# Tag Info

### P and NP classes explanation through lambda-calculus

Turing-machines and $\lambda$-calculus are equivalent only w.r.t. the functions $\mathbb{N} \rightarrow \mathbb{N}$ they can define. From the point of view of computational complexity they seem to ...
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### How do you get the Calculus of Constructions from the other points in the Lambda Cube?

First, to reiterate one of cody's points, the Calculus of Inductive Constructions (which Coq's kernel is based on) is very different from the Calculus of Constructions. It is best thought of as ...
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### What was the original intent for the creation of Lambda calculus?

He wanted to create a formal system for the foundations of logic and mathematics that was simpler than Russell's type theory and Zermelo's set theory. The basic idea was to add a constant $\Xi$ to ...

### Using lambda calculus to derive time complexity?

A recent developpement on this topic: U. dal Lago and B. Accatoli proved that the length of the leftmost-outermost reduction (LOr) of a $\lambda$-term is an invariant (time) cost model for $\lambda$-...
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### How exactly does lambda calculus capture the intuitive notion of computability?

You're in good company. Kurt Gödel criticized $\lambda$-calculus (as well as his own theory of general recursive functions) as not being a satisfactory notion of computability on the grounds that it ...
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### Is there a typed lambda calculus which is consistent and Turing complete?

Alright I'll give a crack at it: In general for a given type system $T$, the following is true: If all well-type terms in the calculus $T$ are normalizing, then $T$ is consistent when viewed as a ...
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### Why do constructivists not seem to care too much about call/cc

Constructive mathematics is not just a formal system but rather an understanding of what mathematics is about. Or to put it differently, not every kind of semantics is accepted by a constructive ...
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### Historic Relationship between Typed Lambda Calculus and Lisp?

First, your friend is wrong about the history of the $\lambda$-calculus. Church created the untyped calculus first, which he intended as a foundation for mathematics. Fairly quickly, it was discovered ...
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### How is Lambda Calculus a specific type of Term Writing system?

The answer is it depends what you mean by Term Rewrite System. When it was introduced, the concept of Term Rewrite Systems, or TRSes, described what is now called first order TRSes, which is simply a ...
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### (How) Could we discover/analyze NP problems in the absence of the Turing model of computation?

You may wish to look at cost semantics for functional languages. These are various computational complexity measures for functional languages that do not pass through any kind of Turing machine, RAM ...
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### Equivalent formulation of complexity theory in Lambda Calculus?

As you point out, the λ-calculus has a seemingly simple notion of time-complexity: just count the number of β-reduction steps. Unfortunately, things are not simple. We should ask: ...

### What is the "question" that programming language theory is trying to answer?

The overall purpose of PLT is to make industrial software engineering (in a general sense) cheaper (also in a general sense), through optimising the most important tool (programming languages) and ...
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### Scott's stochastic lambda calculi

One apparent strength of his approach is that it allows higher-order functions (i.e. lambda terms) to be observable outcomes, which measure theory generally makes quite tricky. (The basic problem is ...

### Do I have to give up the Law of the Excluded Middle in order to Learn $\lambda$-Calculus?

To me, your question seems analogous to saying "I've heard that non-Euclidean geometry requires me to give up Euclid's fifth axiom, which is very useful in many mathematical contexts." You don't have ...
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### Is this behavior in a programming language inconsistent?

Yes, your type inference seems incomplete. This example can be dealt with fairly trivially, by computing the respective type equations, e.g. in the style Hindley/Milner does it. Alpha-renaming the ...