Question & Answer: Define next state function in terms of description and as a state transition diagram or a mathemat…..

Define next state function in terms of description and as a state transition diagram or a mathematical function or pseudo code, define time scale for your model.

Pattern Recognizer (as a register)

A Pattern Recognizer that accepts single digit inputs (0 or a 1) each time unit and outputs a NO (as a 0) until the pattern 101101 has been seen in the stream of inputs then a YES (as a 1) will be output.

A register that hold the last several digits received and shifts in a new digit when received and shift out the oldest digit.


Z = (IZ, SZ, OZ, NZ, RZ)

IZ = {0, 1}

SZ = {x1, x2, x3, x4, x5, x6}; xk={0, 1}, j=1, 2, 3, 4, 5, 6}

OZ = {YES, NO}

NZ={((x, p), y):x ∈SZ;P ∈IZ;y ∈SZ;if x=(x1, x2, x3, x4, x5, x6), then y=(P, x2, x3, x4, x5, x6)}

RZ = {(x, q):x ∈SZ;q ∈OZ ;if x=(1, 0, 1, 1, 0, 1) then q=YES, else q=NO}

Question: ?

Run two system experiments to demonstrate that your pattern recognizer works.

Use the following input trajectories:

f1={(0,1),(1,1),(2,0),(3,1),(4,1),(5,0),(6,1),(7,0),(8,1),(9,1),(10,0),(11,1),(12,1),(13,0), (14,0), (15,0),(16,1)}

f2={(0,0),(1,1),(2,1),(3,1),(4,0),(5,0),(6,1),(7,1),(8,0),(9,0),(10,1),(11,1),(12,0),(13,0), (14,0), (15,1),(16,1)}

Start your system in the state (initial) that represents “None”, or the (0,0,0,0,0,0,0) state, of the digits of the pattern have been observed and run your experiments for 16 time steps

Expert Answer


A thermodynamic system is described by a number of thermodynamic parameters (e.g. temperature, volume, pressure) which are not necessarily independent. The number of parameters needed to describe the system is the dimension of the state space of the system (D). For example, a monatomic gas having a fixed number of particles is a simple case of a two-dimensional system (D = 2). In this example, any system is uniquely specified by two parameters, such as pressure and volume, or perhaps pressure and temperature. These choices are equivalent. They are simply different coordinate systems in the two-dimensional thermodynamic state space. Given pressure and temperature, the volume is calculable from them. Likewise, given pressure and volume, the temperature is calculable from them. An analogous statement holds for higher-dimensional spaces, as described by the state postulate.

Quite generally, a state function is on the form

F ( P , V , T , … ) = 0 , {displaystyle F(P,V,T,ldots )=0,} {displaystyle F(P,V,T,ldots )=0,}

where P denotes pressure, T denotes temperature, V denotes volume, and the ellipsis denotes other possible state variables like particle number N and entropy S. If the state space is two-dimensional as in the above example, one may visualize the state space as a three-dimensional graph (a surface in three-dimensional space). The labels of the axes are not generally unique, since there are more state variables than three in this case, and any two independent variables suffice to define the state.

When a system changes state continuously, it traces out a “path” in the state space. The path can be specified by noting the values of the state parameters as the system traces out the path, perhaps as a function of time, or some other external variable. For example, we might have the pressure P ( t ) {displaystyle P(t)} P(t) and the volume V ( t ) {displaystyle V(t)} V(t) as functions of time from time t 0 {displaystyle t_{0}} t_{0} to t 1 {displaystyle t_{1}} t_{1} . This will specify a path in our two dimensional state space example. We can now form all sorts of functions of time which we may integrate over the path. For example, if we wish to calculate the work done by the system from time t 0 {displaystyle t_{0}} t_{0} to time t 1 {displaystyle t_{1}} t_{1} we calculate

W ( t 0 , t 1 ) = ∫ 0 1 P d V = ∫ t 0 t 1 P ( t ) d V ( t ) d t d t . {displaystyle W(t_{0},t_{1})=int _{0}^{1}P,dV=int _{t_{0}}^{t_{1}}P(t){frac {dV(t)}{dt}},dt.} {displaystyle W(t_{0},t_{1})=int _{0}^{1}P,dV=int _{t_{0}}^{t_{1}}P(t){frac {dV(t)}{dt}},dt.}

It is clear that in order to calculate the work W in the above integral, we will have to know the functions P ( t ) {displaystyle P(t)} P(t) and V ( t ) {displaystyle V(t)} V(t) at each time t {displaystyle t} t , over the entire path. A state function is a function of the parameters of the system which only depends upon the parameters’ values at the endpoints of the path. For example, suppose we wish to calculate the work plus the integral of V d P {displaystyle VdP} VdP over the path. We would have:

Φ ( t 0 , t 1 ) = ∫ t 0 t 1 P d V d t d t + ∫ t 0 t 1 V d P d t d t = ∫ t 0 t 1 d ( P V ) d t d t = P ( t 1 ) V ( t 1 ) − P ( t 0 ) V ( t 0 ) . {displaystyle Phi (t_{0},t_{1})=int _{t_{0}}^{t_{1}}P{frac {dV}{dt}},dt+int _{t_{0}}^{t_{1}}V{frac {dP}{dt}},dt=int _{t_{0}}^{t_{1}}{frac {d(PV)}{dt}},dt=P(t_{1})V(t_{1})-P(t_{0})V(t_{0}).} Phi(t_0,t_1)= int_{t_0}^{t_1}Pfrac{dV}{dt},dt +int_{t_0}^{t_1}Vfrac{dP}{dt},dt =int_{t_0}^{t_1}frac{d(PV)}{dt},dt=P(t_1)V(t_1)-P(t_0)V(t_0).

It can be seen that the integrand can be expressed as the exact differential of the function P ( t ) V ( t ) {displaystyle P(t)V(t)} P(t)V(t) and that therefore, the integral can be expressed as the difference in the value of P ( t ) V ( t ) {displaystyle P(t)V(t)} P(t)V(t) at the end points of the integration. The product P V {displaystyle PV} PV is therefore a state function of the system.

By way of notation, we will specify the use of d to denote an exact differential. In other words, the integral of d Φ {displaystyle dPhi } dPhi will be equal to Φ ( t 1 ) − Φ ( t 0 ) {displaystyle Phi (t_{1})-Phi (t_{0})} Phi(t_1)-Phi(t_0) . The symbol δ will be reserved for an inexact differential, which cannot be integrated without full knowledge of the path. For example, δ W = P d V {displaystyle delta W=PdV} delta W=PdVwill be used to denote an infinitesimal increment of work.

It is best to think of state functions as quantities or properties of a thermodynamic system, while non-state functions represent a process during which the state functions change. For example, the state function P V {displaystyle PV} PV is proportional to the internal energy of an ideal gas, but the work W {displaystyle W} W is the amount of energy transferred as the system performs work. Internal energy is identifiable, it is a particular form of energy. Work is the amount of energy that has changed its form or location.

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