An Infinitary Model of Linear Logic

@inproceedings{Grellois2015AnIM,
  title={An Infinitary Model of Linear Logic},
  author={Charles Grellois and Paul-Andr{\'e} Melli{\`e}s},
  booktitle={FoSSaCS},
  year={2015}
}
In this paper, we construct an infinitary variant of the relational model of linear logic, where the exponential modality is interpreted as the set of finite or countable multisets. We explain how to interpret in this model the fixpoint operator Y as a Conway operator alternatively defined in an inductive or a coinductive way. We then extend the relational semantics with a notion of color or priority in the sense of parity games. This extension enables us to define a new fixpoint operator Y… 
Relational Semantics of Linear Logic and Higher-order Model Checking
TLDR
This article develops a new and somewhat unexpected connection between higher-order model-checking and linear logic and shows how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors in higher- order model- checking.
Finitary Semantics of Linear Logic and Higher-Order Model-Checking
In this paper, we explain how the connection between higher-order model-checking and linear logic recently exhibited by the authors leads to a new and conceptually enlightening proof of the selection
Higher-order parity automata
  • Paul-André Melliès
  • Computer Science
    2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2017
TLDR
The main result is that the acceptance of an infinitary λ-term by a higher-order parity automaton A is decidable, whenever the inf initary κ-term is generated by a finite and simply-typed λY-term.
A Model for Behavioural Properties of Higher-order Programs
TLDR
This work considers simply typed lambda-calculus with fixpoints as a non-interpreted functional programming language and shows how to construct a finitary model recognizing monadic second-order logic (MSOL) properties.
Colored intersection types: a bridge between higher-order model-checking and linear logic
The model-checking problem for higher-order recursive programs, expressed as higher-order recursion schemes (HORS), and where properties are specified in monadic second-order logic (MSO) has received
Modelling Coeffects in the Relational Semantics of Linear Logic
TLDR
All typing system introduced giving a parametric version of the exponential modality of linear logic can be interpreted in the relational category (Rel) of sets and relations, allowing a great variety of exponential comonads in Rel.
A cartesian-closed category for higher-order model checking
  • M. Hofmann, J. Ledent
  • Computer Science
    2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
  • 2017
TLDR
The construction of an abstract lattice from a given Büchi automaton is described and it is shown that there is a Galois insertion between it and the lattice of languages of finite and infinite words over a given alphabet.
Research Internship Proposal Higher-Order Model-Checking and Linear Logic
  • Computer Science
  • 2017
TLDR
It has been proved by Ong in 2006 that MSO formulas can be checked automatically over the infinite trees generated by HORS, and that the decision procedure is n-EXPTIME complete, where n is the order of the HORS representing the infinite tree.
A model for divergence insensitive properties of lambdaY-terms
TLDR
It is shown how to construct models recognizing properties expressed by parity automata that cannot detect divergence, which resemble standard Scott models of latices of monotone functions, but application needs to be modified and the the fixpoint operator should be interpreted as a particular non-extremal fixpoint in a lattice.
Every λ-Term is Meaningful for the Infinitary Relational Model
TLDR
It is proved that infinite types are able to characterize the arity of every λ-terms and that, in the infinitary extension of the relational model, every term has a "meaning" i.e. a non-empty denotation.
...
...

References

SHOWING 1-10 OF 29 REFERENCES
Exponentials with Infinite Multiplicities
TLDR
An exponential functor is defined on the category of sets and relations which allows to define a denotational model of differential linear logic and of the lambda-calculus with resources and it is shown that, when the semi-ring has an element which is infinite, this model does not validate the Taylor formula.
A Calculus of Circular Proofs and Its Categorical Semantics
TLDR
A calculus of "circular proofs": the graph underlying a proof is not a finite tree but instead it is allowed to contain a certain amount of cycles, and each system admits always a unique solution.
Least and Greatest Fixed Points in Linear Logic
TLDR
This work designs a focused proof system that proves complete with respect to the initial system, and establishes weak normalization for it, and shows how these foundations can be applied to intuitionistic logic.
Polarized Proof Nets with Cycles and Fixpoints Semantics
TLDR
A new polarized linear deduction system which handles recursion is defined by extending the cut-rule, in such a way that iteration unrolling is achieved by cut-elimination, and the proof nets counterpart is obtained by allowing oriented cycles, which had no meaning in usual polarized linear logic.
Evaluation is MSOL-compatible
TLDR
This work considers simply-typed lambda calculus with fixpoint operators and shows that evaluation is compatible with monadic second-order logic (MSOL), which means that for a fixed finite vocabulary of terms, the MSOL properties of Bohm trees of terms are effectively MS OL properties of terms themselves.
Using Models to Model-Check Recursive Schemes
TLDR
It is argued that having models capturing some class of properties has several other virtues in addition to providing decidability of the model-checking problem, and provides a construction of such a model for every property expressed by automata with trivial acceptance conditions and divergence testing.
What is a Categorical Model of Intuitionistic Linear Logic?
TLDR
This paper re-addresses the old problem of providing a categorical model for Intuitionistic Linear Logic (ILL) and finds that Seely's model is unsound in that it does not preserve equality of proofs.
Complete axioms for categorical fixed-point operators
  • A. Simpson, G. Plotkin
  • Mathematics
    Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332)
  • 2000
TLDR
A general completeness theorem for iteration operators is proved, relying on a new, purely syntactic characterisation of the free iteration theory, and it is shown how iteration operators arise in axiomatic domain theory.
µ-Bicomplete Categories and Parity Games
TLDR
The category μ-bicomplete is called if every μ-term defines a functor and the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.
Typing Weak MSOL Properties
TLDR
A denotational semantics of λY-calculus is constructed that is capable of computing properties expressed in Weak monadic second-order logic (wMSO).
...
...