Questions tagged [linear-logic]

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11 votes
2 answers
2k views

Can linear logic be used to resolve unexpected hanging/surprise examination paradox?

In the Unexpected Hanging Paradox, the prisoner tries to narrow down their date of execution using seemingly sound logical reasoning. They instead arrive at a contradiction. When the paradox is ...
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4 votes
0 answers
97 views

Full coherence for non-symmetric linearly distributive categories?

1. Context Mac Lane's coherence theorem for monoidal categories can be phrased as "every formal diagram in a monoidal category commutes.“ I am interested in this type of coherence question for ...
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4 votes
0 answers
293 views

Is this linearly distributive category really free?

In Natural deduction and coherence for weakly distributive categories Blute et al. claim to give a presentation of the free (non-symmetric) linearly distributive category $\operatorname{PNet_E}(C)$ on ...
  • 445
2 votes
0 answers
94 views

Empires and the net criterion

Currently, I am struggling to understand the proof of Proposition 2.5 on page 250 (page 22 in the document) of the paper Natural deduction and coherence for weakly distributive categories by Blute, ...
  • 445
4 votes
0 answers
114 views

Correctness criteria for proof nets

In their paper Natural deduction and coherence for weakly distributive categories Blute, Cockett, Seely and Trimble introduce so-called proof circuits (aka two-sided proof structures) for the positive ...
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5 votes
1 answer
181 views

On the correspondence between proof nets and sequents

1. Context While trying to answer my question on the existence of a (useful) graphical calculus for star-autonomous categories, I came across the paper Natural deduction and coherence for weakly ...
  • 445
6 votes
1 answer
228 views

Linear logic and linearly distributive categories

I asked this question ten days ago on MathStackexchange (see here). Despite having placed a bounty on the question, I have not received any answers or comments until now. Following Nick Champion's ...
  • 445
5 votes
0 answers
93 views

Equivalent formulation of linear logic with more axioms and less inference rules

We can formulate classical (sequent) logic with only the structural inference rules including cut, and a collection of axioms like $A, B \vdash A \wedge B$. This is equivalent to the usual sequent ...
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7 votes
1 answer
284 views

Ordered logic is the internal language of which class of categories?

Wikipedia says: The internal language of closed symmetric monoidal categories is linear logic and the type system is the linear type system. "A Fibrational Framework for Substructural and Modal ...
8 votes
1 answer
304 views

Distributivity of ! over?

Has anyone studied a variant of linear logic, or of its semantic counterpart (exponential modalities on linearly distributive categories / $\ast$-autonomous categories / polycategories) for which ...
  • 61.4k
9 votes
0 answers
287 views

Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...
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2 votes
2 answers
351 views

What is the sequent calculus for differential linear logic? [closed]

I have searched, but only managed to turn up the presentation in interaction nets. I'd be equally interested in a categorical model of DiLL.
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6 votes
0 answers
153 views

Lambek calculus, linear logic, and linear algebra

In his 1958 paper, The Mathematics of Sentence Structure, Joachim Lambek introduced the Lambek calculus. In modern terms, it could be understood as a syntax for biclosed monoidal categories, and he ...
6 votes
1 answer
401 views

Proof of ¬(¬1 ⊗ ¬1) in tensorial logic

I believe I once had a proof of this proposition, but it's been lost to the mists of time and old hard drives, so who knows if it was correct, and try as I might I can't seem to reproduce it. Is it ...
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6 votes
0 answers
136 views

Linear logic with storage preserving positives

Has anyone studied a version of linear logic in which the storage modality $!$ preserves the positive connectives and quantifiers $\otimes,\oplus,\exists$? That is, such that we have $!(A\otimes B) = ...
  • 61.4k
12 votes
1 answer
662 views

Is Girard's LU just an embedding of classical and intuitionistic logic into linear logic?

This question is about Girard's system LU, presented in his paper On the unity of logic. Girard starts by giving a "modal" sequent calculus with two zones of both hypotheses and consequents, $\Gamma;\...
  • 61.4k
11 votes
1 answer
496 views

Dioperads vs polycategories

As defined by Gan, a dioperad consists of sets of operations $P(n,m)$ with "$n$ inputs and $m$ outputs", which can be composed by joining one output of one operation to one input of another, giving ...
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0 votes
1 answer
145 views

Differential categories vs McBride's notion of derivative

Has anyone done an analysis to see if Blute, Cockett, and Seely's differential categories suffice to provide a notion of 1-hole context in the symmetric monoidal setting?
2 votes
0 answers
203 views

Can a relationship be constructed between the Coherence space and Phase space semantics of linear logic?

I'm not very familiar with linear logic, so please bear with me, i.e., please "read between the lines" to my underlying question if I don't formulate it rigorously correctly. To help model some of my ...
5 votes
1 answer
496 views

Embedding of classical into intuitionistic linear logic

Following on from this recent question, there is another construction that is well-known, but I don’t know a good primary source for: the Kolmogorov-style double-negation embedding of classical into ...
6 votes
1 answer
355 views

Conservativity of multiplicative linear logic over intuitionistic multiplicative linear logic

It is well known that multiplicative linear logic (MLL) is conservative over intuitionistic multiplicative linear logic (IMLL). In other words, if an IMLL formula is provable in MLL then it is already ...
1 vote
1 answer
86 views

Injecting premises into two implicational premises connected by a tensor (multiplicative conjunction) in linear logic

I have another question regarding linear logic: I want to get to the proof E, using the premises in (1-4). Is this at all possible? 1: $A$ 2: $C$ 3: $(A\multimap B)\otimes(C\multimap D)$ 4: $B\...
1 vote
0 answers
120 views

Dissolution of Tensors

I have a question that might seem odd to linear logic experts (I am somewhat of a novice). I know that two items of the same type can be combined into one premise with a tensor (multiplicative ...
8 votes
3 answers
527 views

Models of intuitionistic linear logic that reflect the resource interpretation

I am interested in models of intuitionistic linear logic, that is, the logic that you get if you take classical linear logic and restrict the set of operators to $\otimes$, $1$, $\multimap$, $\times$, ...
4 votes
3 answers
569 views

Exponentials in the opposite category of finite separable algebras

Let $K$ be a field and $G=Gal(K_s/K)$ is its absolute Galois group. Then, by Galois theory, the category of finite separable algebras over $K$ (denoted by $Sep(K)$) and the category of finite ...
1 vote
1 answer
164 views

Interaction-based approximation for HP-complete λ-theory?

We are looking for a proof or counter-examples for the following hypothesis. Two combinators $M$ and $N$ are solvable and equivalent in the HP-complete sensible $\lambda$-theory iff either $$ \exists ...
1 vote
1 answer
214 views

Hypothesis: interaction-based model for λKβη

We are looking for a proof or counter-examples to the following Hypothesis. In interaction calculus $\langle \varnothing\ |\ \Gamma(M, x) \cup \Gamma(N, x)\rangle \downarrow \langle \varnothing\ |\ ...
4 votes
0 answers
240 views

Is it possible to implement η-reduction in interaction nets?

There are several ways to encode λ-terms in interaction nets; for instance, using the original optimal algorithm by Lamping, or compiling λ-calculus into interaction combinators. However, all the ...
2 votes
0 answers
135 views

Turing-complete primitive interaction systems

Let us call primitive an interaction system with the signature $\Sigma = \{(\rho, 0), (\xi, n)\}, \quad n \geq 2;$ and the only rule being of the form $\rho \bowtie \xi[\rho, \xi(a_1, \dots , a_n), ...
6 votes
2 answers
666 views

Are exponentials in categorical models of linear logic harmful?

Categorical models for linear logic with $\otimes$, $1$, $\&$, $\top$, $\oplus$, $0$, and $\multimap$ are typically symmetric monoidal closed categories (for modeling $\otimes$, $1$, and $\...
7 votes
1 answer
405 views

Looking for papers and articles on the Tarskian Möglichkeit

Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =_{def} \neg A \to A$ (which he ...
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14 votes
1 answer
924 views

How is Fredkin and Toffoli's Conservative Logic related to Linear Logic?

In the answers to this question, Timothy Gowers asks: I've been interested in this question for some time. I haven't put any serious thought into it, so all I can offer is a further question rather ...
4 votes
3 answers
641 views

What is the proper name for "compact closed" multiplicative intuitionistic linear logic?

Multiplicative intuitionistic linear logic (MILL) has only multiplicative conjunction $\otimes$ and linear implication $\multimap$ as connectives. It has models in symmetric monoidal closed ...
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