Symmetric categorial grammar, or Lambek-Grishin calculus (LG), arose out of the generalization of Lambek's syntactic calculus proposed by V.N. Grishin in 1983. LG and ACG share the same compositional architecture. The differences are:

• ACG source and target logics are intuitionistic systems: judgements are of the form X ⊢ A, with A a single conclusion formula. The abstract syntax of LG is a bilinear system with judgements X ⊢ Y allowing multiple conclusions. In LG, one considers a composition operation ('Merge', fusion) and its dual (fission); these two communicate via structure-preserving distributivity principles.
• Whereas ACG specifies linear order and phrase structure on a word-by-word basis in the lexicon, LG encodes these aspects of grammatical structure in its logical constants: the LG source logic is non-commutative and respects constituent structure.
• Compositional interpretations for LG are constrained to pass through a continuation-passing-style translation. The image of the CPS translation is a linear lambda term, in Curry-Howard correspondence with a MILL proof, as in ACG.

The bilinear perspective of LG leads to a re-assessment of mild context-sensitivity: semantic expressivity results from the CPS translation that makes the context an explicit component of the computation; syntactic expressivity is obtained from the distributivity laws that channel the flow of information between the fusion/fission dimensions of grammatical structure.

Key notions to be covered:

• Symmetries: fusion, fission; linear distributivity.
• Bringing in the context: normal derivations, continuation-passing-style translation.
• Graphical calculus: LG proof nets and normal derivations.
• Modeling mild context-sensitivity: LG and the k-MCFG hierarchy.

Tuur and René on Moot's encoding of TAG in LG, and on his analysis of LG in terms of Hyperedge Replacement Grammar; Julia and Vlasta on the category theoretic background of categorial grammar.