First NuSMV model check

There are several tools to apply check LTL formulas (LTL - 3, etc.). Among these, NuSMV is an open source tool that I have been using. This tool uses BDDs or SAT solvers to check properties, but I'll leave these topics for later and now focus on actual model checking a system..

In the tutorial, the following example is given:

MODULE main
VAR
request: boolean;
st: {ready, busy};
ASSIGN
init(st) := ready;
next(st) := case
              st = ready & request : busy;
              TRUE : {ready, busy};
            esac;

The given model contains two variables: request is a boolean and st - representing the state of the system - has two possible values (ready or busy).

In the ASSIGN section the initial and transition relation constraints are defined. The variable st is initialized as ready, and then for each transition:
if in the current state st is ready and there is a request, in the next state the value of st will be busy. Otherwise ("TRUE : ..."), the next value of st will be either ready or busy, non-deterministically chosen.
There are no constraints regarding initial or next values of request.

This model represents the given state machine:

The two states on the left are the initial states, since these are the states satisfying that initially st = ready.
Then, almost every transition is possible, besides when st = ready and there is a request, according to the state machine definition in the next state st must be busy.

We can check the following LTL formula:
\(\textbf{G} (st = ready \implies \textbf{X}\ st = busy)\)

That is, at any state, if st = ready, then in the next state st = busy.

To do so, we add the following to the model:

LTLSPEC
G (st = ready -> X st = busy)

Let's assume we have a file  nusmv1.smv (open with any text editor).

Now (from the command line) we can run NuSMV and check if the guarantee holds.

After having downloaded it, we can execute

NuSMV nusmv1.smv

You may need to access the whole path, for instance (in Windows)

C:\Program Files\NuSMV\bin\NuSMV nusmv1.smv

The output is the following:

-- specification  G (st = ready ->  X st = busy)  is false
-- as demonstrated by the following execution sequence
Trace Description: LTL Counterexample
Trace Type: Counterexample
-> State: 1.1 <-
  request = 1
  st = ready
-> Input: 1.2 <-
-> State: 1.2 <-
  request = 0
  st = busy
-- Loop starts here-> State: 1.3 <-
  st = ready
-> State: 1.4 <--> Input: 1.5 <-
-> State: 1.5 <-
  st = busy
-> Input: 1.6 <-
-> State: 1.6 <-
  st = ready

In the first state of the counterexample, request is true (1) and st is ready. In the next state request is false and st is busy. So far the formula holds.
Then it is inidicated the start of the loop. This is since infinite paths are analyzed, and every counterexample includes a prefix and a loop. The first state of the loop (State 1.3) changes the value of st but not of request (only the variables changed appear). So in State 1.3 request is false and st=ready. However, in the next state st keeps being ready! Our formula is false, since there could be states when the state is ready but the next state is not busy (for instance, when there is no request).

If we change the formula and check

LTLSPEC
G ((request & st = ready) -> X st = busy)

The output obtained is:

-- specification  G ((request & st = ready) ->  X st = busy)  is true

LOTOS - Choice

Some time ago, I've written about the basic events in LOTOS specification (LOTOS - 1). These are given by
g?var: T \(\rightarrow\) an input of type T is expected through the gate g.
g! constant \(\rightarrow\)  an output (constant) is exposed through the gate g.

Additional operators allow building more complex processes:

Sequential composition (;)

Several events can occur in a sequence. For example, g? x: int; g!x+1 represent two events (at two different instants of time). In the first one, it is expected to synchronize on an integer input, while in the second one it is expected to synchronize on the output that is the value received plus one.

We can now define a process P with a gate g that it always receives a value and exposes the value incremented:

process P[g] =
g?x:int ; g!x+1; P[g]

The last element in the sequential composition (P[g]) indicates that a new instance of the process starts when reaching to this point.

Choice ([])

The operator Choice ([]) allows representing the choice between two (or more) paths.

For example,

g! "hello"
[]
g! "world"

allows either exposing "hello" or "world" through the gate g.

Guards ([condition] \(\rightarrow\))

In general, choice conditions are non-deterministic (or more precisely, dependent on the environment). To add conditions, we can use guards combined with the choice operator:

[condition1] \(\rightarrow\) behavior1
[]
[condition2] \(\rightarrow\) behavior2

Then, the behaviors enabled depend on the conditions. For example, let P be a process having been instantiated with a variable x

P[g](x: int)
[x <0] \(\rightarrow\) g! -x; g? y: int; P[g](y)
[]
[x >= 0] \(\rightarrow\) g! x; g? y: int; P[g](y)

This process has an if-else like behavior. It exposes the absolute value of x, receives a new value from the environment (y), and creates a new instance of the process with this value.

Temporal Logic - LTL - 3

In this post we'll see the formal semantics of LTL (LTL - 1, LTL - 2, LTL - Literature):

The model is represented by \(\langle S, I, R, L\rangle\) where \(S\) is the set of states, \(I \subseteq S\) is the set of initial states, \(R:SxS\) is the relation transition definition and \(L: S \rightarrow 2^{AP}\) is the labeling definition where AP is the set of available atomic propositions.

In the example of the last post:

We have three states, \(S = \{s_0,s_1,s_2\}\), with \(I = \{s_0\}\), \(R = \{(s_0,s_1), (s_1,s_2), (s_2, s_2)\}\), \(L = \{s_0\mapsto\{p\},s_1\mapsto\{q\},s_2\mapsto\{p,q,r\}\}\).

A path is an infinite path \(\pi=\pi_0\pi_1\dots\) where  \(\pi_0 \in I\) and for every \(i\), \((\pi_i,\pi_{i+1})\in R\). That is, the first state of the path is an initial state of the model, and for every two consecutive states of the path, there is a relation between them.

Now we define the semantics for the boolean operations and the temporal operators we have mentioned.

\(\pi,i\vDash q\) with \(q \in AP\) if and only if \(q \in L(\pi_i)\). That is, given a path \(\pi\) and index \(i\) of a state within the path an atomic proposition \(q\) is satisfied if and only if \(q\) belongs to the labels of that state.

\(\pi,i\vDash \neg \varphi\) if and only if  \(\pi,i\not\vDash  \varphi\). That is, not a formula is satisfied if the formula itself is not satisfied.

\(\pi,i\vDash  \varphi \land \theta\) if and only if  \(\pi,i\vDash  \varphi\) and \(\pi,i\vDash  \theta\). That is, the conjunction of two linear temporal logic formulas \(\varphi,\theta\) is satisfied if and only if both formulas are satisfied.

\(\pi,i\vDash  \varphi \lor \theta\) if and only if  \(\pi,i\vDash  \varphi\) or \(\pi,i\vDash  \theta\). That is, the disjunction of two linear temporal logic formulas \(\varphi,\theta\) is satisfied if and only at least one of the formulas is satisfied.

Now, we show the semantics of the temporal operators G, F, X, U.

\(\pi,i\vDash \textbf{G}\varphi\) if and only if for every \(j\ge i\), \(\pi,j\vDash \varphi\) , that is, every path starting with every state after \(i\) satisfies \(\varphi\).

\(\pi,i\vDash \textbf{F}\varphi\) if and only if exists \(j\ge i\), \(\pi,j\vDash \varphi\) , that is, there is a state in the future such that the path starting with that state satisfies \(\varphi\).

\(\pi,i\vDash \textbf{X}\varphi\) if and only if \(\pi,i+1\vDash \varphi\) , that is, there is the next state satisfies \(\varphi\).

\(\pi,i\vDash \varphi\textbf{U}\theta\) if and only if exists \(j\ge i\), \(\pi,j\vDash \theta\) and for every \(i\le k < j\), \(\pi,k\vDash \varphi\) , that is, there is a state \(j\) in the future such that the \(\theta\) is satisfied, and every state till \(j\) (non-including) satisfies \(\varphi\).


Or semantics:

Meeting of the Sussex Border Path with Hook Street at Monckton Hook (Andy Potter) / CC BY-SA 2.0

Temporal Logic in Literature

In this post I want to put some phrases from Literature and show (possible) translation to temporal logic (LTL - 1, LTL - 2).

1)
"In such terms Mr Gradgrind always mentally introduced himself, whether to his private circle of acquaintance, or to the public in general." Charles Dickens, Hard Times, Chapter 2

Let's compare these two formulas:

\(F (MrGradgrindMentallyIntroducesHimself)\)

\(G (MrGradgrindMentallyIntroducesHimself)\)

The first one indicates that there is one state in the future in which Mr. Gradgrind mentally introduces himself. The second one indicates that this occurs at every state, matching the keyword always in the sentence.

2)
"There will be the noise of the blows of the whip, which they will give to the horses." Alexandre Dumas, The forty-Five Guardsmen, Chapter 1

Let's compare these two formulas:

\(F\ noiseBlowsOfWhip \land F\ whipToHorses\)

\(F (noiseBlowsOfWhip \land whipToHorses)\)

The first one indicates that there is a state in the future in which there is the noise of the blows of the whip, and there is state in the future in which they whip the horses, while the second one indicates a state in the future in which both things occur. Since we know that both things are related, the whip causes the noise, we know they must be at the same state, and the second formula is preciser.

3)
' This proposal met with general applause, until an old mouse got up and said: "That is all very well, but who is to bell the Cat?" ' Aesop, Aesop's fables, Belling the Cat

A first attempt could to be to write the following formula:

\(Applause \land OldMouseQuestion\)

However, this formula expresses that at the first instant of time both things occur, and the applause may happen later and even later the question. Thus, we may attempt to correct the formula as follows:

\(Applause \land F\ OldMouseQuestion\)

It is better in that it expresses that first the applause occurs and sometime in the future the question by the old mouse. However, the applause goes only to be interrupted by the old mouse (keyword: until). Then we correct it as follows:

\(Applause\ U\ OldMouseQuestion\)

Here represents that the applause goes on until the old mouse asks its question. The only ambiguous part that remains is whether the story starts with the applause or this happens at some moment in the future. Due to our additional knowledge that first there is a discussion, then the proposal to bell the cat and only then the applause, we know that this part does not occur at the first state:

\(F\ Applause\ U\ OldMouseQuestion\)

Modularity 2015 - Submissions: October 10, 2014

Modularity 2015 second round submission deadline: October 10th, 2014

More information: Modularity 2015 - Research Results Track

PSC track submission deadline: 26th September

PSC (Programming for Separation of Concerns) Track of ACM Symposium on Applied Computing deadline: September 26th.

Details on the CFP at http://www.dmi.unict.it/~tramonta/sac/

Temporal Logic - LTL - 2

In the previous post (LTL - 1), we started talking about Linear Temporal Logic, a logic that allows expressing temporal properties about every path of a model.
In particular we have observed the temporal operators (or modalities):
G (Globally)
F (Eventually)
There is a relation between these two operators: Writing that globally something (property) holds, is equivalent to saying that the negation of the property will never hold. In terms of the operators this is expressed as:
G property = not F not property
I take this opportunity to present syntax notations I will use in future posts:
not = \( \neg \)
or = \( \lor \)
and = \( \land \)
Now, other temporal operators of interest are:
X formula (at the next state formula holds)
formula1 U formula2 (formula1 holds until formula2 holds)
For example,

In the given state machine, p q and r are atomic propositions. For example, p could represent x = 1 , q represent y > 0, and r represent that z is within some range of values. These are just examples, for each system, the set of atomic propositions is defined, and with these, the state machine representing the system is built. We will consider that if an atomic proposition does not appear in a state, then that atomic proposition does not hold at that state. For example the first state satisfies \( p \land \neg q \land \neg r \).

When standing at the first state of the model in the example (let's call it M), we can see that the next state satisfies q. That is, \( M \vDash \textbf{X}\ q \). When writing that the model satisfies an LTL formula, we are saying that every path starting from the initial states of the model satisfies the formula.

We can also observe that \(M \vDash  \textbf{X}\ \neg p \) ,\( M \vDash   \neg \textbf{X}\ r \), \( M \vDash  \textbf{X} \textbf{X}\ q \), and \( M \vDash  \textbf{X} \textbf{X}\textbf{X}\ p \). Question to the reader: why?


Thus, X expresses at the next state (from the state we are currently at). For example although the model does not satisfy that at the next state of every path starting from the initial states p will hold, there exists a future state such that its next state will satify p. That is \( M \vDash \textbf{F}\textbf{X}\ p \). And this way the different temporal operators can be combined.

cc license from origin

The remaining operator for this post is U: The idea is that a formula "formula1 U formula2" is satisfied when formula1 holds at every state until some state where formula2 holds. If formula2 holds, then the formula with until holds. If formula2 never holds, then the until formula is not satisifed. In the next blog we will give the more formal definitions of the temporal operators (or modalities) and understand why the express the general idea we have seen in these two posts.