# Syntax And Semantics

### Syntax

The syntax of the PRISM property specification language subsumes various probabilistic temporal logics, including PCTL, CSL, (probabilistic) LTL, PCTL* and CTL. Informally, the syntax can be summarised as follows: a property can be any valid, well-typed PRISM expression, which (optionally) also includes the probabilistic operators discussed previously (`P`, `S` and `R`) and the non-probabilistic (CTL) ones `A` and `E`). This mean that any of the following operators can be used:

• `-` (unary minus)
• `*`, `/` (multiplication, division)
• `+`, `-` (addition, subtraction)
• `<`, `<=`, `>=`, `>` (relational operators)
• `=`, `!=` (equality operators)
• `!` (negation)
• `&` (conjunction)
• `|` (disjunction)
• `<=>` (if-and-only-if)
• `=>` (implication)
• `?` (condition evaluation: `condition ? a : b` means "if `condition` is true then `a` else `b`")
• `P` (probabilistic operator)
• `S` (steady-state operator)
• `R` (reward operator)
• `A` (for-all operator)
• `E` (there-exists operator)

This allows you to write any property expressible in logics such as PCTL and CSL. For example, CSL allows you to nest `P` and `S` operators:

P=? [ F>2 S>0.9[ num_servers >= 5 ] ]

"the probability of it taking more than 2 hours to get to a state from which the long-run probability of at least 5 servers being operational is >0.9"

You can also express various arithmetic expressions such as:

1 - P=? [ F[3600,7200] oper ]

"the probability that the system is not operational at any point during the second hour of operation"

R{"oper"}=? [ C<=t ] / t

"the expected fraction of time that the system is available (i.e. the expected interval availability) in the time interval [0, t]"

P=? [ F fail_A ] / P=? [ F any_fail ]

"the (conditional) probability that component A eventually fails, given that at least one component fails"

### Semantics

We omit a formal presentation of the semantics of the PRISM property language. The semantics of the probabilistic temporal logics that the language incorporates can be found from a variety of sources. See for example the pointers given in the About and Documentation sections of the PRISM website.

It is worth, however, clarifying a few points specific to PRISM. A property is evaluated with respect to a particular state of a model. Depending on the type of the property, this value may either be a Boolean, an integer or a double. When performing model checking, PRISM usually has to actually compute the value for all states of the model but, for clarity, will by default report just a single value. Typically, this is the value for the (single) initial state of the model. For example, this:

P=? [ F "error" ]

will report the probability, from the initial state of the model, of reaching an "error" state. This:

P>0.5 [ F "error" ]

will return `true` if and only if the probability, from the initial state, is greater than 0.5.

Note: This is contrast to older versions of PRISM, which treated numerical and Boolean-valued properties differently in this respect.

For models with multiple initial states, we need to adapt these definitions slightly. In this case, the two properties above will yield, respectively:

• the range of values (over all initial states) of the probability of reaching "error"
• `true` if and only if the probability is greater than 0.5 from all initial states.

You can also ask PRISM to return different values using filters, which are described in the next section.