In the first part of the paper, flow diagrams are introduced to represent inter ah mappings of a set into itself. Although not every diagram is decomposable into a finite numbm of given base diagrams, this becomes hue at a semantical level due to a suitable extension of the given set and of the basic mappings defined in it. Two normalization methods of flow diagrams are given. The first has |hree base diagrams; the second, only two.In the second part of the paper, the second method is applied to 'lhe theory of Turing machines. With every Turing maching provided with a two-way half-tape, ihere is associated a similar machine, doing essentially 'lhe same job, but working on a tape obtained from the first one by interspersing alternate blank squares. The new machine belongs to the family, elsewhere introduced, generated by composition and iteration from the two machines X and R. That family is a proper subfamily of the whole family of Turing machines.
We formalize a technique introduced by B\"{o}hm and Piperno to solve systems of recursive equations in lambda calculus without the use of the fixed point combinator and using only normal forms. To this aim we introduce the notion of a canonical algebraic term rewriting system, and we show that any such system can be interpreted in the lambda calculus by the B\"{o}hm - Piperno technique in such a way that strong normalization is preserved. This allows us to improve some recent results of Mogensen concerning efficient g\"{o}delizations $\godel{~}: \Lambda \rightarrow \Lambda$ of lambda calculus. In particular we prove that under a suitable g\"{o}delization there exist two lambda terms $\bf E$ (self-interpreter) and $\bf R$ (reductor), both having a normal form, such that for every (closed or open) lambda term $M\;$ ${\bf E}\godel{M} \reduce M$ and if $M$ has a normal form $N$, then ${\bf R}\godel{M} \reduce \godel{N}$
Lambda-calculus is extended in order to represent a rather large class of recursive equation systems, implicitly characterizing function(al)s or mappings of some algebraic domain into arbitrary sets. Algebraic equality will then be represented by A~g-convertibility (or even reducibility). It is then proved, under very weak assumptions on the structure of the equations, that there always exist solutions in normal form (Interpretation theorem). Some features of the solutions, like the use of parametric representations of the algebraic constructors, higherorder solutions by currification, definability of functions on unions of algebras, etc., have been easily checked by a first implementation of the mentioned theorem, the CuCh machine. * This work has been pal'tially supported by grahts from ESPRIT BRA 7232 working group "Gentzen" and from MURST 40% (Italy).
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.