Handling epsilon productions in recursive descent parsing

I am working on a recursive descent parser for lambda calculus. In my grammar, after removing left-recursion, I am left with the following two productions:

# APPLICATION  -> ATOM APPLICATION' # APPLICATION' -> ATOM APPLICATION' | ε        

Both APPLICATION and APPLICATION’ correspond to the same type of node in the AST. How should I handle parsing the second production? I know that choosing between two alternatives for a production is performed by looking up the next token in buffer, but the epsilon production confuses me.

EDIT:

Here is current implementation (Ruby) that fails to left-associate applications.

require_relative 'token.rb' require_relative 'ast.rb'  class Parser     def self.call(tokens)         new(tokens).send(:parse_expr)     end      def self.to_proc         method(:call).to_proc     end      private         def initialize(tokens)             @tokens = tokens             @l = -1         end          def parse_expr             if lookahead.type == :tklambda                 return ExprNode.new([parse_abstraction], @l)             else                 return ExprNode.new([parse_application], @l)             end         end          def parse_abstraction             match_token(:tklambda)             id = parse_identifier             match_token(:tkdot)             expr = parse_expr             return AbstrNode.new([id, expr], @l)         end          def parse_identifier             id_token = match_token(:tkid)             val = id_token.value             return IdNode.new(nil, @l, val)         end          def parse_application             left_child = parse_atom             first_atom = [:tklparen, :tkid] # FIRST(ATOM)             # application -> atom application'             while !lookahead.nil? and first_atom.include?(lookahead.type)                 return AppNode.new([left_child, parse_application], @l)             end             # application' -> atom application' | ε             return left_child         end          def parse_atom             if lookahead.type == :tklparen                 match_token(:tklparen)                 expr = parse_expr                 match_token(:tkrparen)                 return AtomNode.new([expr], @l)             else                 return AtomNode.new([parse_identifier], @l)             end         end          def match_token(type = nil)             if !type.nil? and lookahead.type != type                 raise "Unexpected token at #{@l + 1}. Expected: #{type}, "\                       "got: #{lookahead.type}."             end             @l += 1             return @tokens[@l]         end          def lookahead             return @tokens[@l + 1]         end end 

Asymptotic Bound on Minimum Epsilon Cover of Arbitrary Manifolds

Let $ M \subset \mathbb{R}^d$ be a compact smooth $ k$ -dimensional manifold embedded in $ \mathbb{R}^d$ . Let $ \mathcal{N}(\epsilon)$ denote the size of the minimum $ \epsilon$ cover $ P$ of $ M$ ; that is for every point $ x \in M$ there exists a $ p \in P$ such that $ \| x – p\|_{2}$ .

Is it the case that $ \mathcal{N}(\epsilon) \in \Theta\left(\frac{1}{\epsilon^k}\right)$ ? If so, is there a reference?

Proving equivalence between $\epsilon$ based & $lub$ definition of supremum.

Based on $ \epsilon$ have a new definition of supremum:

Let there be a nonempty set $ X$ with supremum $ s$ , then $ X\cap(s – \epsilon, s]\ne \emptyset, \,\, \forall \epsilon\gt 0$ .

The conventional definition is given by:

Let $ X$ be a nonempty set of real numbers. The number $ s$ is called the supremum of $ X$ if $ s$ is an upper bound of $ X$ and $ s \le y$ for every upper bound of $ X$ .

Let, the conventional definition be denoted by ‘Def. 1’, while the new definition by ‘Def. 2’.

Have two questions below. I need help in attempting them.

Q. 1 : Need show that the two definitions are equivalent by proving the following two conditional statements:

(i) If $ s = sup(X)$ , as given by Defn. 1, then $ s$ is the supremum, as given by Defn. 2. Here, assume that Defn. 1 holds, and use this assumption to prove that Defn. 2 holds.

Let $ s’$ is supremum as per Defn. 2. Also, the relation between the magnitudes of $ s,s’$ is unknown, & need be established.

$ s$ will have set $ X$ elements in the range $ (s-\epsilon, s]$ if $ s-s’ \lt \epsilon$ , by the below proof:

Let us assume that $ s-s’ \ne 0 $ , let $ s-s’=k.\epsilon, k\lt 1$ , then $ s = s’+k.\epsilon \implies s -\epsilon = s’+(k-1).\epsilon \implies s -\epsilon \lt s’$ .

$ s-\epsilon\lt s’\implies \exists x \in X: X\cap (s – \epsilon, s]\ne \emptyset$ .
But, Def. 2 can take any $ \epsilon\gt 0$ to ensure $ \exists x \in X: X\cap (s’ – \epsilon, s’]\ne \emptyset$ .
So, if Def. 1 is to have ability to take any $ \epsilon\gt 0$ , need the lower bound of $ (s – \epsilon, s]$ to equal at least to $ s’ – \epsilon$ .
But, $ s – \epsilon= s’+(k-1)\epsilon \ge s- \epsilon, \forall k, 0\lt k\lt 1$ .
So, the only possible value is $ k=0$ to have the lower bound of $ (s – \epsilon, s]$ equal to $ s’ – \epsilon$ .

But, by this cannot impose any restriction on the upper bound $ s$ (of Def. 1) to equal $ s’$ (of Def. 2).

(ii) If $ s = sup(X)$ , as given by Defn. 2, then $ s$ is the supremum, as given by Defn. 1. Here, assume that Defn. 2 holds, and use this assumption to prove that Defn. 1 holds.

Let us modify for consistency with part (i) sake, $ s$ replaced by $ s’$ .

If Defn. 2 holds, then the upper bound of the interval is bounded by $ s’$ , which is also the last element that can possibly be (if, $ s’\in X$ ) in $ X$ . For Defn. 1 to hold, the upper bound must then be the same as the upper bound of Defn. 2, i.e. $ s’$ .

Q. 2: What is the practical significance of showing that these two definitions are logically equivalent?

The step (i) of showing that if Defn. 1 holds, then Defn. 2 holds, leads to having the lower bound of $ (s – \epsilon, s]=s’ – \epsilon$ .

The step (ii) of showing that if Defn. 2 holds, then Defn. 1 holds, leads to having the upper bound of $ (s – \epsilon, s]=s’$ per bound implies the max. value is $ s=1$ at $ n= \infty$ . All possible values of $ n$ are covered in the interval $ (s-\epsilon,s]$ with $ n=\infty$ at the upper bound.

Proving there is an eigenvalue $\lambda$ for which $|\lambda – b_{jj}| < \epsilon \sqrt{n}$


Let $ A$ be an $ n\times n$ real symmetric matrix. By applying Jacobi’s method, suppose we have generated an orthogonal matrix $ R$ and a symmetric matrix $ B$ such that the equality

$ $ B = R^{T}AR $ $

holds. Moreover, suppose the inequality $ |b_{ij}| < \epsilon$ holds for all $ i \neq j$ .

Show that for each $ j = 1, 2, \ldots, n$ , there is at least one eigenvalue $ \lambda$ of $ A$ such that $ |\lambda – b_{jj}| < \epsilon \sqrt{n}$ holds.


This is an exercise that I am doing to study for my final exam. So, I’ve just recently learned Jacobi’s method, and I know that the eigenvalues and eigenvectors are related to the matrices $ B$ and $ R$ ; however, I have no idea how to use those results to prove an inequality. I also have no idea how to get the $ \sqrt{n}$ term in there. I would greatly appreciate any help in this exercise.

Thanks

Solving $u_{\epsilon \eta} = \frac{ u_\epsilon – u_eta}{4 (\epsilon – \eta) } .$

I’m trying to solve the following PDE

$ $ y^2 u_{xx} – u_{yy} = 0.$ $

I’ve found its canonical form as $ $ u_{\epsilon \eta} = \frac{ u_\epsilon – u_eta}{4 (\epsilon – \eta) } .$ $

However, I’m having trouble in solving this PDE. I’ve tried solving with mathematica, and that gave me

$ $ u = \frac{ x + y^3}{ 3} ,$ $ but, I need to know how to obtain such a result from the canonical form of the given PDE.

Use of the reverse triangle inequality in epsilon proof

I’m new to analysis and trying to prove something about a converging series.

Now I want to get from $ |x_{n}-\bar{x}| < \frac{|\bar{x}|}{2}$ to the following statement $ |x_{n}| > \frac{|\bar{x}|}{2}$ using the reverse triangle inequality, but I just don’t seem to get it right.

As for as my knowledge goes, the reverse inequality states that $ ||b|-|a|| \leq |b|-|a|$ . Any suggestions on how to apply this?

PS: it is a bout a converging sequence $ x_{n}$ with limit $ \bar{x}$ .

Given least upper bound $\alpha$ for $\{\ f(x) : x \in [a,b] \ \}$, $\forall \epsilon > 0 \ \exists x$ s.t. $\alpha – f(x) < \epsilon$

I can’t figure out how all of this follows. Taken from Ch.8 of Spivak’s Calculus.

If $ \alpha$ is the least upper bound of $ \{\ f(x) : x \in [a,b] \ \}$ then, $ $ \forall \epsilon > 0 \ \exists x\in [a,b] \ \ \ \ \ \ \ \alpha – f(x) < \epsilon$ $ This, in turn, means that $ $ \frac{1}{\epsilon} < \frac{1}{\alpha – f(x)}$ $

Anti-concentration: upper bound for $P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2Z_i^2 \ge \epsilon)$

Let $ \mathbb S_{n-1}$ be the unit sphere in $ \mathbb R^n$ and $ z_1,\ldots,z_n$ be a i.i.d sample from $ \mathcal N(0, 1)$ .

Question

Given $ \epsilon > 0$ (may be assumed to be very small), what is a reasonable upper bound for the tail probability $ P(\sup_{a \in \mathbb S_{n-1}}\sum_{i=1}^na_i^2z_i^2 \ge \epsilon)$ ?

Observations

  • Using ideas from this other answer (MO link), one can establish the non-uniform anti-concentration bound: $ P(\sum_{i=1}^na_i^2z_i^2 \le \epsilon) \le \sqrt{e\epsilon}$ for all $ a \in \mathbb S_{n-1}$ .

  • The uniform analogue is another story. May be one can use covering numbers ?