Small-angle approximation

In mathematical logic, a ground term of a formal system is a term that does not contain any free variables.

Similarly, a ground formula is a formula that does not contain any free variables. In first-order logic with identity, the sentence In computational number theory and computational algebra, Pollard's kangaroo algorithm (aka Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper as his better-known rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.

The algorithm

Suppose $G$ is a finite cyclic group of order $n$ which is generated by the element $\alpha$, and we seek to find the discrete logarithm $x$ of the element $\beta$ to the base $\alpha$. In other words, we seek $x\in Z_{n}$ such that $\alpha ^{x}=\beta$. The lambda algorithm allows us to search for $x$ in some subset $\{a,\ldots ,b\}\subset Z_{n}$. We may search the entire range of possible logarithms by setting $a=0$ and $b=n-1$, although in this case Pollard's rho algorithm is more efficient. We proceed as follows:

1. Choose a set $S$ of integers and define a pseudorandom map $f:G\rightarrow S$.

2. Choose an integer $N$ and compute a sequence of group elements $\{x_{0},x_{1},\ldots ,x_{N}\}$ according to:

• $x_{0}=\alpha ^{b}\,$
• $x_{i+1}=x_{i}\alpha ^{f(x_{i})}{\mbox{ for }}i=0,1,\ldots ,N-1$

3. Compute

$d=\sum _{i=0}^{N-1}f(x_{i})$.

Observe that:

$x_{N}=x_{0}\alpha ^{d}=\alpha ^{b+d}\,.$

4. Begin computing a second sequence of group elements $\{y_{0},y_{1},\ldots \}$ according to:

• $y_{0}=\beta \,$
• $y_{i+1}=y_{i}\alpha ^{f(y_{i})}{\mbox{ for }}i=0,1,\ldots ,N-1$

and a corresponding sequence of integers $\{d_{0},d_{1},\ldots \}$ according to:

$d_{n}=\sum _{i=0}^{n-1}f(y_{i})$.

Observe that:

$y_{i}=y_{0}\alpha ^{d_{i}}=\beta \alpha ^{d_{i}}{\mbox{ for }}i=0,1,\ldots ,N-1$

5. Stop computing terms of $\{y_{i}\}$ and $\{d_{i}\}$ when either of the following conditions are met:

A) $y_{j}=x_{N}$ for some $j$. If the sequences $\{x_{i}\}$ and $\{y_{j}\}$ "collide" in this manner, then we have:

$x_{N}=y_{j}\Rightarrow \alpha ^{b+d}=\beta \alpha ^{d_{j}}\Rightarrow \beta =\alpha ^{b+d-d_{j}}{\pmod {n}}\Rightarrow x\equiv b+d-d_{j}{\pmod {n}}$

and so we are done.

B) $d_{i}>b-a+d$. If this occurs, then the algorithm has failed to find $x$. Subsequent attempts can be made by changing the choice of $S$ and/or $f$.

Complexity

Pollard gives the time complexity of the algorithm as ${\scriptstyle O({\sqrt {b-a}})}$, based on a probabilistic argument which follows from the assumption that f acts pseudorandomly. Note that when the size of the set {a, …, b} to be searched is measured in bits, as is normal in complexity theory, the set has size log(b − a), and so the algorithm's complexity is ${\scriptstyle O({\sqrt {b-a}})=O(2^{{\frac {1}{2}}\log(b-a)})}$, which is exponential in the problem size. For this reason, Pollard's lambda algorithm is considered an exponential time algorithm. For an example of a subexponential time discrete logarithm algorithm, see the index calculus algorithm.

Naming

The algorithm is well known by two names.

The first is "Pollard's kangaroo algorithm". This name is a reference to an analogy used in the paper presenting the algorithm, where the algorithm is explained in terms of using a tame kangaroo to trap a wild kangaroo. Pollard has explained that this analogy was inspired by a "fascinating" article published in the same issue of Scientific American as an exposition of the RSA public key cryptosystem. The article described an experiment in which a kangaroo's "energetic cost of locomotion, measured in terms of oxygen consumption at various speeds, was determined by placing kangaroos on a treadmill".

The second is "Pollard's lambda algorithm". Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda ($\lambda$). The shorter stroke of the letter lambda corresponds to the sequence $\{x_{i}\}$, since it starts from the position b to the right of x. Accordingly, the longer stroke corresponds to the sequence $\{y_{i}\}$, which "collides with" the first sequence (just like the strokes of a lambda intersect) and then follows it subsequently.

Pollard has expressed a preference for the name "kangaroo algorithm", as this avoids confusion with some parallel versions of his rho algorithm, which have also been called "lambda algorithms".

See also

• Rainbow table

References

Template:Number-theoretic algorithms x (x=x) is a ground formula.

A ground expression is a ground term or ground formula.

Examples

Consider the following expressions from first order logic over a signature containing a constant symbol 0 for the number 0, a unary function symbol s for the successor function and a binary function symbol + for addition.

• s(0), s(s(0)), s(s(s(0))) ... are ground terms;
• 0+1, 0+1+1, ... are ground terms.
• x+s(1) and s(x) are terms, but not ground terms;
• s(0)=1 and 0+0=0 are ground formulae;
• s(1) and ∀x: (s(x)+1=s(s(x))) are ground expressions.

Formal definition

What follows is a formal definition for first-order languages. Let a first-order language be given, with $C$ the set of constant symbols, $V$ the set of (individual) variables, $F$ the set of functional operators, and $P$ the set of predicate symbols.

Ground terms

Ground terms are terms that contain no variables. They may be defined by logical recursion (formula-recursion):

1. elements of C are ground terms;
2. If f∈F is an n-ary function symbol and α₁, α₂, ..., α_n are ground terms, then f(α₁, α₂, ..., α_n) is a ground term.
3. Every ground term can be given by a finite application of the above two rules (there are no other ground terms; in particular, predicates cannot be ground terms).

Roughly speaking, the Herbrand universe is the set of all ground terms.

Ground atom

A ground predicate or ground atom or ground literal is an atomic formula all of whose argument terms are ground terms.

If p∈P is an n-ary predicate symbol and α₁, α₂, ..., α_n are ground terms, then p(α₁, α₂, ..., α_n) is a ground predicate or ground atom.

Roughly speaking, the Herbrand base is the set of all ground atoms, while a Herbrand interpretation assigns a truth value to each ground atom in the base.

Ground formula

A ground formula or ground clause is a formula without free variables.

Formulas with free variables may be defined by syntactic recursion as follows:

1. The free variables of an unground atom are all variables occurring in it.
2. The free variables of ¬p are the same as those of p. The free variables of p∨q, p∧q, p→q are those free variables of p or free variables of q.
3. The free variables of In computational number theory and computational algebra, Pollard's kangaroo algorithm (aka Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist J. M. Pollard, in the same paper as his better-known rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.

The algorithm

Suppose $G$ is a finite cyclic group of order $n$ which is generated by the element $\alpha$, and we seek to find the discrete logarithm $x$ of the element $\beta$ to the base $\alpha$. In other words, we seek $x\in Z_{n}$ such that $\alpha ^{x}=\beta$. The lambda algorithm allows us to search for $x$ in some subset $\{a,\ldots ,b\}\subset Z_{n}$. We may search the entire range of possible logarithms by setting $a=0$ and $b=n-1$, although in this case Pollard's rho algorithm is more efficient. We proceed as follows:

1. Choose a set $S$ of integers and define a pseudorandom map $f:G\rightarrow S$.

2. Choose an integer $N$ and compute a sequence of group elements $\{x_{0},x_{1},\ldots ,x_{N}\}$ according to:

• $x_{0}=\alpha ^{b}\,$
• $x_{i+1}=x_{i}\alpha ^{f(x_{i})}{\mbox{ for }}i=0,1,\ldots ,N-1$

3. Compute

$d=\sum _{i=0}^{N-1}f(x_{i})$.

Observe that:

$x_{N}=x_{0}\alpha ^{d}=\alpha ^{b+d}\,.$

4. Begin computing a second sequence of group elements $\{y_{0},y_{1},\ldots \}$ according to:

• $y_{0}=\beta \,$
• $y_{i+1}=y_{i}\alpha ^{f(y_{i})}{\mbox{ for }}i=0,1,\ldots ,N-1$

and a corresponding sequence of integers $\{d_{0},d_{1},\ldots \}$ according to:

$d_{n}=\sum _{i=0}^{n-1}f(y_{i})$.

Observe that:

$y_{i}=y_{0}\alpha ^{d_{i}}=\beta \alpha ^{d_{i}}{\mbox{ for }}i=0,1,\ldots ,N-1$

5. Stop computing terms of $\{y_{i}\}$ and $\{d_{i}\}$ when either of the following conditions are met:

A) $y_{j}=x_{N}$ for some $j$. If the sequences $\{x_{i}\}$ and $\{y_{j}\}$ "collide" in this manner, then we have:

$x_{N}=y_{j}\Rightarrow \alpha ^{b+d}=\beta \alpha ^{d_{j}}\Rightarrow \beta =\alpha ^{b+d-d_{j}}{\pmod {n}}\Rightarrow x\equiv b+d-d_{j}{\pmod {n}}$

and so we are done.

B) $d_{i}>b-a+d$. If this occurs, then the algorithm has failed to find $x$. Subsequent attempts can be made by changing the choice of $S$ and/or $f$.

Complexity

Pollard gives the time complexity of the algorithm as ${\scriptstyle O({\sqrt {b-a}})}$, based on a probabilistic argument which follows from the assumption that f acts pseudorandomly. Note that when the size of the set {a, …, b} to be searched is measured in bits, as is normal in complexity theory, the set has size log(b − a), and so the algorithm's complexity is ${\scriptstyle O({\sqrt {b-a}})=O(2^{{\frac {1}{2}}\log(b-a)})}$, which is exponential in the problem size. For this reason, Pollard's lambda algorithm is considered an exponential time algorithm. For an example of a subexponential time discrete logarithm algorithm, see the index calculus algorithm.

Naming

The algorithm is well known by two names.

The first is "Pollard's kangaroo algorithm". This name is a reference to an analogy used in the paper presenting the algorithm, where the algorithm is explained in terms of using a tame kangaroo to trap a wild kangaroo. Pollard has explained that this analogy was inspired by a "fascinating" article published in the same issue of Scientific American as an exposition of the RSA public key cryptosystem. The article described an experiment in which a kangaroo's "energetic cost of locomotion, measured in terms of oxygen consumption at various speeds, was determined by placing kangaroos on a treadmill".

The second is "Pollard's lambda algorithm". Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda ($\lambda$). The shorter stroke of the letter lambda corresponds to the sequence $\{x_{i}\}$, since it starts from the position b to the right of x. Accordingly, the longer stroke corresponds to the sequence $\{y_{i}\}$, which "collides with" the first sequence (just like the strokes of a lambda intersect) and then follows it subsequently.

Pollard has expressed a preference for the name "kangaroo algorithm", as this avoids confusion with some parallel versions of his rho algorithm, which have also been called "lambda algorithms".

See also

• Rainbow table

References

Template:Number-theoretic algorithms x p and Template:In-text citations Banded Waveguides Synthesis is a physical modeling synthesis method to simulate sounds of dispersive sounding objects, or objects with strongly inharmonic resonant frequencies efficiently. It can be used to model the sound of instruments based on elastic solids such as vibraphone and marimba bars, singing bowls and bells. It can also be used for other instruments with inharmonic partials, such as membranes or plates. For example simulations of tabla drums and cymbals have been implemented using this method. Because banded waveguides retain the dynamics of the system, complex non-linear excitations can be implemented. The method was originally invented in 1999 by Georg Essl and Perry Cook to synthesize the sound of bowed vibraphone bars Template:Harv.

In the case of the standard one-dimensional wave equation $y_{tt}=c^{2}y_{xx}$ disturbances of all frequencies travel with the same constant speed $c$. In dispersive media the traveling speed of disturbances depends on their frequency and we get $c(\omega )$ where $\omega$ is the frequency of the disturbance. Many physical systems are dispersive, for example the elastic beams described by the Euler-Bernoulli beam equation $y_{tt}=ky_{xxxx}$ where $k$ is a material constant.

Banded waveguides model dispersive behavior by splitting the propagation of disturbances into frequency bands. Each frequency band is modeled using a band-limited version of the standard digital waveguide method. Each frequency band is tuned to the resonant frequencies of the sounding object to be modeled to avoid any discretization error at the dominant and audible frequencies Template:Harv.

Banded waveguide synthesis is implemented in most available sound synthesis libraries and programs such as:

• STK
• ChucK
• pd and Max/MSP via PerCoLate
• RTCMix
• SuperCollider
• CSound
• Common Lisp Music

Sources

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Template:Sound synthesis types x p are the free variables of p except x.

References

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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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• First-Order Logic: Syntax and Semantics