An introduction to differential linear logic: proof-nets, models and antiderivatives
@article{Ehrhard2017AnIT, title={An introduction to differential linear logic: proof-nets, models and antiderivatives}, author={Thomas Ehrhard}, journal={Mathematical Structures in Computer Science}, year={2017}, volume={28}, pages={995 - 1060} }
Differential linear logic enriches linear logic with additional logical rules for the exponential connectives, dual to the usual rules of dereliction, weakening and contraction. We present a proof-net syntax for differential linear logic and a categorical axiomatization of its denotational models. We also introduce a simple categorical condition on these models under which a general antiderivative operation becomes available. Last, we briefly describe the model of sets and relations and give a…
71 Citations
Mackey-complete spaces and power series – a topological model of differential linear logic
- Mathematics, PhilosophyMathematical Structures in Computer Science
- 2016
A denotational model of Intuitionist Linear Logic which is also a differential category, with usual syntactic differentiation and where interpretations of proofs decompose as a Taylor expansion is described.
A Logical Account for Linear Partial Differential Equations
- Mathematics, PhilosophyLICS
- 2018
D-DiLL is introduced, a deterministic refinement of DiLL with a D-exponential, for which it exhibits a cut-elimination procedure, and a categorical semantics.
Turing Machines and Differential Linear Logic
- Computer Science, Mathematics
- 2017
It is shown that for proofs of the appropriate form, the derivatives obtained via the coalgebraic structure agree with those from elementary calculus, allowing one to write Taylor expansions of proofs.
Categorical models of Linear Logic with fixed points of formulas
- Mathematics2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021
A categorical semantics of µLL, a version of propositional Linear Logic with least and greatest fixed points extending David Baelde’s propositional µMALL with exponentials, shows that µLL enjoys a denotational form of normalization of proofs.
Encodings of Turing machines in linear logic
- Computer ScienceMathematical Structures in Computer Science
- 2020
This paper explicitly compute the denotations under this semantics of an interesting class of proofs in linear logic, introduced by Girard: the encodings of step functions of Turing machines.
Call-By-Push-Value from a Linear Logic Point of View
- Mathematics, Computer ScienceESOP
- 2016
A fully polarized version of this calculus which turns out to be a term language for a large fragment of LLP and refines lambda-mu is introduced and an adequacy theorem is proved.
Convenient antiderivatives for differential linear categories
- MathematicsMathematical Structures in Computer Science
- 2020
Abstract Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a…
Convenient antiderivatives for differential linear categories
- MathematicsMath. Struct. Comput. Sci.
- 2020
This paper shows that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives and shows that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential categories with antidervatives.
Coherent differentiation
- Computer Science, PhilosophyArXiv
- 2021
This work introduces a categorical framework for differentiation which does not require additivity and is compatible with deterministic models such as coherence spaces and probabilistic modelssuch as Probabilistic coherence Spaces.
A Profunctorial Scott Semantics
- Mathematics, Computer ScienceFSCD
- 2020
It is proved that morphisms in the co-Kleisli bicategory correspond to the concept of strongly finitary functors (sifted colimits preserving functors) between presheaf categories.
References
SHOWING 1-10 OF 56 REFERENCES
The Cut-Elimination Theorem for Differential Nets with Promotion
- MathematicsTLCA
- 2009
The authors prove the Cut-Elimination Theorem for the whole DiLL: every differential net that is sequentializable can be reduced to a cut-free net.
The conservation theorem for differential nets
- MathematicsMathematical Structures in Computer Science
- 2015
The conservation theorem for differential nets – the graph-theoretical syntax of the differential extension of Linear Logic (Ehrhard and Regnier's DiLL) is proved, which turns the quest for strong normalisation into one for non-erasing weak normalisation (WN), and indeed this result is used to prove SN of simply typed DiLL.
Realizability Proof for Normalization of Full Differential Linear Logic
- Mathematics, Computer ScienceTLCA
- 2011
This work provides an extension of this proof that embrace Full Differential Linear Logic (a logic that can describe both single-use resources and inexhaustible resources) and is modular enough so that further extensions (to second order, to additive constructs or to any other independent feature that can be dealt with using realizability) come for free.
On Köthe sequence spaces and linear logic
- MathematicsMathematical Structures in Computer Science
- 2002
This work provides a simple setting in which typed λ-calculus and differential calculus can be combined and gives a few examples of computations.
Finiteness spaces
- Mathematics, Computer ScienceMathematical Structures in Computer Science
- 2005
A new denotational model of linear logic based on the purely relational model, where webs are equipped with a notion of ‘finitary’ subsets satisfying a closure condition and proofs are interpreted as finitary sets is investigated.
Full Abstraction for Resource Calculus with Tests
- MathematicsCSL
- 2011
We study the semantics of a resource sensitive extension of the lambda-calculus in a canonical reflexive object of a category of sets and relations, a relational version of the original Scott D…
Confluence of Pure Differential Nets with Promotion
- Computer Science, MathematicsCSL
- 2009
It is proved that pure differential nets are Church-Rosser modulo such equivalences, which generalizes to linear logic regular proof nets, and uses a result of finiteness of developments, given by strong normalization when blocking a suitable notion of "new" cuts.
Linear Domains and Linear Maps
- MathematicsMFPS
- 1993
This work studies the symmetric monoidal closed category LIN of linear domains, which is a denotational model of linear logic, and its objects are inverse limits of finite, bounded complete posets with respect to projection-embedding pairs preserving all suprema.
Transport of finiteness structures and applications
- MathematicsMathematical Structures in Computer Science
- 2016
It is shown how to apply this construction to prove the existence of least fixpoints for particular functors in the category of finiteness spaces, which include the functors involved in a relational interpretation of lazy recursive algebraic datatypes along the lines of the coherence semantics of system T.