Linear logic is a type of logic that was developed by [[Jean-Yves Girard]] in the 1980s. It is a refinement of classical logic that is particularly suited for reasoning about resources and computation. Here, the notion of resource is directly related to the notion of knowing too much or knowing too little, as stated in [[Plato's Problem]] and [[Orwell's Problem]]. In linear logic, propositions are treated as resources that can be consumed or used up. Unlike classical logic, which allows the duplication and deletion of propositions, linear logic introduces new connectives and rules that enforce the proper usage of resources. The connectives in linear logic include: 1. Tensor (∗): It represents the combination or fusion of two resources. 2. Par (⊕): It represents the choice or alternative between two resources. 3. With (&): It represents the conjunction or combination of two resources. 4. Implication (⊸): It represents the consumption or use of a resource. To ensure proper resource management, linear logic introduces rules such as weakening (allowing weakening of propositions), contraction (allowing duplication of propositions), and exchange (allowing reordering of propositions). # Linear Logic and Polynomial Functors Now, how does [[linear logic]] relate to [[polynomial functors]]? [[Polynomial functors]] are a concept from [[category theory]] that provides a framework for defining and studying recursive data types. They represent data structures built up from base types using operations such as product and sum. The connection between [[linear logic]] and [[polynomial functors]] arises from their shared focus on managing resources and computation. The connectives in linear logic correspond to operations on polynomial functors. For example: - Tensor (∗) corresponds to product: it combines two data structures into one. - Par (⊕) corresponds to sum: it represents a choice between different data structures. - With (&) corresponds to product: it combines multiple data structures into one. - Implication (⊸) corresponds to an exponential: it represents repeated application or recursion on a data structure. By understanding this connection, researchers have been able to apply techniques from linear logic to study recursion schemes, higher-order computation, and resource management in the context of polynomial functors. This has led to advancements in formal verification, programming language theory, and type systems. ## Logical and Polynomial Perspectives [[Linear logic]] and [[polynomial functors]] are two different concepts in mathematics, but there are some connections between them. 1. Categorical Perspective: Both linear logic and polynomial functors can be studied from a categorical perspective. Linear logic can be understood as a fragment of [[intuitionistic logic]] with a special focus on resource management, while polynomial functors are a concept in [[category theory]] that generalizes the notion of polynomial maps between sets. Category theory provides a common framework to study both linear logic and polynomial functors. 2. Type Theory: Linear logic has been used as the foundation for various type systems, such as linear types or session types, which provide a way to enforce resource usage in programming languages. Polynomial functors have been used in the development of higher-order type theories, such as System Fω or dependent type theories, which allow for the definition of complex types using recursion and iteration. 3. Substructural Logics: Linear logic is an example of a substructural logic, where the rules for logical connectives are restricted to reflect certain structural properties of resources. Similarly, polynomial functors can be seen as capturing some structural properties of functions or mappings between sets. Both linear logic and polynomial functors provide a way to reason about and manipulate structures with restricted or controlled behavior. 1. Computational Complexity: Linear logic has been studied in the context of computational complexity theory, particularly in relation to proof complexity and program analysis. Polynomial functors have connections to computational complexity through their relationship with recursion schemes and their use in defining data structures with efficient algorithms for operations like map or fold. 2. The Lens of Resource Accounting: Linear Logic is all about resource [[accounting]]. In this speech:[[@WhatAreWeTracking2022|What are we tracking? How Applied Category Theory puts thinking on rails]], [[David Spivak]] repeated stated that [[mathematics]] in general, and [[category theory]]'s approach to problem is also related to accounting in computational resources. Overall, while linear logic and polynomial functors are distinct concepts in mathematics, they share common themes related to structure, computation, and resource management. The categorical perspective and the use of [[Type Theory]] provide connections between these two areas of study. # Linear Logic and Type Systems Linear logic and type systems are related in the sense that both are concerned with reasoning about resources and their usage. Linear logic is a logical system developed by [[Jean-Yves Girard]] in the 1980s, which places emphasis on resource management and consumption. It introduces a new way of reasoning about propositions and their proofs, by distinguishing between linear resources (which can be used exactly once) and unrestricted resources (which can be used multiple times). This allows for more precise modeling of systems where resources are limited or need to be carefully managed. Type systems, on the other hand, are a fundamental part of programming languages that help ensure type safety and correctness. They classify values into different types and enforce rules about how they can be used and combined. Type systems help catch errors at compile-time by checking for inconsistencies in the usage of values. # Conclusion [[Linear logic]] has been influential in the development of type systems for programming languages. One application is linear types, which extend traditional type systems with linear resources. Linear types impose restrictions on how values are used, tracking ownership or permission to access resources. This can help prevent certain kinds of bugs like use-after-free or resource leaks. In this way, linear logic provides a theoretical foundation for understanding and designing type systems that go beyond traditional notions of typing. By incorporating ideas from linear logic into type systems, programming languages can provide better guarantees about resource management and usage, leading to more reliable software. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Linear Logic") ```