PATH VERIFICATION

Thus the path verification condition is required to be true whenever the initial condition A holds and the path Is followed and condition B holds at the end of the path it is required to be false whenever the initial condition A holds and the path is followed but condition B is false at the end of the path. 

If the flowchart P has a loop-free graph - if P is a tree - then the construction of W(P,A,B) is now quite simple. If P is loop-free there are only a finite number of paths 0, ...,on from START to STOP which are consistent and hence execution sequences. The input condition A(X) is a function only of the inputs, of course, while the output condition B(X,Y) can be regarded as a function of the input and of the final values of all the program variables (some of these values, of course, may play no role in the statement of the condition). Notice that under these conditions, when is a complete execution sequence from START to STOP, the path verification condition VCPjO AB, ) for any interpretation I is a function of the input X alone. 

We wish to see that for any choice of X = a and Y = b each path verification condition in W(P,A,B,I) holds. That is, we must examine each V(P,a,At r,I)(a,b) where o is a consistent path from tagged point t to tagged point r not passing through any other tagged point en route from t to r. If the hypothesis of the conditional expression V(P,a,A, A r,I)(a,b) is false, then the verification condition is vacuously true. If it is true, then A (a,b) is true and by definition of Aj, A(a) is true and computation (P,I,a) at some point enters tagged point t with Y = b. Further a is the continuation of this computation and reaches r with Y = b. So there is certainly a time when computation (P,I,a) reaches r with this specification of Y. Now if r is not a STOP statement, inductive assertion was assigned by our definition and thus Ar(a,b ) holds by definition. 

We have already seen that each of these path verification conditions holds for all choices of x,y15y2 and z. Since V x(U a V) is always logically equivalent to (V x U) a (V x V), we lave shown that W(P,A,B,I) holds for all x,yy,y2 and z... 

There are 7 paths to verify one from START to STOP not passing through a, one from START to a, one from a to B, one from a to STOP and two from 0 back to B and one from S up to a. (Why is there no need to verify the path for which DIMENSION X(N) is false ) We leave It to the reader to check out these path verification conditions. 

Notice that this reverses the way we have been treating assignment statements in path verification conditions. Essentially we have been saying that to prove that B holds of the values after assignment x - E if A holds previously, one must... 

Observe that we have in this procedure worked out some of the steps previously left to the THEOREM PROVER, The previous procedure involves having the progranmer select a set of inductive assertions and critical points, and then feed this into the computer parts a VERIFICATION CONDITION GENERATOR and a THEOREM PROVER. In this alternative construction we still need inductive assertions as the nature of the Rule of Iteration for WHILE statements shows. Now the inductive assertions are fed directly into the THEOREM PROVER which las been augmented by the special axioms and rules D0,D1,D2,D3 and D4 in addition to all of the usual arithmetic axioms, rules of inference, rules for handling identities and special axioms for the primitives in question (such as the factorial axioms in our example). In effect the THEOREM PROVER works backwards from the output condition and the various inductive assertions using DO - D3 to find what amounts to path verification conditions -... 

It does not seem clear which procedure is easier to implement or more efficient (if and when it works ). Both can be augmented to handle other constructions in any particular programming language. The first procedure is augmented by changing the PATH VERIFICATION CONDITION GENERATOR and the second by adding additional axioms or rules to the THEOREM PROVER. 

Tool path verification graphically simulating the tool motion to ensure that the tool path has no collision with the fixture and the machine and produces correct part geometry. 

Polyurethane board can include a low density foam block for applications from styling models to cutter path verification, and styling. Low density master models from which vacuum-consolidated RP tools and components can be produced. 

These statistics suggested that the data fit the measurement model reasonably well (df=1.327 TL1=0.993 CFI=0.995 NFI=0.981 RMSEA=0.048). It indicated that all of the measurement items in the theoretical model also significantly loaded on the constructs on which they were loading. Table 2 shows the result of path verification and explains the proposed hypotheses ... 

This can be illustrated by showing the net work involved in various adiabatic paths by which one mole of helium gas (4.00 g) is brought from an initial state in whichp = 1.000 atm, V= 24.62 1 [T= 300.0 K], to a final state in whichp = 1.200 atm, V= 30.7791 [T= 450.0 K]. Ideal-gas behaviour is assumed (actual experimental measurements on a slightly non-ideal real gas would be slightly different). Infomiation shown in brackets could be measured or calculated, but is not essential to the experimental verification of the first law. 

Despite these limitations, mobile monitoring systems have been used to obtain useful information, such as the verification and tracking of the St. Louis, Missouri, urban plume. The measurement of a well-defined urban plume spreading northeastward from St. Louis is shown in Fig. 15-2 (7). These data were collected by a combination of instrumented aircraft and mobile vans. Cross-sectional paths were flown by the aircraft at increasing distances downwind. Meteorological conditions of low wind speed in the same direction helped to maintain this urban plume in a well-defined... 

We assume that we are given a flowchart scheme P and a path a in P. This path may be from START to STOP or it may be from any statement in P to any other statement in P. We assume that we are given an initial predicate A and final predicate B. We wish to construct mechanically - i.e. by an algorithm which can be implemented on a computer - a VERIFICATION CONDITION V(P,a,A,B) with the following property. We assune now that input variables X and program variables Y are disjoint. 

The verification condition we shall construct will be vacuously true if a is not in fact a consistent path or if A does not hold at the start of the path. 

Observe that if a is not a consistent path, then the hypothesis in formula V(P,o,A,B) is always false, i.e., inconsistent, and so V(P,0,A,B) is always true. Hence in this case, V(P,a,A,B) trivially satisfies the definition of a verification condition. 

I(V(P,cr,A,B)) which can be regarded as the corresponding verification condition for path o in program (P,I) with initial condition 1(A) and terminal condition KB). ... 

We consider three paths. Path is the path from a to g which reaches 6 only at the end and otherwise does not pass through 0. Path a2 is the path from 0 back to 6 once while a3 is the direct one step path from B to y Path encounters no tests so the Q sets are empty and the verification condition V(P,0-, Aa,Ag) is constructed as ... 

Path a2 encounters test T(y2) and is followed only when the outcome of the test is FALSE, so y2 is in Q(a2,T,FALSE). Thus the verification condition V(P,o2jAg) for path a2 is ... 

To illustrate what is happening, we write out the paths and verification conditions in terms of A, B and it is left to the reader to check out the actual conditions. The set of tagged points is S = START, STOEL, STCSP2, STOP3, Tagl. ... 

Verification of the destruction of mustard in HD hydrolysates has not presented the same technical challenges as VX, but it does require the use of NMR analysis, which takes four to six hours to measure both mustard and sulfonium ions (U.S. Army, 1998b). Verification of agent destruction also constitutes a critical path item in the operational cycle of each facility. Currently, analysis of each batch of hydrolysate takes six hours, provided that reliable analytical results are obtained from the first analysis. Thus, reducing the time required to verify agent destruction in process streams would significantly improve the overall processing efficiency and schedule. 

Avoid using pipe threads whenever possible because they create long leak paths. Long, thin leak paths require long time periods for leak verifications. If pipe threads must be used, do not use Teflon tape because it voraciously absorbs helium and will later thwart the use of a helium leak detector. Rather, use a small amount of thread sealant paste that contains Teflon (see Fig. 7.54). Glyptol, a common temporary leak sealant, is sometimes used as a pipe sealant. This sealant is not recommended because it ages and, after some six months or so, can develop a leak. 

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