complexity theory – Inhabitation of STLC is in PSPACE

Urzyczyn: Inhabitation in Typed Lambda-Calculi (A syntactic approach) gives a proof that STLC inhabitation problem is in PSPACE (section 2, lemma 1). I don’t understand certain aspects of the proof:

Lemma: There is an alternating polynomial time algorithm to determine whether a given type A is inhabited in a given basis $Gamma$ in the STLC.

Proof.If a type is inhabited, it is inhabited by a term in a long normal form.

Question 1: what is a long normal form.

To determine if there exists a term $M$ in a long normal, satisfying $Gamma vdash M:A$ we proceed as follows:

  • If $A = A_1 to A_2$ then $M$ must be an abstraction $M = lambda x:A_1. M’$. Thus, we look for an $M’$ satifying $Gamma, x:A_1 vdash M’:A_2$.

  • If $A$ is a type variable, then $M$ is an application of a variable to a sequence of terms.

Question 2: I thought there weren’t type variables in the STLC.

We nondeterministically choose a variable z, declared in $Gamma$ to be of type $A_1 rightarrow ldots rightarrow A_n rightarrow A$. If there is no such variable , we reject. If $n = 0$ then we accept. If $n > 0$, we answer in parallel the questions if $A_i$ are inhabited in $Gamma$.

Question 3: it doesn’t matter the actual typing of $z$ in $Gamma$ right? as long as we consume it and don’t use it again in this step.

This alternating procedure is repeated as long as there are new questions of the form $Gamma vdash ? : A$. We can assume that all types in $Gamma$ are different. At each step of the procedure, the basis $Gamma$ either stays the same or expands. Thus the number of steps does not exceed the squared number of subformulas of types in $Gamma,A$.

Question 4: why? could someone spell out some steps of the reasoning here?

number theory – Every root of $x^n-1$ is simple in $ mathbb{Z}_p[x]$

Let $p$ be a prime number s.t $p$ doesn’t divide $n$. Show that every roots of $x^n-overline{1}$ is simple in $mathbb{Z}_p$

If $overline {a} in mathbb{Z}_p$ is a root of $x^n – overline{1}$ then $a^n equiv 1$ $mod p$ and $gdc(a,p)=1$. By Fermat’s little theorem we have $a^{p-1} equiv 1$ $modp$.

Ok, now I neeed to prove that if $overline{b} in mathbb{Z}_p$ is root of $x^n – overline{1}$, that is, $b^n equiv 1 $ $mod p$, then $overline {a} = overline{b}$, that is, $a equiv b$ $ mod p$. Can you give me a way to solve that?

probability theory – Extending the concept of distribution function to any totally or partially ordered measurable space

Let $(Omega,mathcal A, P)$ be a probability space.

Let $(S,Sigma)$ be a measurable space.

Let $X:(Omega,mathcal A) to (S, Sigma)$ be a random variable.

Then $X$ has probability measure $mu_X = P circ X^{-1}$, also called the distribution of $X$.

In Wikipedia and other Google search, the concept of a distribution function $F_X: S to (0,1)$ in measure theory and probability theory seems to be limited to the case where $S = Re$, where

$$F_X(t) = P({omega in Omega : X(omega) leq t})$$

However, the above definition seems to work fine for any totally ordered $S$. Actually, since only $leq$ is involved, it seems that we really only need $S$ to be partially ordered.

Q: Is it OK to extend the definition of distribution function to any totally or partially ordered $S$? If so, is there a text (journal article or classic book) where distribution functions are so defined? If not, why not?

database theory – Find candidate keys – What are the steps

I have these following functional dependencies I figured out:

DM -> RA
RDT -> AM
AD -> RM

I got with a software to calculate what the candidate keys were to this:

{R, D, T}
{A, D, T}
{M, D, T}

But I don’t know HOW i should do this manually to figure out this. Not to use the actual software. What the steps are to solving this. First I thought I should do something like this to figure out the candidate keys:

DM+ = DMRA
RDT+ = RDTAM
AD+ = ADRM

But from what I understand is that only the RDT+ is giving all the attributes for it to be a candidate key? I am so confused by this. How should I think to pick it out from these functional depedencies?

ct.category theory – Lie monoids as monoids internal to the category of smooth manifolds?

This question can be thought as a complement to this one.

Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups, seem to deserve a much more complicated definition (see, for instance, ‘Lie semigroups and their applications’, by Hilgert and Neeb, section 1.4).

Briefly, these are thought as closed subsemigroups of Lie groups, satisfying an extra property. This property, on its turn, is related to the infinitesimal counterpart of the notion of Lie semigroup (in the above reference, the notion of ‘Lie wedge’, whose definition, consequently, must precede that of a Lie semigroup).

What kind of difficulties appear if one tries to define a Lie monoid simply as a monoid internal to the category of smooth manifolds (or some related category)?

A LITTLE BIT OR FURTHER DISCUSSION

Lie groupoids, on their turn, can be defined as groupoids internal to the category of smooth manifolds. Is there an analogous notion of ‘Lie category’, in which morphisms are allowed not to be isomorphisms? Of course, the same question holds for its infinitesimal counterpart.

I tried to find some reference dealing with such a notion, but couldn’t. Though, it seems to be a reasonable one to consider even within the realm of Lie groupoid theory. For example, if one wants to allow distinct objects to have distinct automorphism groups, but still be connected by morphisms, this notion seems to be a necessary step.

In particular, that’s the case if one wants to allow morphisms between distinct objects to be not only isomorphisms between their automorphism groups, but also covering maps between them. I can’t think right now of a concrete example coming, say, from Physics, but it sounds possible that the ‘internal symmetries’ of a system might ‘collapse’ in this particular way.

Besides that, exactly as Lie groupoids can be considered as natural generalizations of Lie groups (even if this shouldn’t be considered the most appropriate point of view, for many reasons…), the ‘Lie categories’ would be natural generalizations of Lie monoids. Indeed, a ‘Lie category’ with one object would amount precisely to a Lie monoid.

Any references will be appreciated.

computability – Halting problem theory vs. practice

There are only 2 types of infinite programs:

  1. Those that repeat their own state after a point (cyclical)
  2. Those that grow indefinitely in used memory

Those in 1st type, follow this pattern:

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Where there is a pair of distinct indices i and j such that xi = xj, and after which the cycle repeats itself again (thanks to the deterministic nature of programs). In this case the inputs x, contain the whole memory and variables used by the algorithm, plus the current instruction pointer.

Cycle detection algorithms work very well in practice for this type and can prove that a given cyclical program will never finish, usually after a small number of steps, for most random programs.

Proving those in the 2nd type is where the challenge is. One could argue that type 2 can never exist in reality (as all computers have finite memory) but that is not very useful in practice because the memory used may grow very slowly for a regular computer to ever be full. A simple example of that is a binary counter that never stops and never repeats its full state completely.

rt.representation theory – Canonical commutation relations-bounded vs. unbounded picture

Suppose that $Q,P$ are self-adjoint operators which satisfy the relation $$(1) (Q,P)=iI$$ One can easily show that in this case $P,Q$ cannot be bounded. However one can find unbounded operators (multiplication by $x$ and $frac1i frac{d}{dx}$) satisfying this relation. When dealing with unbounded operators one encounters problems with the domains so in order to avoid them one proceed as follows: as any self-adjoint operator gives rises to the one parameter group of unitary operators via $P mapsto (e^{itP})_{t in mathbb{R}}$ one can formulate this problem in terms of this one-parameter groups. The canonical commutation relation takes the form $$(2) V(s)U(t)=e^{its}U(t)V(s)$$ where $U$ corresponds to $Q$ while $V $ corresponds to $P$. One form of the Stone-von Neumann theorem states that any irreducible representation of $(2)$ (say on the space $mathcal{H}$) is unitary equivalent to the operators of multiplication by $x$ and $frac1i frac{d}{dx}$ (in more details: there exists a unitary $W:L^2(mathbb{R}) to mathcal{H}$ such that $W^{-1}U(t)W=e^{itQ}$ and $W^{-1}V(s)W=e^{isP}$ where $P=frac1i frac{d}{dx}$ and $Q=M_x$. As far as I know for the relation $(1)$ this is no longer true (probably due to issues with the domains). So I would like to clarify what is the exact relation between $(1)$ and $(2)$.

Suppose that if $U(t)$ and $V(s)$ are one parameter groups of unitaries with generators $Q$ and $P$ resp. Is it true that $P,Q$ satisfy $(1)$ if and only if $U(t)$ and $V(s)$ satisfy $(2)$?

nt.number theory – Updates on a least prime factor conjecture by Erdos

In the 1993 article “Estimates of the Least Prime Factor of a Binomial Coefficient,” Erdos et al. conjectured that
$$operatorname{lpf} {N choose k} leq max(N/k,29).$$

Where $operatorname{lpf}(x)$ denotes the smallest prime factor of $x$.

I am posting here to ask whether any progress has been made toward this conjecture; ie. has this been proven for an integer greater than $29$?

graph theory – Sequences of degrees by iterated vertex removal

Let $G=(V, E)$ be a graph of order $n$. Let $(v_1, dots, v_n)$ be a sequence of vertices. Let $G_i=G({v_{i+1}, dots, v_n})$ be the induced subgraph on the last $n-i$ vertices. Let $d_i$ be the degree of $v_i$ in $G_i$. We then obtain a sequence of non-negative integers $(d_1, dots, d_n)$. Of course, from a graph $G$, we can obtain a family of such integer sequences depending on the sequence of vertices we start with.

I wonder whether such integer sequences have been studied, as there are some natural and interesting questions regarding such sequences. For example, my original motivation is to obtain, for a given graph, such a sequence that is maximum with respect to the majorization order. A natural strategy would be to choose $v_1$ with maximum degree in $G_0$, and $v_2$ with maximum degree in $G_1$, and so on. But first there is an ambiguity about several vertices with the maximum degree in $G_0$. Second, it is not even a priori clear to me that whether we should stick to vertices with maximum degree…

Thank you for any information and suggestions.