Decision methods for linearly ordered Heyting algebras

This paper presents a simple decision method for positively quantified formulae of the classical firstorder theory LOHA of linearly ordered Heyting algebras, where by positively quantified we mean that universal quantifiers appear only in positive positions and existential quantifiers appear only in negative positions. Π1-formulae are examples. In particular, word problems, either in LOHA or in the more restrictive theory LOL of linearly ordered lattices, with or without bounds 0 and 1, are solvable by this method, as are word problems in Gödel algebras (i.e. those Heyting algebras that satisfy the condition ∀xy.(x∨y = 1), but are not necessarily linearly ordered). Since zero-order Gödel-Dummett logic LC has free Gödel algebras as its Lindenbaum algebras [17], the method can be used to decide formulae in that logic, e.g. by interpreting each formula A as a term h(A) of LOHA and showing that 1 6 h(A) is derivable. Our presentation is, as befits the subject, in algebraic terminology. Key technical contributions are already made in the logical setting by various authors, as discussed below; the algebraic presentation however is novel, compact and easily implemented. It is also slightly more general, in the absence of a method to interpret such quantified formulae into Gödel-Dummett logic. The decidability of the full first-order theory was first shown by Kreisel [21]; this was later rediscovered ([12], [24]), as pointed out in [29]. We do not claim to cover the full first-order theory; but we give a method that offers relative simplicity in important but restricted cases. The first-order definability of lattice operations means that (e.g.) any atom r 6 s∨t can be rewritten using a quantifier as ∀x.((s 6 x∧t 6 x) ⊃ r 6 x). The same applies to the exponentiation operator in Heyting algebras, where (e.g.) r 6 t can be rewritten as ∀x.(s∧x 6 r ⊃ x 6 t). However, applied to an atomic formula in negative position, this creates a universal formula in negative position; in effect, this increases the logical complexity of the problem to be solved. In the linear case, use of quantifiers can be avoided, as shown below, and so the logical complexity is not increased. In fact, for reasons associated with space complexity, discussed below, there are good reasons for using some rewritings by means of quantifiers, observing the positivity constraint overall, by a different method.