...aren't those mutually exclusive?
I feel a mix of "those are obviously different complexity levels" and "is it like C pre-processor turing-completeness situation?"
> Generically, reductions stronger than polynomial-time reductions are used, since all languages in P (except the empty language and the language of all strings) are P-complete under polynomial-time reductions.
The lambda calculus, by its simplicity as just a rewriting language, makes it "obvious" how effective computability emerges from very little.
Do you think there is an "obvious" way to simulate the lambda calculus using origami crease patterns more directly? For example, a cyclic tag system or even rule 110 configuration simulating the lambda calculus without indirection through Turing machines.
[1] https://www.cs.mcgill.ca/~jking/papers/origami.pdf
[2] https://www.pythabacus.com/Origami%20Fractions/folding.htm
Knuth's Art of Computer Programming was built around assembly language for a fantasy computer which is inspired more or less by the Turing machine (program counter is an index into a program 'state', instructions transform a data 'state' and transition to a different program 'state') whereas Structure and Interpretation of Computer Programs is more inspired by Church.
The pinnacle of undergraduate CS education, I think, is compilers, which is where those approaches are ultimately unified on a practical level (you make a machine that transforms one to the other) but the introductory course for the non-professional programmer or the person who aspires to writing compilers someday is still pretty controversial.
Is it?
I think most people who have heard of the topic are familiar with the Church-Turing thesis and know that both definitions of effective calculability are equivalent.
My preference is mostly a matter of taste I think. I admire how little there is to the lambda calculus definition and how computability somehow emerges through construction and definition (which admittedly are not simple). It nicely shows that you need very little "machinery" to get a powerful computational system.
Turing machines by comparaison seem somewhat contrieved with their infinite tape, head and register even if I realise that in a lot of way they are closer to an actual computer.
gnabgib•1y ago