LNL polycategories and doctrines of linear logic

Shulman, Michael

doi:10.46298/lmcs-19(2:1)2023

Cited by 2 publications

(1 citation statement)

References 36 publications

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“…The notation is motivated by the fact that if the functor q : E → B has sufficient fibrational structure, then the minimal lift may be decomposed as a fiberwise coproduct of pushforwards (cf. [MZ13, p.13], [Shu,Thm. 4.28]).…”

Section: Addendum a Gcfls As Initial Models Of Gcfgsmentioning

confidence: 99%

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

Melliès¹,

Zeilberger²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We begin by explaining how any context-free grammar encodes a functor of operads from a freely generated operad into a certain "operad of spliced words". This motivates a more general notion of CFG over any category $C$, defined as a finite species $S$ equipped with a color denoting the start symbol and a functor of operads $p : Free[S] \to W[C]$ into the operad of spliced arrows in $C$. We show that many standard properties of CFGs can be formulated within this framework, and that usual closure properties of CF languages generalize to CF languages of arrows. We also discuss a dual fibrational perspective on the functor $p$ via the notion of "displayed" operad, corresponding to a lax functor of operads $W[C] \to Span(Set)$. We then turn to the Chomsky-Sch\"utzenberger Representation Theorem. We describe how a non-deterministic finite state automaton can be seen as a category $Q$ equipped with a pair of objects denoting initial and accepting states and a functor of categories $Q \to C$ satisfying the unique lifting of factorizations property and the finite fiber property. Then, we explain how to extend this notion of automaton to functors of operads, which generalize tree automata, allowing us to lift an automaton over a category to an automaton over its operad of spliced arrows. We show that every CFG over a category can be pulled back along a ND finite state automaton over the same category, and hence that CF languages are closed under intersection with regular languages. The last important ingredient is the identification of a left adjoint $C[-] : Operad \to Cat$ to the operad of spliced arrows functor, building the "contour category" of an operad. Using this, we generalize the C-S representation theorem, proving that any context-free language of arrows over a category $C$ is the functorial image of the intersection of a $C$-chromatic tree contour language and a regular language.

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Section: Addendum a Gcfls As Initial Models Of Gcfgsmentioning

confidence: 99%

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

Melliès¹,

Zeilberger²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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Clones, closed categories, and combinatory logic

Saville

2024

Lecture Notes in Computer Science

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We explain how to recast the semantics of the simply-typed $$\uplambda $$ λ -calculus, and its linear and ordered variants, using multi-ary structures. We define universal properties for multicategories, and use these to derive familiar rules for products, tensors, and exponentials. Finally we outline how to recover both the category-theoretic syntactic model and its semantic interpretation from the multi-ary framework. We then use these ideas to study the semantic interpretation of combinatory logic and the simply-typed $$\uplambda $$ λ -calculus without products. We introduce extensional SK-clones and show these are sound and complete for both combinatory logic with extensional weak equality and the simply-typed $$\uplambda $$ λ -calculus without products. We then show such SK-clones are equivalent to a variant of closed categories called SK-categories, so the simply-typed $$\uplambda $$ λ -calculus without products is the internal language of SK-categories.

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LNL polycategories and doctrines of linear logic

Cited by 2 publications

References 36 publications

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

Parsing as a lifting problem and the Chomsky-Sch\"utzenberger representation theorem

Clones, closed categories, and combinatory logic

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