We introduce the notion of bisimulation for first-order Kripke models. It is defined as a relation that satisfies certain zig-zag conditions involving back-and-forth moves between nodes of Kripke models and, simultaneously, between the domains of their underlying structures. As one of our main results, we prove that if two Kripke models bisimulate to a certain degree, then they are logically equivalent with respect to the class of formulae of the appropriate complexity. Two applications of the notion introduced in the paper (...) are given. First, we consider bisimulations in the context of Kripke models of Heyting Arithmetic. Next, we discuss the notion of elementary submodel of a Kripke model and provide a suitable example. (shrink)

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich Theorem, which characterizes the (...) sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies $\forall_n (= \Pi^0_n)$ and $\exists_n (=\Sigma^0_n)$ , and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic. (shrink)