Monad , definition

More specifically, the basic notions of a Turing Machine, of computable functions and of undecidable properties are needed for Chapter VI (Decision Problems) the definitions of recursive, primitive recursive and partial recursive functions are helpful for Section F of Chapter IV and two of the proofs in Chapter VI. The basic facts regarding regular sets, context-free languages and pushdown store automata are helpful in Chapter VIII (Monadic Recursion Schemes) and in the proof of Theorem 3.14. For Chapter V (Correctness and Program Verification) it is useful to know the basic notation and ideas of the first order predicate calculus a highly abbreviated version of this material appears as Appendix A. 

DEFINITION A program scheme P is monadic if all functions and tests appearing... 

DEFINITION For a monadic scheme P with a single output variable, the value language of P is the language... 

DEFINITION A recursion scheme is monadic if all its functions, basis, predicate and defined, are monadic. 

DEFINITION Let and p be monadic basis function letters and let T be a mona-... 

The pushdown stores considered in the previous chapter contain as individual items members of any domain. In the case of monadic schemes we can get away with a simpler and stricter definition of a store. We can regard a store as a special variable u whose value under any interpretation must be a member of T for a fixed vocabulary T and to which we can apply as functions only POP(u) (which erases the rightmost symbol, if any, of u regarded as the top of the stack) and PUSH(u,A) (which adds A to the top (right) of u for any A in T ). The only... 

We have shown that in the monadic case one simple pushdown store suffices. Similar to this definition of the augmentation of a flowchart scheme by a simple pushdown store one can define a counter as a reserved variable u whose values can only be non-negative integers and to which can only be applied the functions u + 1 and u - 1 and the predicate u = 0. As in the case of an added pushdown store, all assignments to or by u must be independent variable - that is u f(v) and v + f(u) are forbidden for v u and any f. ... 

To give a uniform definition, let us represent TRUE by 1 and FALSE by 0 and define the interpreted value language L (S) of a monadic recursion scheme S with r tests T-, ...,T as the set of all words of the farm ... 

THEOREM 8.4 The reversal of the interpreted value language of a monadic recursion scheme with r tests is an (r+1)-definite deterministic context-free language. 

Now (L (Pg)) is not 2-definite (or even k-definite for any k ) and so P3 cannot be translated into any strongly equivalent monadic recursion scheme. 

On this definition statement (1) expresses an external relation, since it can be inferred from the statements A has 10 dollars and B has 5 dollars. On the other hand we may defy anyone to come up with a similar reduction of the relational statement (2) to statements only ascribing monadic predicates to the relata. ... 

In the last chapter he refers to the dynamist school that opposes the atomic defenders, namely in the consideration that the chemical atoms are really extensible and actually indivisible. This would imply that prime atoms could also be indivisible and extensible. He deduces that, in this field, the position of the atomists is no safer, because if, on the one hand, the inextensible monad is incomprehensible, the philosophical atom, indivisible by definition, is counterintuitive, as indivisibility cannot coexist with extension. He also quotes du Bois Raymond s position on these subjects, namely if inside certain limits, the theory of atoms rends excellent services to the physico-mathematical analysis of phenomena and even until a certain point turns it indispensable, since we surpass these limits we exaggerate its... 

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