 Instead of the basic types e and t of classical extensional Montague grammar, DMG has basic types \(e, t\) and \(m (m\) for marker). States pick out entities for markers, so they can be viewed as objects of type \(m \rightarrow e\). Abbreviating \(m \rightarrow e\) as \(s\) (for “state”), we call objects of type \(s \rightarrow s \rightarrow t\) state transitions. The variable \(P\) in the DMG translation of “a man” has type \(m \rightarrow s \rightarrow s \rightarrow t\), so VP meanings have been lifted from type \(e \rightarrow t\) to this type. Note that \(\rightarrow\) associates to the right, so \(m \rightarrow s \rightarrow s \rightarrow t\) is shorthand for \(m \rightarrow(s \rightarrow(s \rightarrow t))\). Indeed, DMG can be viewed as the result of systematic replacement of entities by markers and of truth values by state transitions. A VP meaning for “is happy” is a function that maps a marker to a state transition. The state transition for marker \(u_i\) will check whether the input state maps \(u_i\) to a happy entity, and whether the output context equals the input context. The variables \(a\), a\('\) range over states, and the expression \((u_i \mid x)a\) denotes the result of resetting the value of \(u_i\) in \(a\) to \(x\), so the old value of \(u_i\) gets destroyed (destructive assignment). The anaphoric index \(i\) on reference marker \(u_i\) is introduced by the translation. In fact, the translation starts from the indexed indefinite noun phrase “a man\(_i\)”. The connection between Montagovian compositionality and dynamic semantics as well as the basic Montagovian and dynamic ingredients are much more transparent and streamlined in the typed Logic of Change proposed in Muskens 1991, 1995, 1996. Because of this, Muskens’s Compositional DRT is probably the de facto standard and starting point for current research in compositional dynamic semantics. An alternative treatment is given in Incremental Typed Logic (ITL), an extension to typed logic of a “stack semantics” that is based on variable free indexing and that avoids the destructive assignment problem. The basic idea of the stack semantics for DPL, developed in Vermeulen 1993, is to replace the destructive assignment of ordinary DPL, which throws away old values when resetting, by a stack-valued one that allows old values to be reused. Stack-valued assignments assign to each variable a stack of values, the top of the stack being the current value. Existential quantification pushes a new value on the stack, but there is also the possibility of popping the stack to reuse a previously assigned value. Eijck’s 2000 ITL is in fact a typed version of stack semantics, using a single stack.