Finite Projective Formulas
Nick Arevadze
Abstract:
The notion of projective formula was introduced by Ghilardi [8] in
1999. Let us denote by P_n a set of fixed p_1, ..., p_n propositional
variables and by \Phi_n all equivalence classes of intuitionistic
formulas with variables in P_n. Consider a substitution \sigma : P_n
-> \Phi_n and extend it to all of \Phi_n by \sigma(\phi(p1, ..., pn))
= \phi(\sigma(p_1), ..., \sigma(p_n)). Now, a formula \phi \in \Phi_n
is called projective if there exists a substitution \sigma : \Phi_n ->
\Phi_n such that |- \sigma(\phi) and \phi |- \psi <-> \sigma(\psi),
for all \psi \in \Phi_n. In this paper we study projective formulas
from the relational and algebraic semantical point of view.
We show a close connection between projective formulas and projective
Heyting algebras (for definition see Section 4). Namely, to each
finitely generated projective Heyting algebra there corresponds a
projective formula; to non-isomorphic finitely generated projective
algebras there correspond nonequivalent projective formulas, but there
can be non-equvalent projective formulas which correspond to
isomorphic projective algebras. To a fixed n-generated projective
Heyting algebra H there correspond as many projective formulas as
there are di®erent retractions between H and \Phi_n (\Phi_n being the
free n-generated Heyting algebra and H its retract). We have a
one-to-one correspondence between projective formulas and (H; ir)
couples, where H is a projective algebra, and i; r the retractions.
Ghilardi (together with the results of Dick de Jongh and Albert
Visser) has shown in [8] that projective formulas are (the same as)
the exact and extendible formulas introduced earlier by de Jongh and
Visser. A formula \phi \in \Phi_n is called extendible if for all
finite, rooted models M_1, ..., M_k which force \phi, there can be
defined a valuation on the model M - obtained by adding a point as a
new root to the disjoint union of the M_i's { in such a way that M |=
\phi.
A subset A of the n-universal model M_n (see Section 3) is called
admissible if there exists a formula \phi \in \Phi_n such that A = {w
\in M_n : w |= \phi}. A set A is called extendible if for any
anti-chain in it, A contains at least one element totally covered by
this anti-chain. Now, a formula is extendible i® its corresponding
admissible set is extendible. Then we proceed with characterizing the
admissible extendible subsets of the n-universal model. Here the
problem of the characterization of infinite admissible extendible sets
seems rather complicated for n greater than one; we just give an
alternative proof of the fact that there is an infinite number of
them.
However, it is easier to approach the finite admissible extendible
subsets of the n-universal model which correspond to the so-called
finite projective formulas. All finite subsets of the n-universal
model are admissible (Grigolia [11], de Jongh [5],[6]) and a necessary
condition for them to be extendible is that their widths should be
less than or equal to two and their depths less than or equal to n +
1. Using this fact, we write out all the finite extendible subsets of
the 2-universal model - there are 26 such - and present their
corresponding finite projective formulas.
In addition, we give a combinatorial formula which is a counter of the
number of finite projective formulas of n variables. A computer
program which realizes this combinatorial formula is attached to the
thesis. We have conducted calculations for n ranging from 1 to 6. The
finite formulas of one variable were found by Dick de Jongh (see [4]),
there are four of them. As we mentioned above, there are 26 finite
formulas of two variables; for n = 3 there are 256 finite formulas;
for n = 4: 3386; for n = 5: 55984 and for n = 6: 1110506.