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.