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Questions tagged [linear-logic]

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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
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1answer
318 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 ...
6
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0answers
95 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) = ...
10
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1answer
333 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;\...
7
votes
1answer
265 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 ...
0
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1answer
120 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?
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0answers
181 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
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1answer
355 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
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1answer
281 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
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1answer
80 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\...
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0answers
111 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 ...
6
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3answers
392 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
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3answers
538 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
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1answer
159 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
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1answer
208 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\ |\ ...
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0answers
212 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 ...
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0answers
126 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), ...
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2answers
527 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
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1answer
358 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 ...
14
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1answer
816 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
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3answers
534 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 ...