and shake up traditional logic by treating propositions as consumable resources. This fresh approach allows for more precise reasoning about resource usage, making it super useful in computer science and programming.
These logics introduce new connectives and rules that capture resource-sensitive properties. They've found applications in areas like memory management, quantum computing, and program verification, showing how proof theory can tackle real-world problems.
Linear Logic and Substructural Logics
Overview of Linear Logic
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Linear logic is a substructural logic developed by in 1987
Differs from classical logic by treating propositions as resources that are consumed when used
Provides a more fine-grained analysis of proofs and computation
Allows for reasoning about resource-sensitive properties (memory usage, time complexity)
Substructural Logics and Resource Sensitivity
Substructural logics are logical systems that lack or restrict some of the structural rules of classical logic (weakening, contraction, exchange)
is a key feature of substructural logics
Propositions are treated as resources that are consumed when used in a proof
Allows for more precise reasoning about resource usage and allocation
Resource interpretation of linear logic
Each proposition represents a single-use resource
Proofs correspond to processes that consume and produce resources
Connectives in Linear Logic
Multiplicative Connectives
(⊗) and disjunction (\parr)
A⊗B represents the simultaneous availability of resources A and B
A\parrB represents a choice between resources A and B
(⊸)
A⊸B represents a process that consumes A to produce B
: 1 (unit) and ⊥ (bottom)
Additive Connectives
(&) and disjunction (⊕)
A&B represents a choice between resources A and B, but only one is used
A⊕B represents the availability of either resource A or B
Additive constants: ⊤ (top) and 0 (zero)
Exponentials
Exponential modalities: ! (of course) and ? (why not)
!A allows for unlimited duplication and discarding of the resource A
?A allows for unlimited duplication of the resource A, but not discarding
reintroduce some of the structural rules in a controlled manner
Allows for embedding of classical logic within linear logic
Proof Systems for Linear Logic
Sequent Calculus for Linear Logic
is a proof system that operates on sequents of the form Γ⊢Δ
Γ and Δ are multisets of formulas
Intuitive reading: consuming the resources in Γ produces the resources in Δ
Inference rules for each connective and exponential
Rules for multiplicative and additive connectives reflect their resource interpretation
Rules for exponentials allow for controlled use of structural rules
Proof Nets
are a graphical representation of proofs in linear logic
Provide a more abstract and parallel view of proofs compared to sequent calculus
Correctness criteria for proof nets (acyclicity and connectedness)
Ensures that proof nets correspond to valid proofs in linear logic
Cut elimination in proof nets corresponds to the execution of proofs as processes
Other Substructural Logics
Relevance Logic
requires that the antecedent and consequent of an implication are relevant to each other
Rejects the principle of explosion (A∧¬A⊢B) and the principle of monotonicity (Γ⊢A implies Γ,B⊢A)
Applications in computer science (type systems, information retrieval)
Affine Logic
is a variant of linear logic that allows for discarding of resources, but not duplication
Obtained by adding the weakening rule to linear logic
Used in the study of quantum computation and quantum information theory
Bunched Implications
(BI) is a substructural logic that combines intuitionistic logic with a multiplicative conjunction
Provides a logical foundation for reasoning about resource composition and separation
Applications in program verification (separation logic) and computer security (access control)