = \{k \in \mathbb{N} : 1 \leq k \leq n\} 
\llbracket 1,n \rrbracket = \{k \in \mathbb{N} : 1 \leq k \leq n\}

Defining a matrix with double bracket notation.

A=⟦aij⟧n×nA = \llbracket a_{ij} \rrbracket_{n \times n} 
A = \llbracket a_{ij} \rrbracket_{n \times n}