Formal languages

In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with meanings or semantics. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In logic and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way. The field of formal language theory studies primarily the purely syntactical aspects of such languages—that is, their internal structural patterns. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages. In the 17th century, Gottfried Leibniz imagined and described the characteristica universalis, a universal and formal language which utilised pictographs. During this period, Carl Friedrich Gauss also investigated the problem of Gauss codes.

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Related exercises (6)

Asymmetrical pendulum

A pendulum is made up of a marble of mass m connected to a fixed point O with a massless string of constant length. A nail is placed at a distance d from point O, vertically below, such that the pendulum as lengthxa0L when it goes right and lengthxa0du2005+u2005L when it goes left (see diagram). The marble is placed such that the string makes an anglexa0α (α

Car suspension problem

A car’s suspension is comfortable when it has a vertical oscillation period that our bodies are used to: the oscillation period of walking, Tu2004=u20040.8u2005s.

Find the spring constant k of the suspension in a car of mass $M = 1540\,{\rm kg}$ .
The car lowers when we place a trunk of mass $m = 70\,{\rm kg}$ in it. Find its vertical displacement.

To make the car more comfortable, we add a shock absorber/dampener that creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)

Excitation of a vertical harmonic oscillator

A mass m is suspended on a spring of natural length l₀ and of elastic constant k. The spring is attached to a piston. The vertical motion of the piston is described by the equation s(t)u2004=u2004s₀cosu2006(ωt)xa0.

Give explicit expressions for the external forces and determine the equation of motion.
Show that the equation of motion can be recast as, $\ddot{x}\left(t\right) + c_1\,x\left(t\right) = c_1\,s\left(t\right)\ ,$ where x(t)u2004=u2004z(t)u2005+u2005c₂ with z(t) the vertica

Conceptual question on angular momentum

A propeller-driven plane (the propeller, on the plane’s nose, as its angular momentum vector pointing in the plane’s motion direction) is horizontally flying and preparing to turn left. If the pilot does not compensate, will the plane tend to rise up or sink down upon turning? Explain your answer in terms of the angular momentum. A spinning top in the shape of a pine cone, as a small groove in which a piece of string is wound. The string is attached to a stick on which a child pulls to throw the top, in ra

Piston and Rod

The system is modelled by a wheel turning with a constant angular speed θ̇ and of a rod of length L where one extremity is fixed to the wheel and the other can slide on an axis passing through the centre of the wheel.

Determine the expression of the velocity V_(P) of an arbitrary point P of the rod in Cartesian coordinates as a function of the angles θ and α.
Show that the velocity V_(P) can be written as V_(P)u2004=u2004V_(A)u2005+u2005α̇u2005×u2005APxa0.
Show that the velocity V

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Related exercises (6)

Asymmetrical pendulum

Car suspension problem

A car’s suspension is comfortable when it has a vertical oscillation period that our bodies are used to: the oscillation period of walking, Tu2004=u20040.8u2005s.

Find the spring constant k of the suspension in a car of mass $M = 1540\,{\rm kg}$ .
The car lowers when we place a trunk of mass $m = 70\,{\rm kg}$ in it. Find its vertical displacement.

To make the car more comfortable, we add a shock absorber/dampener that creates a friction force F⃗u2004=u2004u2005−u2005bv⃗_(z), where v⃗_(z)

Excitation of a vertical harmonic oscillator

Give explicit expressions for the external forces and determine the equation of motion.
Show that the equation of motion can be recast as, $\ddot{x}\left(t\right) + c_1\,x\left(t\right) = c_1\,s\left(t\right)\ ,$ where x(t)u2004=u2004z(t)u2005+u2005c₂ with z(t) the vertica

Conceptual question on angular momentum

Piston and Rod

Determine the expression of the velocity V_(P) of an arbitrary point P of the rod in Cartesian coordinates as a function of the angles θ and α.
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Show that the velocity V

Formal languages

Chat with Graph Search

standard bookkeeping

interactivity library:

first our usual bookkeeping

standard bookkeeping

interactivity library:

first our usual bookkeeping

The Isoperimetric Problem in Regular Trees