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#cs

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I'd welcome advice on lambda calculus.

---

Given context:

Γ ≡ β : ∗, x : Πα : ∗. α

I need to find 3 examples of terms that inhabit β→β

... that aren't β-convertible to each other, and are in β-normal form.

---

my example 1:

λz:β .z

my example 2:

λβ .xβ

for a 3rd example.. I'm playing with ideas like

x(β→β)

since (β→β) is of type * using the formation rule, but not sure..

---

My old introduction was very outdated, so it's time to reintroduce myself:
#introduction

Hi 👋, I’m Laura.

I am a transfeminine person, somewhat in the middle of my transition. 🏳️‍⚧️ #trans #transbubble

A major part of my time I spend as a Postdoc in computer science, working on embedded AI and low-power IoT communication. #cs #TinyML #IoT #academia #science

Outside of work, I am active in the local #queer center (board member, GER: Vorstand), I enjoy playing board games, and I listen to too many #podcasts.

advice needed !

I need to find an inhabitant of β given the context

Γ ≡ β : ∗, x : Πα : ∗. α

My brain is telling me that I am free to choose pretty much any type to inhabit β because the context doesn't impose any restriction.

I could have "int" or "bool" or "int → bool", etc and they would all be ok.

Am I right? I'm suffering beginner lack of confidence.

Trying to get my head around this..

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In simple type theory the following term isn't typeable

xx

But in second order type theory the following is

λx.xx

Eg

λx:Πα.*.α→α . x(σ→σ)(xσ)

---

What is unfamiliar is the different arguments to each x. Looks wrong but I guess isn't.

⛓️💥 Broken Tokens? Your Language Model can Secretly Handle Non-Canonical Tokenizations

arxiv.org/abs/2506.19004

arXiv logo
arXiv.orgBroken Tokens? Your Language Model can Secretly Handle Non-Canonical TokenizationsModern tokenizers employ deterministic algorithms to map text into a single "canonical" token sequence, yet the same string can be encoded as many non-canonical tokenizations using the tokenizer vocabulary. In this work, we investigate the robustness of LMs to text encoded with non-canonical tokenizations entirely unseen during training. Surprisingly, when evaluated across 20 benchmarks, we find that instruction-tuned models retain up to 93.4% of their original performance when given a randomly sampled tokenization, and 90.8% with character-level tokenization. We see that overall stronger models tend to be more robust, and robustness diminishes as the tokenization departs farther from the canonical form. Motivated by these results, we then identify settings where non-canonical tokenization schemes can *improve* performance, finding that character-level segmentation improves string manipulation and code understanding tasks by up to +14%, and right-aligned digit grouping enhances large-number arithmetic by +33%. Finally, we investigate the source of this robustness, finding that it arises in the instruction-tuning phase. We show that while both base and post-trained models grasp the semantics of non-canonical tokenizations (perceiving them as containing misspellings), base models try to mimic the imagined mistakes and degenerate into nonsensical output, while post-trained models are committed to fluent responses. Overall, our findings suggest that models are less tied to their tokenizer than previously believed, and demonstrate the promise of intervening on tokenization at inference time to boost performance.
#ai#llm#cs

Help needed !

Show there is a t such that ⊥: t where

⊥ ≡ Πα:*.α

I've only ever seen the type binder Π used in λ2 types of terms, not terms themselves.

It would be like saying ... show there is a t such that (α → α):t

What am I missing?

Did I get this right?

Given these four declarations:

α : ∗
β : ∗
f : α →β
x : α

The following are the only valid λ2-contexts:

α : *
α : *, x : α
β : *
α: *, β : *
α: *, β : *, f: α→β

(here λ2 is second order lambda calculus, and α, β are type variables, not object variables)

Help 😶‍🌫️!

The textbook says that types don't matter in β-reduction.

I've been bashing my head for 3 days and still can't see it.

In the following

(λx:σ.M) N

surely the type of N must match the type of x ??

When the textbook talks about β-substition (not reduction) then it says types matter because you can't substitute A for B if the types don't match.

What am I missing?

The University of Manchester joins two new national research hubs to drive sustainable manufacturing

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europesays.com/uk/196859/

When I was in #CS grad school, back in the early 1990s, #wavelets were hot in 3D volumetric CG—oh, those SIGGRAPH symposia on the topic. At the same time in #EE, loads of papers were published on their efficacy in DSP. Just about everyone in EE and CS seemed to have published at least one paper on wavelets. Fun times. But the current state of wavelet academic #research seemed to have dried up.

I don't quite understand why wavelet transform has not supplanted Fourier transform in many #engineering and #computing application domains, considering its estimable time-frequency locality and its prodigious multi-resolution analysis capabilities, compared to Fourier analysis.

I am but a mere "maths carpenter". So, what am I missing, I wonder.

Why is there no consideration of base cases ?

This is the textbook's explanation of "structural induction". The subsequent worked examples don't refer to base cases either.

Is it because they are always trivially true? I doubt this, myself.