Clausal Language (ver. 5.81.16, by P.J. Voda, J. Komara, J. Kluka)
[CL] Syntax. See the module Mclsyntax in the file mclsyntax.cl
.
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[CL] Remark. Catalan pairing function - see the module
.
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[CL] Remark. Clausal language is allowed.
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Chapter. Recursive Functions.
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Section. Beyond Primitive Recursion.
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Subsection. Universal Function for Primitive Recursive Functions is not
Primitive Recursive.
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Arithmetization of p.r. function symbols. See also the module
.
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Primitive recursive indices. See also the predicate
holding of well-formed p.r. indices in the module
.
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Effective operations on p.r. indices.
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Exercise. [Uloha (1 bod)] Define the function
such that
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Exercise. Define the binary primitive recursive function
satisfying
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Primitive recursive terms.
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Arithmetization of p.r. terms. See also the module
.
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Denotation of p.r. terms.
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Explicit definitions. [Uloha (1 bod)] Define the binary function
such that
-
for every
, if the number
is the code of a p.r. term
with all its free variables indicated then the application
yields a p.r. index of the n-ary p.r. function
explicitly defined by
or equivalently, we wish to have
for every
-tuple
.
Remark. You will need the unary function
yielding the arity of the p.r. function symbol
. The function is defined in the module Mprf .
Remark. Clausal language allowed.
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Exercise. Find p.r. indices of
,
and
with the help of
.
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Iteration. Define a function
such that
In other words, if
is a p.r. index of a unary function then
is a p.r. index of its iteration.
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Exercise. [Uloha (1 bod)] Find a p.r. index of
which corresponds to the p.r. derivation of addition as an iteration of the
successor function.
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Parameterless primitive recursion. Define a function
satisfying
In other words, if
is a constant and
is a p.r. index of a binary function
then
is a p.r. index of the unary function
defined by following parameterless primitive recursion:
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Exercise. Find a p.r. index of
which corresponds to the derivation of the predecessor function by parameterless
recursion.
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Exercise. [Uloha (1 bod)] Find a p.r. index of
which corresponds to the p.r. derivation of modified subtraction as an iteration
of the predecessor function.
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Parametric function (Kleene). Define the function
such that
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Exercise. [Premiova uloha (1 bod)] Prove the Kleene's
s-m-n theorem for the case
and
by providing a ternary primitive recursive function
satisfying
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Exercise. Define a primitive recursive function
satisfying
In other words, if
is a p.r. index of a binary function then
is a p.r. index of its diagonalization.
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Exercise. [Premiova uloha (1 bod)] Show that the binary function
is not primitive recursive, that is
or equivalently
by providing a primitive recursive function
satisfying
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