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ABSTRACT:In this paper typical properties of large random Boolean ANDrOR formulas are investigated. Such formulas with n variables are viewed as rooted binary trees chosen from the uniform distribution of all rooted binary trees on m nodes, where n is fixed and m tends to infinity. The leaves are labeled by literals and the inner nodes by the connectives ANDrOR, both uniformly at random. In extending the investigation to infinite trees, we obtain a close relation between the formula size complexity of any given Boolean function f and the probability of its occurrence under this distribution, i.e., the negative logarithm of this probability differs from the formula size complexity of f only by a polynomial factor.

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Product Logic Π is an axiomatic extension of Hájek's Basic Fuzzy Logic BL coping with the 1-tautologies when the strong conjunction & and implication → are interpreted by the product of reals in [0, 1] and its residuum respectively. In this paper we investigate expansions of Product Logic by adding into the language a countable set of truth-constants (one truth-constant r for each r in a countable Π-subalgebra C of [0, 1]) and by adding the corresponding book-keeping axioms for the truth-constants. We first show that the corresponding logics Π(C) are algebraizable, and hence complete with respect to the variety of Π(C)-algebras. The main result of the paper is the canonical standard completeness of these logics, that is, theorems of Π(C) are exactly the 1-tautologies of the algebra defined over the real unit interval where the truth-constants are interpreted as their own values. It is also shown that they do not enjoy the canonical strong standard completeness, but they enjoy it for finite theories when restricted to evaluated Π-formulas of the kind r → ϕ, where r is a truth-constant and ϕ a formula not containing truth-constants. Finally we consider the logics Π ∆ (C), the expansion of Π(C) with the well-known Baaz's projection connective ∆, and we show canonical finite strong standard completeness for them.

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