Non-recursive

Retrieved Source of formula and raw material data for a non-recursive modelling procedure Formula characteristics Raw material characteristics ... 

The table below illustrates these issues by comparing how a recursive subroutine must handle data which is available from a database, such as the cost of a raw material, data that is calculated for the formulated product, such as PBR, and data for intermediate products. (The variable names shown in the table are part of the example procedure given in the appendix.) Compare with the previous table for a non-recursive modelling procedure s data structure. 

The correction for non-uniform instrumental broadening in SEC is solved through a non-recursive matrix stochastic technique. To this effect, Tung s equation ( ) must be reformulated in matrix form, and the measurements assumed contaminated with zero-mean noise. 

The Structurally Recursive Method is then expanded, and a second, non-recursive algorithm fw the manipulator inertia matrix is derived from it A finite summation, which is a function of individual link inertia matrices and columns of the propriate Jacobian matrices, is defined fw each element of the joint space inertia matrix in the Inertia Projection Method. Further manipulation of this expression and application of the composite-rigid-body inertia concept [42] are used to obtain two additional algwithms, the Modified Composite-Rigid-Body Method and the Spatial Composite-Rigid-Body Method, also in the fourth section. These algorithms do make use of recursive expressions and are more computationally efficient. 

Parks, T.W. and McClellan, J.H. 1972a. Chebyshev approximations for non recursive digital filters with linear phase. IEEE Trans. Circuit Theory CT-19 189-194. 

Sometimes, the line between what and how is non-existent for instance, the specification of sameFirst/2 is non-recursive, but it nevertheless also incorporates a way of how to solve the problem. 

Several aspects of logic algorithm design can be discussed now. We first propose a useful terminology for logic algorithm classification, and then show in what sense minimal cases and non-recursive non-minimal cases, though syntactically similar, are totally different concepts. 

As the logic algorithms of this book exhibit, not all non-minimal cases are recursive. Indeed, prefix-traversal logic algorithms have non-recursive non-minimal cases. But some structural cases of logic algorithms in Chapter 4, or [Deville 90], seem hard to classify. There are several potential reasons to this. 

A first reason is that non-recursive non-minimal cases of prefix-scan logic algorithms can often be merged with their minimal cases. The resulting algorithm looks like it has no minimal case. It exhibits cases whose structural forms are not mutually exclusive, and is thus easy to detect as being a rewriting of the canonical version. 

The form L = L J is not minimal, as it overlaps with the other form. It rather results from a merger of the minimal case and the non-recursive non-minimal case. ... 

Another reason for non-recursive non-minimal cases is recursion elimination by partial evaluation. The result is a non-minimal case that looks like a minimal case. This is hard to detect, since the cases still exhibit mutually exclusive structural forms. There is no limit to creating non-recursive non-minimal cases by partial evaluation. 

It is important to understand that such logic algorithms with non-recursive, non-minimal cases are the result of re-writing canonical logic algorithms, rather than unpleasant aberrations. 

Let s illustrate the notion of specifications by examples and properties by a few sample specifications (others may be found in Section 14.4.2). The chosen language for properties is, strictly for the sake of illustration, non-recursive normal clauses that have a head with the predicate of the examples. Universal quantifiers are usually dropped for convenience. Also, negative examples are not allowed. Note that most of this Part II is independent of such choices. Our claim is that such properties and examples, if carefully chosen, embody the minimal knowledge that doesn t give away the solution, but is sufficient for successful algorithm design. 

For non-recursive literals, since they are assumed to be primitives, it is possible to compute the witnesses without knowing We thus only need an oracle for the specified predicate rin. A candidate mechanized oracle is one that performs deduction using the specification EP(r) as knowledge about rIn, Such a deductive oracle is sound provided EP(f) is consistent with % Other oracles can be imagined, performing analogical reasoning, say. In the sequel we just assume the existence of such an oracle, but do not require it to be sound or complete. A more practical version of the previous definition is thus ... 

But this schema is very lengthy, and doesn t sufficiently show the commonalities between the recursive and the non-recursive sub-cases. We thus syntactically merge... 

For a non-recursive property Pj, a normalized successful derivation is an instance of the following derivation schema ... 

Establishing the complexity of the method is quite intricate, unless some assumptions are made. Thus, assume that properties are non-recursive, and that all primitives used in T r) are deterministic, and let c be the number of disjuncts in LA r), p the number of properties of P r), and t the number of recursive atoms in LA(r). There are p proofs to be made. Each proof-tree has only two choice-points, namely the DCI resolution of the head of the initial goal, where there are c possibilities, and the DCI resolution of the t recursive atoms, where there are p - 1 possibilities, all other proof steps being deterministic. Each proof tree thus has size 0(cp ). Hence, the time complexity of the method is 0(cp ). This assumes that there is a fixed maximum number of atoms for the definitions of the used primitives. Indeed, if that number is a function of c, t, or p, then this complexity analysis doesn t hold. 

Let s close in now on the MSG Method. Intuitively, its objective is to infer a logic algorithm of a predicate r n, given a finite set of examples of rhi The method should be applicable if the intended relation (from which the examples are extracted) can be expressed by a logic algorithm that is defined solely in terms of the =/2 primitive (hence is non-recursive, among others). This is feasible iff, in the intended relation, some parameters are somehow syntactically constructed from some other parameters. 

In terms of alternative techniques to the MSG Method, there is of course the whole literature on empirical learning, and inductive logic programming (ILP), as surveyed in Chapter 3. But it should be noted that the MSG Method only aims at the synthesis of a sub-class of concept descriptions, namely non-recursive algorithms that are implemented in terms of the =/2 primitive only, and whose intended relation, though unknown as a whole, is however known to feature a given dataflow pattern between... 

Within a restricted setting, the MSG Method infers, from a finite set of general examples, a non-recursive logic algorithm that is defined in terms of the =/2 primitive only, and that is correct wrt a natural extension of the given examples. The underlying algorithm is non-deterministic. 

This amounts to the support of binary predicates, single induction parameters, a single minimal form, a single non-minimal form, and non-recursive non-minimal cases. 

An instantiation of the Solve (respectively SolveNonMinj ) predicate-variable computes, in the minimal case (respectively the non-recursive non-minimal case), the value of the other parameter Y from the induction parameter X. Step 5 yields LA r) by using similar methods to those of Steps 6 and 7. 

For instance, in case L is the empty list, its compression C is the empty list as well. This is performed by the atomic formula C = []. There is no non-recursive non-minimal case, and hence no need to instantiate some SolveNonMinj. LA (compress) is thus as follows ... 

This amounts to splitting the non-minimal case into two non-mandatory cases, called the non-recursive non-minimal case and the recursive non-minimal case, respectively. For convenience, we drop the qualifier non-minimal from these two names. Recursion may be useless in the sense that the recursively computed TY are not needed for the computation of Y. Useless recursion wouldn t affect the correctness of a logic algorithm, though. Its elimination is thus rather a matter of algorithm optimization. The following objectives of the strategy ... 

Task H In the non-recursive case, delete every disjunct that is an expansion wit an example that is also covered by the recursive case. Indeed, every example covered by the recursive case in LA r) is also covered by the non-recursive case (but the converse doesn t hold) this amounts to eliminating unwanted redundancy ... 

Task I In the recursive case, delete every disjunct where the general example procComp ,,yj) would have no admissible alternatives wit a = / = 1, and expand the non-recursive case wit the corresponding example Indeed, if yj is not somehow constructed in terms of tyj, then recursion is useless for testing y. 

This amounts so far to splitting the sequence of examples S2 (see Step 2) into two complementary (but possibly empty) sub-sequences 21 and S22> such that the examples of 21 covered by the disjuncts of the non-recursive case, whereas the examples... 

The objective at Step 5 is to instantiate the predicate-variables Solve and SolveNonMini of the divide-and-conquer schema. The number v of sub-cases of the non-recursive case must also be found. This amounts to transforming LA (r) into LA5(r) such that it is covered by the following schema ... 

A first idea is to use the MSG Method (see Chapter 10). Note that, in both the minimal and the non-recursive case, Y may be totally independent of X. Indeed, Y could be constructed in terms of ... 

Task P Split E into two complementary (but possibly empty) sub-sets E and E, such that every example of E is admissible wrt a=0 and b =1. The set E is relative the non-recursive examples of r n where Y is effectively constmcted from all the constituents of the decomposition of X, while the set E is relative the non-recursive examples of r n where Y is not constructed from all the constituents of ... 

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