Tuple Relational Calculus – Find names of persons who own at least one house in each city in Canada

Using tuple relational calculus, Find names of persons who own at least one house in each city in Canada.

Bolded are keys.

City(city-name,country-name,area,population)

House(hno,#rooms,stno,owner-name)

Street(stno,city-name,length)

This is what I came up with:

{ h.owner-name | House(h) and forAll c (City(c) and c.name=”Canada”) -> (Exists s(Street(s) and s.city-name = c.name) and Exists x(House(x) and x.stno = s.stno and x.owner-name = h.owner-name) ) }

How I read this aloud: “For all Canadian cities, there exists a street with a HOUSE x that has the same owner as the HOUSE h.”

Is this correct? I’ve peen pulling my hair out trying to understand tuple relational calculus.

Is $\Gamma \vdash x x : T$ possible in the simply typed lambda calculus?

Is $ \Gamma \vdash x x : T$ possible?

This problem appears on page 104 of Benjamin Pierce’s “Types and Programming Languages”.

My conclusion is that it is was the case then we would get $ x: T_1 \to T_2$ and $ x: T_1$ and by some axiom, these types are not equal.

The problem is identifying this axiom but I fear it might be possible to have this equality…

Any hints?

Converting Lambda Calculus functionality to predicate logic syntax

I am trying to validate the simplest possibly notion of a formal system as relations between finite strings. I know that Lambda Calculus has the expressive power of a Turing Machine:

<λexp> ::= < var > | λ . <λexp> | ( <λexp> <λexp> )

I was thinking that it might be possible so somehow convert the functionality of lambda calculus into syntax that is closer to predicate logic by defining named functions that take finite string arguments and return finite string values.

Does anyone here have any ideas on this?

Does type-1 lambda calculus exist?

I’m interested in the intersection of linguistics and computer science, I’ve been reading on Chomsky hierarchy, and would like to know if there exist lambda calculus types that are equivalent to the Chomsky types, especially the type-1 that’s context-sensitive and has the linear-bounded non-deterministic Turing machine as an automation equivalent on the wiki.

What’s the proof complexity of E-KRHyper (E-hyper tableau calculus)?

Before the question, let me explain better what is E-KRHyper:

E-KRHyper is a theorem proving and model generation system for first-order logic with equality. It is an implementation of the E-hyper tableau calculus, which integrates a superposition-based handling of equality into the hyper tableau calculus (source: System Description: E-KRHyper).

I am interested in the complexity of system E-KRHyper because it is used in the question-answer system Log-Answer (LogAnswer – A Deduction-Based Question Answering System (System Description)).

I have found a partial answer:

our calculus is a non-trivial decision procedure for this fragment (with equality), which captures the complexity class NEXPTIME (source: Hyper Tableaux with Equality).

I don’t understand much of complexity theory so my question is:

What is the complexity of a theorem to be proved in terms of the number of axioms in the database and in terms of some parameter of the question to be answered?

Lambda Calculus as a branch of set theory

This answer to a question about whether C is the mother of all languages contained an interesting tidbit that I am curious about:

The functional paradigm, for example, was developed mathematically (by Alonzo Church) as a branch of set theory long before any programming language ever existed.

Is this true? What is the link between these topics that is so fundamental as to make lambda Calculus an outgrowth of set theory? The best I can come up with is that standard mathematical functions possess domains and codomains.

How/when is calculus used in Computer Science?

Many computer science programs require two or three calculus classes.

I’m wondering, how and when is calculus used in computer science? The CS content of a degree in computer science tends to focus on algorithms, operating systems, data structures, artificial intelligence, software engineering, etc. Are there times when Calculus is useful in these or other areas of Computer Science?

lambda calculus as set-theoretic operations

It is possible to interpret typed lambda calculus a-la Church as logical operations (because of Curry-Howard correspondence). Also, there is a isomorphism between logical and set-theoretic operations. So, is it possible to direct interpret lambda application as union of sets, and lambda abstraction as subset relation or something like this? Which set-theoretic operations corresponds to lambda application and abstraction?

Rreference Request: book on stochastic calculus (not finance)

I am looking at looking at fractional Gaussian/Brownian noise from a signal theoretic and engineering point of view. In particular, I am looking at the math behind what defines these noise processes and what consequences this has on the physics, either generating them or consuming these noise signals.

As an engineer by training I am familiar with both (real/multivariate/complex) calculus and basic probability theory and also stochastic signals. But most of what I am doing now is where fractional calculus and stochastic calculus meet (Hic sunt dracones… literally). I think I can get my way around most of the fractional calculus part, but for the stochastic calculus I am in need of better understanding of how it works.

What I am looking for is a book (or lecture notes) that not only give me an understanding and intuition how stochastic calculus works (ie. how to apply it), but I also need the proofs in order to tell what I am allowed to do with the theorems and what not. Measure theory shouldn’t be much of a problem, as I have two mathematicians at hand who can explain things, if I get stuck.