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Partial Orders, Residuation, and First-Order Linear Logic

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Natural Language Processing in Artificial Intelligence—NLPinAI 2020

Part of the book series: Studies in Computational Intelligence ((SCI,volume 939))

Abstract

We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent allows us to define many useful logical operators. In addition, the partial order constraints improve the efficiency of proof search.

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Notes

  1. 1.

    To show this in full detail would require us to do the simple but tedious job of proving that this definition satisfies the monotonicity and Application/Co-Application principles of Table 2.

  2. 2.

    A more elegant solution for ensuring confluence would replace the right-hand side of the \( p \) and \( u \) contractions by the left-hand side of the \( c \) contraction.

  3. 3.

    In the literature on finite state automata it is common to refer to sequences of symbols produced by such an automaton as “words”. However, we reserve “words” to refer to elements in the lexicon of a type-logical grammar and exclusively use “string” for a sequence of symbols produced by a finite state automaton.

  4. 4.

    A closed form solution for this recurrence is the following [26].

    $$ p[n] = \left\lceil \frac{2(2^n -1)}{3} \right\rceil $$

    .

  5. 5.

    This analysis also makes an unexpected empirical claim: the treatment of parasitic gapping in type-logical grammars using the linear logic exponential ‘!’ would require the exponential to have scope over the quantified variable representing the empty string. We therefore need to claim that parasitic gapping can only happen with atomic formulas.

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Authors and Affiliations

  1. LIRMM, CNRS, Université de Montpellier, 860 rue Saint Priest, 34095, Montpellier, France

    Richard Moot

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Correspondence to Richard Moot .

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  1. Department of Algebra and Logic, Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Sofia, Bulgaria

    Roussanka Loukanova

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Moot, R. (2021). Partial Orders, Residuation, and First-Order Linear Logic. In: Loukanova, R. (eds) Natural Language Processing in Artificial Intelligence—NLPinAI 2020. Studies in Computational Intelligence, vol 939. Springer, Cham. https://doi.org/10.1007/978-3-030-63787-3_2

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