The atoms of the planning instance, Atms**, are the (finitely many) expressions formed**
by applying the relation symbols in Rs to the objects in Os **(respecting arities).**
Init consists of two parts: **Init**logical is a set of literals formed from the atoms in **Atms**.

Initnumeric **is a set of propositions asserting the initial values of a subset of the primitive**
numeric expressions of the domain. These assertions each assign to a single primitive numeric expression a constant real value. The goal condition is a proposition that can include both atoms formed from the relation symbols and objects of the planning instance and numeric propositions between primitive numeric expressions and numbers.

As **is a collection of action schemas (non-durative actions) each expressed in the syntax**
of pddl**. The primitive numeric expression schemas and atom schemas used in these action**
schemas are formed from the function symbols and relation symbols (used with appropriate arities) defined in the domain applied to objects in Os **and the schema variables.**
The semantics shows how instantiated action schemas can be interpreted as state transitions, in a similar way to the familiar state transition semantics defined by **Lifschitz. An**
important difference is that states can no longer be seen as simply sets of propositions, but must also account for the numeric expressions appearing in the planning instance and the time at which the state holds. This is achieved by extending the notion of state.

Definition 2 Logical States and States Given the finite collection of atoms for a planning instance I, AtmsI , a logical state is a subset of AtmsI **. For a planning instance**
with dimension dim, a state is a tuple in (R, P(**Atms**I ), R
dim
⊥) where R⊥ = R ∪ {⊥} and
⊥ denotes the undefined value. The first value is the time **of the state, the second is the**
logical state and the third value is the vector of the dim values of the dim **primitive numeric**
expressions in the planning instance.

The initial state for a planning instance is (0**, Init**logical, x) where x **is the vector of values**
in R⊥ corresponding to the initial assignments given by **Init**numeric **(treating unspecified**
values as ⊥).

Undefined values are included in the numeric ranges because there are domains in which some terms start undefined but can nevertheless be initialised and exploited by actions.

To interpret actions as state transition functions it is necessary to achieve two steps.

Firstly, since (in pddl2.1**) plans are only ever constructed from fully instantiated action**
schemas, the process by which instantiation affects the constructs of an action schema must be defined and, secondly, the machinery that links primitive numeric **expressions to elements** of the vector of real values in a state and that allows interpretation of the numeric **updating**
behaviours in action effects must be defined. Since the mechanisms that support the second of these steps also affect the process in the first, the treatment of numeric effects is described first.

Definition 3 Assignment Proposition **The syntactic form of a numeric effect consists**
of an assignment operator (assign, increase, decrease, scale-up or scale-down**), one** primitive numeric expression, referred to as the lvalue**, and a numeric expression (which** is an arithmetic expression whose terms are numbers and primitive numeric expressions), referred to as the **rvalue**.