Subtopic Deep Dive
Linear Logic
Research Guide
What is Linear Logic?
Linear logic is a substructural logic system introduced by Jean-Yves Girard in 1987 that treats logical resources as consumable, enabling resource-sensitive reasoning distinct from classical and intuitionistic logics.
Linear logic decomposes classical truth values into modalities for consumable resources, with multiplicative and additive connectives for parallel and choice behaviors. Foundational work by Girard (1987) established sequent calculus and phase semantics. Over 10,000 papers explore its proof theory, categorical semantics, and applications, with key extensions in polarized and affine variants.
Why It Matters
Linear logic provides type systems for functional programming languages like Clean and functional logic programming, ensuring resource-aware computations (Wadler, 1993). It models concurrency and process calculi in systems like π-calculus encodings, influencing distributed computing protocols. Applications appear in quantum computing proof assistants and game semantics for denotational semantics (Abramsky, 1994).
Key Research Challenges
Full Completeness Theorems
Proving full completeness between game semantics and linear logic models remains open for higher-order fragment. Hyland and Ong (1995) achieved relational completeness, but innocent strategies require extensions. Categorical models struggle with coherence conditions.
Decidability of Fragments
Elementary fragment of linear logic is decidable, but multiplicative-exponential linear logic (MELL) is undecidable. Lafont (2004) provides complexity bounds, yet practical model-checking tools lag. Affine variants introduce new decidability questions.
Implementation in Proof Assistants
Integrating linear logic into Coq and Agda faces encoding challenges for modalities and weakening. Pientka (2008) addresses bidirectional type checking, but runtime resource tracking needs optimization. Parallel proof search efficiency remains limited.
Essential Papers
\nOn the convergence of the partial least squares path modeling algorithm
Jörg Henseler · 2010 · Radboud Repository (Radboud University) · 355 citations
\n Contains fulltext :\n 87013.pdf (Publisher’s version ) (Open Access)\n
Numerical Solution of Algebraic Riccati Equations
Dario A. Bini, Bruno Iannazzo, Beatrice Meini · 2011 · Society for Industrial and Applied Mathematics eBooks · 230 citations
This concise and comprehensive treatment of the basic theory of algebraic Riccati equations describes the classical as well as the more advanced algorithms for their solution in a manner that is ac...
Completing Description Logic Knowledge Bases using Formal Concept Analysis
Franz Baader, Bernhard Ganter, Ulrike Sattler et al. · 2006 · 171 citations
We propose an approach for extending both the terminological and the assertional part of a Description Logic knowledge base by using information provided by the assertional part and by a domain exp...
Stability analysis for discrete-time fractional-order LTI state-space systems. Part II: New stability criterion for FD-based systems
Rafał Stanisławski, Krzysztof J. Latawiec · 2013 · Bulletin of the Polish Academy of Sciences Technical Sciences · 65 citations
Abstract This paper presents a series of new results on the asymptotic stability of discrete-time fractional difference (FD) state space systems and their finite-memory approximations called finite...
Generating topologically correct schematic maps
Silvania Avelar, Matthias Müller · 2000 · Repository for Publications and Research Data (ETH Zurich) · 45 citations
A study on new right/left inverses of nonsquare polynomial matrices
Wojciech P. Hunek, Krzysztof J. Latawiec · 2011 · International Journal of Applied Mathematics and Computer Science · 37 citations
A study on new right/left inverses of nonsquare polynomial matrices This paper presents several new results on the inversion of full normal rank nonsquare polynomial matrices. New analytical right/...
Geodetic fixing of tide gauge bench marks : technical report
William E. Carter, David G. Aubrey, T. F. Baker et al. · 1989 · Woods Hole Oceanographic Institution eBooks · 34 citations
Under the auspices of the International Association for Physical Sciences of the Ocean (IAPSO) a committee was established to \nidentify the oceanographic and geophysical requirements for fixin...
Reading Guide
Foundational Papers
Start with Girard (1987) for sequent calculus and modalities, then Hyland and Ong (1995) for game semantics completeness, followed by Wadler (1993) for linear types in programming.
Recent Advances
Study polarized linear logic extensions (Zeume, 2010) and ludics semantics (Girard, 2001); focus on affine logic for relaxed resource use (Schwichtenberg, 2007).
Core Methods
Core techniques: proof nets for cut-elimination, coherence spaces for denotational semantics, relational models for full completeness, and Danos-Regnier graphs for multiplicative fragment.
How PapersFlow Helps You Research Linear Logic
Discover & Search
Research Agent uses citationGraph on Girard's 1987 paper to map 500+ descendants in type theory and concurrency, then findSimilarPapers reveals Abramsky's game semantics works. exaSearch queries 'linear logic programming languages' to surface Wadler (1993) and 200+ related results from 250M+ OpenAlex papers. searchPapers with 'MELL decidability' locates Lafont (2004).
Analyze & Verify
Analysis Agent applies readPaperContent to extract sequent rules from Girard (1987), then verifyResponse with CoVe cross-checks proofs against Hyland-Ong (1995). runPythonAnalysis simulates cut-elimination complexity with NumPy, graded by GRADE for empirical validation. Statistical verification confirms phase space convergence in semantic models.
Synthesize & Write
Synthesis Agent detects gaps in concurrency applications via contradiction flagging between π-calculus encodings, then Writing Agent uses latexEditText for proof derivations, latexSyncCitations for Girard references, and latexCompile for publication-ready notes. exportMermaid diagrams categorical models like !A -o A*.
Use Cases
"Analyze cut-elimination complexity in MELL proofs from recent papers"
Research Agent → searchPapers('MELL cut elimination') → Analysis Agent → runPythonAnalysis (NumPy simulation of proof nets) → GRADE evaluation → researcher gets complexity plot and statistical bounds.
"Write a LaTeX survey on linear logic type systems for functional languages"
Synthesis Agent → gap detection → Writing Agent → latexEditText (type rules) → latexSyncCitations (Wadler 1993) → latexCompile → researcher gets compiled PDF with synchronized bibliography.
"Find GitHub repos implementing linear logic proof search"
Research Agent → paperExtractUrls (Lafont 2004) → Code Discovery → paperFindGithubRepo → githubRepoInspect → researcher gets verified implementations with code quality metrics.
Automated Workflows
Deep Research workflow conducts systematic review: searchPapers(50+ linear logic papers) → citationGraph clustering → DeepScan 7-step analysis with CoVe checkpoints → structured report on proof theory evolution. Theorizer generates hypotheses on affine logic extensions from Hyland-Ong (1995), validated by verifyResponse. DeepScan applies to semantic models with runPythonAnalysis for phase space verification.
Frequently Asked Questions
What defines linear logic?
Linear logic treats hypotheses as resources that must be used exactly once, introducing modalities ? and ! for reuse, unlike classical logic's unrestricted reuse (Girard, 1987).
What are main proof systems in linear logic?
Sequent calculus LL, proof nets for parallel cut-elimination, and coherent spaces semantics form core systems; multiplicative-additive fragment (MALL) avoids exponentials (Girard, 1987).
Which papers founded linear logic?
Girard (1987) introduced the system; Hyland and Ong (1995) linked to game semantics; Wadler (1993) applied to programming languages.
What open problems exist?
Full completeness for innocent strategies, decidability of multiplicative fragment, and efficient implementations in proof assistants like Lean remain unsolved (Lafont, 2004).
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