Specifications by Axioms

We first define a possible language for specifications by axioms, and this in the logic programming framework. Later in this section, we list alternative approaches. 

Definition 2-1 An axiomatic specification (or specification by axioms) of a procedure for predicate rhi consists of the union of a set of first-order axioms about rhi and the axiomatic specifications of all the non-primitive predicates used in these axioms. 

The first building block is the actual specification formalism used to elaborate the input to algorithm synthesis. An arbitrary choice has been made here to investigate synthesis from incomplete specifications. Starting from the pros and cons of specifications by examples, and of specifications by axioms, as outlined in Part I, we define, in Section 6.1, a specification approach that is based on examples and properties. Then, in Section 6.2, we illustrate this approach on a few sample problems. Future work and related work are discussed in Section 6.3 and Section 6.4, respectively, before drawing some conclusions in Section 6.5. 

When contrasting the pros and cons of specifications by axioms (as seen in Section 2.1) and specifications by examples (as seen in Section 3.1), it turns out that these specification approaches are quite complementary, each alleviating the drawbacks of the other by its own advantages. This gives rise to the idea of combining these two specification approaches so as to preserve only their benefits, while diminishing their disadvantages. But we must bear in mind the fundamental difference between the two approaches, namely specification completeness and specification incompleteness. 

Merely amalgamating the two formalisms into specifications by axioms and examples is thus not a good idea, as the incomplete example-set would only play an illustrative side-role to the complete axiom-set. 

In terms of related work, specifications by examples are surveyed in Section 3.1. Also, the notion of specifying property can be traced back to the notion of specifying axiom, and specifications by axioms are surveyed in Section 2.1. In this section, we only survey research on extending example-based specifications by another incomplete information source. 

But, on the other hand, one could aim at incomplete specifications and add some stronger form of axioms to the examples so as to overcome the weaknesses of specifications by examples only. We call properties [Hener and Deville 92, 93ab] such a strong form of axioms, and only require them to be written in (some subset of) logic. Indeed, until Part III, we do not restrict ourselves to any syntax or required computational content of properties. We only assume that properties are an incomplete source of information. Actually, in case properties were a complete source of information, most of the results hereafter would remain valid, but not always be relevant. 

Figure 8 shows the trace of the application of Axiom- , Axiom-2, Rule-l, Rule-2, Rule-3, and Rule-4 in our specific scheduling problem. The trace consists of a repeated pattern of rule applications Rule-4 followed by either Rule-2 or Rule-3. At each step the intrasituational rule converts an end-time to a start-time, and then the start-time is matched to the... 

The axiom that environmental exposures are never to pure chemicals is matched by another that the mixtures are frequently so complex as to defy description. The few observations suggest that effects seen from mixtures may be due to one or two specific neurotoxic agents. Judgment must be exercised to curb bias and accept the most plausible attribution as the above studies illustrate. 

Types of Automation Theorem Provers - fully formal machine-checked proofs, in which the theorem prover attempts to produce a formal proof, given a description of tbe system, a set of logical axioms and a set of inference rules. Model Checkers - automated proof of model against tbe specification, in which the model checker verifies certain properties by means of a search of possible states of a system. 

A polynomial algorithm based on the five axioms, algorithm MSG [11,12] yields a mathematically rigorous but the simplest superstructure, i.e., the maximal structure. The maximal structure of synthesis problem comprises all the combinatoriaUy feasible structures capable of yielding the specified products from the specified raw materials. Certainly the optimal network or structure is among these feasible structures. These flowsheets range from the simplest to the most complex, or complete, which is represented by the maximal structure itself. Obviously, the optimal structure in terms of a specific objective function, often the cost, is contained in the maximal structure nevertheless, the simplest is not necessarily the optimal. 

We define a bridge between EMOF- and Ecore- metamodels and NEREUS. The NEREUS specification is completed gradually. First, the signature and some axioms of classes are obtained by instantiating reusable schemes. Associations are transformed by using a reusable component ASSOCIATION. Next, OCL specifications are transformed using a set of transformation rules and a specification that reflects all the information of MOF metamodels is constructed. 

Second, specific reasons must exist in order to believe that the proposed process is incomputable. Since solving the halting problem is known to be incomputable and adding axioms is incomputable by definition (otherwise they would be theorems), then specific evidence indicates that the proposed process is incomputable. 

The hard part then comes in testing the theory. Because the results are incomputable, and not even likely reducible to a probability distribution, testing it is more difficult. In the case of computable causes, a specific end-point prediction can be established by computation, and then the result can be validated against that computation. In this case, the result is not computable, and therefore validation is more difficult. Validation will often be based on the qualitative description of the process rather than a quantitative prediction. Parts of it may still be quantifiable, but only with difficulty. For instance, to test the example presented, a method of identifying and counting the number of axioms within a person s mind is needed in order to come up with a quantifiable prediction. However, since this is not possible, it can only be tested based on secondary quantifications. Thus, testability on proposed oracles becomes much more dialectic. 

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