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Turing-completeness totally free
In this paper, I show that general recursive definitions can be represented in the free monad which supports the ‘effect’ of making a recursive call, without saying how these calls should be executed. Diverse semantics can be given within a total framework by suitable monad morphisms. The Bove-Capretta construction of the domain of a general recursive function can be presented datatype-generically as an instance of this technique. The paper is literate Agda, but its key ideas are more broadly transferable
On Hilbert's Tenth Problem
Using an iterated Horner schema for evaluation of diophantine polynomials, we
define a partial -recursive "decision" algorithm decis as a "race" for a
first nullstelle versus a first (internal) proof of non-nullity for such a
polynomial -- within a given theory T extending Peano Arithmetique PA. If T is
diophantine sound, i.e., if (internal) provability implies truth -- for
diophantine formulae --, then the T-map decis gives correct results when
applied to the codes of polynomial inequalities . The
additional hypothesis that T be diophantine complete (in the syntactical sense)
would guarantee in addition termination of decis on these formula, i.e., decis
would constitute a decision algorithm for diophantine formulae in the sense of
Hilbert's 10th problem. From Matiyasevich's impossibility for such a decision
it follows, that a consistent theory T extending PA cannot be both diophantine
sound and diophantine complete. We infer from this the existence of a
diophantine formulae which is undecidable by T. Diophantine correctness is
inherited by the diophantine completion T~ of T, and within this extension
decis terminates on all externally given diophantine polynomials, correctly.
Matiyasevich's theorem -- for the strengthening T~ of T -- then shows that T~,
and hence T, cannot be diophantine sound. But since the internal consistency
formula Con_T for T implies -- within PA -- diophantine soundness of T, we get
that PA derives \neg Con_T, in particular PA must derive its own internal
inconsistency formula
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