# Binary diagram wiki

It is not so simple to convert from an arbitrary network of logic gates to a BDD [ citation needed ] unlike the and-inverter graph. The size of the BDD is determined both by the function being represented and the chosen ordering of the variables.

It is of crucial importance to care about variable ordering when applying this data structure in practice. The problem of finding the best variable ordering is NP-hard. There are functions for which the graph size is always exponential — independent of variable ordering.

Many logical operations on BDDs can be implemented by polynomial-time graph manipulation algorithms: However, repeating these operations several times, for example forming the conjunction or disjunction of a set of BDDs, may in the worst case result in an exponentially big BDD. From Wikipedia, the free encyclopedia. Bell System Technical Journal, Boute, "The Binary Decision Machine as a programmable controller". Rudell and Randal E. IT University of Copenhagen.

Binary decision diagram Directed acyclic graph Directed acyclic word graph. List of data structures. Retrieved from " https: Diagrams Graph data structures Model checking Boolean algebra. Additional thermodynamic quantities may each be illustrated in increments as a series of lines - curved, straight, or a combination of curved and straight. Each of these iso- lines represents the thermodynamic quantity at a certain constant value. It is possible to envision three-dimensional 3D graphs showing three thermodynamic quantities.

Such a 3D graph is sometimes called a p — v — T diagram. The equilibrium conditions are shown as curves on a curved surface in 3D with areas for solid, liquid, and vapor phases and areas where solid and liquid, solid and vapor, or liquid and vapor coexist in equilibrium. A line on the surface called a triple line is where solid, liquid and vapor can all coexist in equilibrium.

The critical point remains a point on the surface even on a 3D phase diagram. For water, the 3D p — v — T diagram is seen here: An orthographic projection of the 3D p — v — T graph showing pressure and temperature as the vertical and horizontal axes collapses the 3D plot into the standard 2D pressure—temperature diagram.

When this is done, the solid—vapor, solid—liquid, and liquid—vapor surfaces collapse into three corresponding curved lines meeting at the triple point, which is the collapsed orthographic projection of the triple line. Other much more complex types of phase diagrams can be constructed, particularly when more than one pure component is present.

In that case, concentration becomes an important variable. Phase diagrams with more than two dimensions can be constructed that show the effect of more than two variables on the phase of a substance.

Phase diagrams can use other variables in addition to or in place of temperature, pressure and composition, for example the strength of an applied electrical or magnetic field, and they can also involve substances that take on more than just three states of matter.

One type of phase diagram plots temperature against the relative concentrations of two substances in a binary mixture called a binary phase diagram , as shown at right. Such a mixture can be either a solid solution , eutectic or peritectic , among others. These two types of mixtures result in very different graphs. Another type of binary phase diagram is a boiling-point diagram for a mixture of two components, i.

For two particular volatile components at a certain pressure such as atmospheric pressure , a boiling-point diagram shows what vapor gas compositions are in equilibrium with given liquid compositions depending on temperature. In a typical binary boiling-point diagram, temperature is plotted on a vertical axis and mixture composition on a horizontal axis.

A simple example diagram with hypothetical components 1 and 2 in a non- azeotropic mixture is shown at right. The fact that there are two separate curved lines joining the boiling points of the pure components means that the vapor composition is usually not the same as the liquid composition the vapor is in equilibrium with. See Vapor—liquid equilibrium for more information. In addition to the above-mentioned types of phase diagrams, there are thousands of other possible combinations.

Some of the major features of phase diagrams include congruent points, where a solid phase transforms directly into a liquid. There is also the peritectoid , a point where two solid phases combine into one solid phase during cooling. The inverse of this, when one solid phase transforms into two solid phases during cooling, is called the eutectoid. The x-axis of such a diagram represents the concentration variable of the mixture. Phase diagrams can use other variables in addition to or in place of temperature, pressure and composition, for example the strength of an applied electrical or magnetic field, and they can also involve substances that take on more than just three states of matter.

One type of phase diagram plots temperature against the relative concentrations of two substances in a binary mixture called a binary phase diagram , as shown at right. Such a mixture can be either a solid solution , eutectic or peritectic , among others. These two types of mixtures result in very different graphs. Another type of binary phase diagram is a boiling-point diagram for a mixture of two components, i.

For two particular volatile components at a certain pressure such as atmospheric pressure , a boiling-point diagram shows what vapor gas compositions are in equilibrium with given liquid compositions depending on temperature.

In a typical binary boiling-point diagram, temperature is plotted on a vertical axis and mixture composition on a horizontal axis. A simple example diagram with hypothetical components 1 and 2 in a non- azeotropic mixture is shown at right. The fact that there are two separate curved lines joining the boiling points of the pure components means that the vapor composition is usually not the same as the liquid composition the vapor is in equilibrium with.

See Vapor—liquid equilibrium for more information. In addition to the above-mentioned types of phase diagrams, there are thousands of other possible combinations. Some of the major features of phase diagrams include congruent points, where a solid phase transforms directly into a liquid. There is also the peritectoid , a point where two solid phases combine into one solid phase during cooling.

The inverse of this, when one solid phase transforms into two solid phases during cooling, is called the eutectoid. The x-axis of such a diagram represents the concentration variable of the mixture. As the mixtures are typically far from dilute and their density as a function of temperature is usually unknown, the preferred concentration measure is mole fraction.

A volume-based measure like molarity would be inadvisable. Polymorphic and polyamorphic substances have multiple crystal or amorphous phases, which can be graphed in a similar fashion to solid, liquid, and gas phases. Some organic materials pass through intermediate states between solid and liquid; these states are called mesophases. Attention has been directed to mesophases because they enable display devices and have become commercially important through the so-called liquid-crystal technology.

Phase diagrams are used to describe the occurrence of mesophases. From Wikipedia, the free encyclopedia. For the use of this term in mathematics and physics, see phase space. Phase Diagrams and Heterogeneous Equilibria: The Physics of Phase Transition: General Chemistry 4th ed. The Study of Matter Prentice Fourth ed. Principles of General Chemistry.