Showing papers on "Linear logic published in 2023"

An algebraic investigation of Linear Logic

Abstract: In this paper we investigate two logics from an algebraic point of view. The two logics are: MALL (multiplicative-additive Linear Logic) and LL (classical Linear Logic). Both logics turn out to be strongly algebraizable in the sense of Blok and Pigozzi and their equivalent algebraic semantics are, respectively, the variety of Girard algebras and the variety of girales. We show that any variety of girales has equationally definable principale congruences and we classify all varieties of Girard algebras having this property. Also we investigate the structure of the algebras in question, thus obtaining a representation theorem for Girard algebras and girales. We also prove that congruence lattices of girales are really congruence lattices of Heyting algebras and we construct examples in order to show that the variety of girales contains infinitely many nonisomorphic finite simple algebras.

Graphical Proof Theory I: Multiplicative Linear Logic Beyond Cographs

Abstract: Cographs are a class of (undirected) graphs, characterized by the absence of induced subgraphs isomorphic to the four-vertices path, showing an intuitive one-to-one correspondence with classical propositional formulas. In this paper we study sequent calculi operating on graphs, as a generalization of sequent calculi operating on formulas, therefore on cographs. We mostly focus on sequent systems with multiplicative rules (in the sense of linear logic, that is, linear and context-free rules) extending multiplicative linear logic with connectives allowing us to represent modular decomposition of graphs by formulas, therefore obtaining a representation of a graph with linear size with respect to the number of its vertices. We show that these proof systems satisfy basic proof theoretical properties such as initial coherence, cut-elimination and analyticity of proof search. We prove that the system conservatively extend multiplicative linear logic with and without mix, and that the system extending the former derives the same graphs which are derivable in the deep inference system GS from the literature. We provide a syntax for proof nets for our systems by extending the syntax of Retor\'e's RB-structures to represent graphical connectives. A topological characterization of those structures encoding correct proofs is given, as well as a sequentialization procedure to construct a derivation from a correct structure. We conclude the paper by discussing how to extend those linear systems with the structural rules of weakening and contraction, providing a sequent system for an extension of classical propositional logic beyond cographs.

Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic

Abstract: This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist's B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established.

Explorations in Subexponential non-associative non-commutative Linear Logic

Abstract: In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, considering a classical one-sided multi-succedent classical version of the system, following the exponential-free calculi of Buszkowski's and de Groote and Lamarche's works, where the intuitionistic calculus is shown to embed faithfully into the classical fragment.

Coherent differentiation

Abstract: Abstract The categorical models of differential linear logic (LL) are additive categories and those of the differential lambda-calculus are left-additive categories because of the Leibniz rule which requires the summation of two expressions. This means that, as far as the differential lambda-calculus and differential LL are concerned, these models feature finite nondeterminism and indeed these languages are essentially non-deterministic. We introduce a categorical framework for differentiation which does not require additivity and is compatible with deterministic models such as coherence spaces and probabilistic models such as probabilistic coherence spaces.

Planning for Temporally Extended Goals in Pure-Past Linear Temporal Logic

Abstract: We study classical planning for temporally extended goals expressed in Pure-Past Linear Temporal Logic (PPLTL). PPLTL is as expressive as Linear-time Temporal Logic on finite traces (LTLf), but as shown in this paper, it is computationally much better behaved for planning. Specifically, we show that planning for PPLTL goals can be encoded into classical planning with minimal overhead, introducing only a number of new fluents that is at most linear in the PPLTL goal and no spurious additional actions. Based on these results, we implemented a system called Plan4Past, which can be used along with state-of-the-art classical planners, such as LAMA. An empirical analysis demonstrates the practical effectiveness of Plan4Past, showing that a classical planner generally performs better with our compilation than with other existing compilations for LTLf goals over the considered benchmarks.

Controller Synthesis for Unknown-Mode Linear Systems with an Epistemic variant of LTL

Abstract: Linear temporal logic (LTL) with the knowledge operator, denoted as KLTL, is a variant of LTL that incorporates what an agent knows or learns at run-time into its specification. Therefore it is an appropriate logical formalism to specify tasks for systems with unknown components that are learned or estimated at run-time. In this paper, we consider a linear system whose system matrices are unknown but come from an a priori known finite set. We introduce a form of KLTL that can be interpreted over the trajectories of such systems. Finally, we show how controllers that guarantee satisfaction of specifications given in fragments of this form of KLTL can be synthesized using optimization techniques. Our results are demonstrated in simulation and on hardware in a drone scenario where the task of the drone is conditioned on its health status, which is unknown a priori and discovered at run-time.

Dagger linear logic and categorical quantum mechanics

Abstract: This thesis develops the categorical proof theory for the non-compact multiplicative dagger linear logic, and investigates its applications to Categorical Quantum Mechanics (CQM). The existing frameworks of CQM are categorical proof theories of compact dagger linear logic, and are motivated by the interpretation of quantum systems in the category of finite dimensional Hilbert spaces. This thesis describes a new non-compact framework called Mixed Unitary Categories which can accommodate infinite dimensional systems, and develops models for the framework. To this end, it builds on linearly distributive categories, and $*$-autonomous categories which are categorical proof theories of (non-compact) multiplicative linear logic. The proof theory of non-compact dagger-linear logic is obtained from the basic setting of an LDC by adding a dagger functor satisfying appropriate coherences to give a dagger-LDC. From every (isomix) dagger-LDC one can extract a canonical "unitary core" which up to equivalence is the traditional CQM framework of dagger-monoidal categories. This leads to the framework of Mixed Unitary Categories (MUCs): every MUC contains a (compact) unitary core which is extended by a (non-compact) isomix dagger-LDC. Various models of MUCs based on Finiteness Spaces, Chu spaces, Hopf modules, etc., are developed in this thesis. This thesis also generalizes the key algebraic structures of CQM, such as observables, measurement, and complementarity, to MUC framework. Furthermore, using the MUC framework, this thesis establishes a connection between the complementary observables of quantum mechanics and the exponential modalities of linear logic.

Quantum Algorithm for Multiplicative Linear Logic

Abstract: This paper describes a quantum algorithm for proof search in sequent calculus of a subset of Linear Logic using the Grover Search Algorithm. We briefly overview the Grover Search Algorithm and Linear Logic, show the detailed steps of the algorithm and then present the results obtained on quantum simulators.

Linear Logic Properly Displayed

Abstract: We introduce proper display calculi for intuitionistic, bi-intuitionistic and classical linear logics with exponentials, which are sound, complete, conservative, and enjoy cut elimination and subformula property. Based on the same design, we introduce a variant of Lambek calculus with exponentials, aimed at capturing the controlled application of exchange and associativity. Properness (i.e., closure under uniform substitution of all parametric parts in rules) is the main technical novelty of the present proposal, allowing both for the smoothest proof of cut elimination and for the development of an overarching and modular treatment for a vast class of axiomatic extensions and expansions of intuitionistic, bi-intuitionistic, and classical linear logics with exponentials. Our proposal builds on an algebraic and order-theoretic analysis of linear logic and applies the guidelines of the multi-type methodology in the design of display calculi.

Graded Differential Categories and Graded Differential Linear Logic

Abstract: In Linear Logic ($\mathsf{LL}$), the exponential modality $!$ brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic ($\mathsf{DiLL}$) is an extension of Linear Logic which includes additional rules for $!$ which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of $\mathsf{GDiLL}$.

A Curry-Howard Correspondence for Linear, Reversible Computation

Abstract: In this paper, we present a linear and reversible programming language with inductives types and recursion. The semantics of the languages is based on pattern-matching; we show how ensuring syntactical exhaustivity and non-overlapping of clauses is enough to ensure reversibility. The language allows to represent any Primitive Recursive Function. We then give a Curry-Howard correspondence with the logic {\mu}MALL: linear logic extended with least fixed points allowing inductive statements. The critical part of our work is to show how primitive recursion yields circular proofs that satisfy {\mu}MALL validity criterion and how the language simulates the cut-elimination procedure of {\mu}MALL.

Complexity of the Lambek Calculus with One Division and a Negative-Polarity Modality for Weakening

Abstract: In this paper, we consider a variant of the Lambek calculus allowing empty antecedents. This variant uses two connectives: the left division and a unary modality that occurs only with negative polarity and allows weakening in antecedents of sequents. We define the notion of a proof net for this calculus, which is similar to those for the ordinary Lambek calculus and multiplicative linear logic. We prove that a sequent is derivable in the calculus under consideration if and only if there exists a proof net for it. We present a polynomial-time algorithm for deciding whether an arbitrary given sequent is derivable in this calculus.