scholarly journals Cut-Elimination for Full Intuitionistic Linear Logic

1996 ◽  
Vol 3 (10) ◽  
Author(s):  
Torben Braüner ◽  
Valeria De Paiva
Keyword(s):  
Linear Logic ◽  
Formal System ◽  
Independent Interest ◽  
Full Detail ◽  
Cut Elimination ◽  
Full Proof ◽  
Intuitionistic Linear Logic

We describe in full detail a solution to the problem of proving the cut elimination theorem for FILL, a variant of (multiplicative and exponential-free) Linear Logic<br />introduced by Hyland and de Paiva. Hyland and de Paiva's work used a term assignment<br />system to describe FILL and barely sketched the proof of cut elimination. In this paper, as well as correcting a small mistake in their paper and extending the<br />system to deal with exponentials, we introduce a different formal system describing the intuitionistic character of FILL and we provide a full proof of the cut elimination<br />theorem. The formal system is based on a notion of dependence between formulae within a given proof and seems of independent interest. The procedure for<br />cut elimination applies to (classical) multiplicative Linear Logic, and we can (with care) restrict our attention to the subsystem FILL. The proof, as usual with cut<br />elimination proofs, is a little involved and we have not seen it published anywhere.

scholarly journals Reconciling Lambek’s restriction, cut-elimination and substitution in the presence of exponential modalities

2020 ◽  
Vol 30 (1) ◽  
pp. 239-256 ◽  
Author(s):  
Max Kanovich ◽  
Stepan Kuznetsov ◽  
Andre Scedrov
Keyword(s):  
Linear Logic ◽  
Cut Elimination ◽  
Lambek Calculus ◽  
Full Extent ◽  
Intuitionistic Linear Logic

Abstract The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality while keeping Lambek’s restriction. Interestingly enough, we show that for any system equipped with a reasonable exponential modality the following holds: if the system enjoys cut elimination and substitution to the full extent, then the system necessarily violates Lambek’s restriction. Nevertheless, we show that two of the three conditions can be implemented. Namely, we design a system with Lambek’s restriction and cut elimination and another system with Lambek’s restriction and substitution. For both calculi, we prove that they are undecidable, even if we take only one of the two divisions provided by the Lambek calculus. The system with cut elimination and substitution and without Lambek’s restriction is folklore and known to be undecidable.

Girard translation and logical predicates

2000 ◽  
Vol 10 (1) ◽  
pp. 77-89 ◽  
Author(s):  
MASAHITO HASEGAWA
Keyword(s):  
Short Proof ◽  
Linear Logic ◽  
Lambda Calculus ◽  
Independent Interest ◽  
Proof Technique ◽  
Typed Lambda Calculus ◽  
Intuitionistic Linear Logic

We present a short proof of a folklore result: the Girard translation from the simply typed lambda calculus to the linear lambda calculus is fully complete. The proof makes use of a notion of logical predicates for intuitionistic linear logic. While the main result is of independent interest, this paper can be read as a tutorial on this proof technique for reasoning about relations between type theories.

scholarly journals Towards a typed Geometry of Interaction

2010 ◽  
Vol 20 (3) ◽  
pp. 473-521 ◽  
Author(s):  
ESFANDIAR HAGHVERDI ◽  
PHILIP SCOTT
Keyword(s):  
Linear Logic ◽  
Independent Interest ◽  
Cut Elimination ◽  
Mathematical Framework ◽  
Monoidal Categories ◽  
Original Theory ◽  
Categorical Setting ◽  
Abstract Notion ◽  
The Relationship

Girard's Geometry of Interaction (GoI) develops a mathematical framework for modelling the dynamics of cut elimination. We introduce a typed version of GoI, called Multiobject GoI for both multiplicative linear logic (MLL) and multiplicative exponential linear logic (MELL) with units. We present a categorical setting that includes our previous (untyped) GoI models, as well as more general models based on monoidal *-categories. Our development of multiobject GoI depends on a new theory of partial traces and trace classes, which we believe is of independent interest, as well as an abstract notion of orthogonality (which is related to work of Hyland and Schalk). We develop Girard's original theory of types, data and algorithms in our setting, and show his execution formula to be an invariant of cut elimination (under some restrictions). We prove soundness theorems for the MGoI interpretation (for Multiplicative and Multiplicative Exponential Linear Logic) in partially traced *-categories with an orthogonality. Finally, we briefly discuss the relationship between our GoI interpretation and other categorical interpretations of GoI.

Substructural fuzzy logics

2007 ◽  
Vol 72 (3) ◽  
pp. 834-864 ◽  
Author(s):  
George Metcalfe ◽  
Franco Montagna
Keyword(s):  
Linear Logic ◽  
Substructural Logics ◽  
Unit Interval ◽  
Residuated Lattices ◽  
Cut Elimination ◽  
Algebraic Semantics ◽  
Fuzzy Logics ◽  
The Real ◽  
Gentzen Systems ◽  
Intuitionistic Linear Logic

AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].

Multiple conclusion linear logic: cut elimination and more

2020 ◽  
Vol 30 (1) ◽  
pp. 157-174 ◽  
Author(s):  
Harley Eades III ◽  
Valeria de Paiva
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Direct Proof ◽  
Cut Elimination ◽  
Tensor Model ◽  
Double Negation ◽  
Proof Nets ◽  
Small Change ◽  
Definition Of ◽  
Intuitionistic Linear Logic

Abstract Full intuitionistic linear logic (FILL) was first introduced by Hyland and de Paiva, and went against current beliefs that it was not possible to incorporate all of the linear connectives, e.g. tensor, par and implication, into an intuitionistic linear logic. Bierman showed that their formalization of FILL did not enjoy cut elimination as such, but Bellin proposed a small change to the definition of FILL regaining cut elimination and using proof nets. In this note we adopt Bellin’s proposed change and give a direct proof of cut elimination for the sequent calculus. Then we show that a categorical model of FILL in the basic dialectica category is also a linear/non-linear model of Benton and a full tensor model of Melliès’ and Tabareau’s tensorial logic. We give a double-negation translation of linear logic into FILL that explicitly uses par in addition to tensor. Lastly, we introduce a new library to be used in the proof assistant Agda for proving properties of dialectica categories.

scholarly journals Relations and Non-commutative Linear Logic

1991 ◽  
Vol 20 (372) ◽  
Author(s):  
Carolyn Brown ◽  
Doug Gurr
Keyword(s):  
Intuitionistic Logic ◽  
Arbitrary Number ◽  
Linear Logic ◽  
Expressive Power ◽  
Cut Elimination ◽  
Natural Class ◽  
Structural Rules ◽  
Restricted Sense ◽  
Intuitionistic Linear Logic ◽  
Natural Way

<p>Linear logic differs from intuitionistic logic primarily in the absence of the structural rules of weakening and contraction. Weakening allows us to prove a proposition in the context of irrelevant or unused premises, while contraction allows us to use a premise an arbitrary number of times. Linear logic has been called a ''resource-conscious'' logic, since the premises of a sequent must appear exactly as many times as they are used.</p><p>In this paper, we address this ''experimental nature'' by presenting a non-commutative intuitionistic linear logic with several attractive properties. Our logic discards even the limited commutativityof Yetter's logic in which cyclic permutations of formulae are permitted. It arises in a natural way from the system of intuitionistic linear logic presented by Girard and Lafont, and we prove a cut elimination theorem. In linear logic, the rules of weakening and contraction are recovered in a restricted sense by the introduction of the exponential modality(!). This recaptures the expressive power of intuitionistic logic. In our logic the modality ! recovers the non-commutative analogues of these structural rules. However, the most appealing property of our logic is that it is both sound and complete with respect to interpretation in a natural class of models which we call relational quantales.</p>

Linear logic propositions as session types

2014 ◽  
Vol 26 (3) ◽  
pp. 367-423 ◽  
Author(s):  
LUÍS CAIRES ◽  
FRANK PFENNING ◽  
BERNARDO TONINHO
Keyword(s):  
Sequent Calculus ◽  
Linear Logic ◽  
Type System ◽  
Type Systems ◽  
Cut Elimination ◽  
Session Types ◽  
Deadlock Freedom ◽  
Session Type ◽  
Π Calculus ◽  
Intuitionistic Linear Logic

Throughout the years, several typing disciplines for the π-calculus have been proposed. Arguably, the most widespread of these typing disciplines consists of session types. Session types describe the input/output behaviour of processes and traditionally provide strong guarantees about this behaviour (i.e. deadlock-freedom and fidelity). While these systems exploit a fundamental notion of linearity, the precise connection between linear logic and session types has not been well understood.This paper proposes a type system for the π-calculus that corresponds to a standard sequent calculus presentation of intuitionistic linear logic, interpreting linear propositions as session types and thus providing a purely logical account of all key features and properties of session types. We show the deep correspondence between linear logic and session types by exhibiting a tight operational correspondence between cut-elimination steps and process reductions. We also discuss an alternative presentation of linear session types based on classical linear logic, and compare our development with other more traditional session type systems.

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

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