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.

Temporal Logic - LTL - 1

Temporal logic allows expressing properties that a system should satisfy along time. We will first talk about linear temporal logic which expresses properties over every possible path. For example, let's consider the following model:

Let r be the proposition representing that the system is running and h the proposition representing the the system has halted.

Then we could check whether it always holds that this system will always halt. According to the state machine representation this is false: We can consider the path r,r,r,r,... and never get to the halt state.

With linear temporal logic we can express the property "the system will always halt" and other properties referring to time.

For example, the operator G formula (globally) indicates that from a given state, every state state satisfies formula (formula can be any linear temporal logic formula.

For instance, the formula G (r or h) is true. Every state of every path starting from the initial state satisfies that r or h hold. However, the formula G r is not satisfied by every path.

The operator F formula indicates that formula will be true at some state in the future.

The given example does not satisfy F h (because of the given counterexample where the second state is not reached..

However the formula F r, holds (a state satisfying r is reached at the first state)

In future posts we will consider some other examples and other operators.

Call for papers: Modularity 2015

Modularity 2015

 Call for Papers: Modularity '15 Research Results

         March 16-19, 2015, Ft. Collins, Colorado, USA

First round due date: August 4, 2014. Second round due date: October 10, 2014.

Modularity is a key property for scalability of software processes, including debugging, maintenance, reasoning, and testing. Modularity influences system diversity, dependability, performance, evolution, the structure and the dynamics of the organizations that produce systems, human understanding and management of systems, and ultimately system value. Yet the nature of and possibilities for modularity, limits to modularity, the mechanisms needed to achieve it in given forms, and its costs, benefits, and tradeoffs remain poorly understood. Significant advances in modularity thus are possible and promise to yield breakthroughs in our ability to conceive, design, develop, validate, integrate, and evolve modern information systems and their underlying software artifacts.

Modularity at the semantic as well as the syntactic level is a key enabler for the expression of high quality software systems. One of the most important techniques for complexity reduction is abstraction. Novel concepts and abstraction mechanisms, including but not limited to aspect-oriented techniques, are a focus point for improvements in the support for modularity. The scope of this effort covers all perspectives on software systems in all their life-cycle phases, for instance application domain analysis, programming language constructs, formal proofs of system properties, program state visualization in debuggers, performance improvements in compiler algorithms, etc. As the premier international conference on modularity, Modularity'15 continues to advance our understanding of these issues and the expressive power of known techniques.

The Modularity'15 conference invites full, scholarly papers of the highest quality on new ideas and results. Papers are expected to contribute significant new research results with rigorous and substantial validation of specific technical claims, based on scientifically sound reflections on experience, analysis, experimentation, or formal models. Compelling new ideas are especially welcome, which means that the requirements in the areas of validation and maturity are higher for papers that contribute more incremental results.

Modularity'15 is deeply committed to publishing works of the highest caliber. To this aim, two separate paper submission deadlines and review stages are offered. A paper accepted in any round will be published in the proceedings and presented at the conference. Promising papers submitted in the first round that are not accepted may be invited to be revised and resubmitted for review by the same reviewers in the second round. Authors of such invited resubmissions are asked to also submit a letter explaining the revisions made to the paper to address the reviewers' concerns. While there is no guarantee that an invited resubmission will be accepted, this procedure (similar to major revisions requested by journals) is designed to help authors of promising work get their papers into the conference. Submission to both rounds is open for all, and authors who submit to the first round may of course choose to resubmit a revised version in the second round without such an invitation, in which case new reviewers may be appointed. Finally, the same paper cannot be simultaneously submitted to other conferences or journals. In case of doubt, please contact the Program Chair.

Research Topics

Modularity'15 is looking for papers in all areas of software modularity. Topics of interest include, but are not limited to, the following:

* Varieties of modularity: Context orientation; feature orientation; generative programming; aspect orientation; software product lines; traits; families of classes; meta-programming and reflection; components; view-based development.

* Programming languages: Support for modular abstraction in: language design; verification, specification, and static program analysis; compilation, interpretation, and runtime support; formal languages and calculi; execution environments and dynamic weaving; dynamic languages; domain-specific languages.

* Software design and engineering: Requirements and domain engineering; architecture; synthesis; evolution; metrics and evaluation; empirical studies of existing software; economics; testing and verification; semantics; composition and interference; traceability; methodologies; patterns.

* Tools: Crosscutting views; refactoring; evolution and reverse engineering; aspect mining; support for new language constructs.

* Applications: Data-intensive computing; distributed and concurrent systems; middleware; service- oriented computing systems; cyber-physical systems; networking; cloud computing; pervasive computing; runtime verification; computer systems performance; system health monitoring; enforcement of non-functional properties.

* Complex systems: Works that explore and establish connections across disciplinary boundaries, bridging to such areas as biology, economics, education, infrastructure such as buildings or transport systems, and more.

* Composition. Component-based software engineering; Composition and adaptation techniques; Composition algebras, calculi, and type systems; Model-driven composition; Dynamic composition and reconfiguration; Large-scale component-based systems; Cloud, service-oriented architectures; Business process orchestration; Visual composition environments; Performance optimization of composite systems.

Important Dates

* First round:

Submission: August 4, 2014 (23:59 Baker Island / UTC-12)

Notification: September 14, 2014

* Second round:

Submission: October 10, 2014 (23:59 Baker Island / UTC-12)

Notification: December 7, 2014

* Camera ready: February 15, 2015

Submission Guidelines

Modularity'15 is deeply committed to eliciting works of the highest caliber. To this aim, two separate paper submission deadlines and review stages are offered. A paper accepted in any round will be published in the proceedings and presented at the conference. Promising papers submitted in the first round that are not accepted may be invited to be revised and resubmitted for review by the same reviewers in the second round. Authors of such invited resubmissions are asked to also submit a letter explaining the revisions made to the paper to address the reviewers' concerns. While there is no guarantee that an invited resubmission will be accepted, this procedure (similar to major revisions requested by journals) is designed to help authors of promising work get their papers into the conference. Submission to both rounds is open for all, and authors who submit to the first round may of course choose to resubmit a revised version in the second round without such an invitation, in which case new reviewers may be appointed. Finally, the same paper cannot be simultaneously submitted to other conferences or journals. In case of doubt, please get in touch with the Program Chair.

All submissions must conform to both the ACM Policy on Prior Publication and Simultaneous Submissions and the SIGPLAN Republication Policy.

Papers are to be submitted electronically to CyberChair.

Please use http://cyberchairpro.borbala.net/modularitypapers/submit/ to submit your paper. Submissions should use the SIGPLAN Proceedings Format using 9 point font. Please include page numbers in your submission as this will be helpful for the reviewers and also for you when reading their reviews. (If your submission is written using LaTeX, please set the preprint option in the LaTeX \documentclass command to generate page numbers.) Please also ensure that your submission is legible when printed on a black and white printer. In particular, please check that colors remain distinct and font sizes are legible.

To ensure that papers stay focused on their core contributions, the main part of the paper should be no longer than 12 pages. There is a 4 page limit for appendices, and, therefore, for the overall submission must be less than 16 pages. If the paper is accepted, the final submission will be limited to 16 pages, including appendices.

However, it is the responsibility of the authors to keep the reviewers interested and motivated to read their submission. Reviewers are under no obligation to read all or even a substantial portion of a paper if they do not find the initial part of the paper compelling. The committee will not accept a paper if it is unclear that the paper will fit in the Modularity'15 Proceedings.

All submitted papers are peer-reviewed. Accepted papers from all tracks will appear in the Modularity'15 Proceedings in the ACM Digital Library. Submissions will be judged on the potential impact of the ideas and the quality of the presentation.

Program Committee Chair

Gary T. Leavens, University of Central Florida, USA (leavens@cs.ucf.edu)

Program Committee

Don Batory, University of Texas at Austin

Eric Bodden, Fraunhofer SIT and TU Darmstadt

Paulo Borba, Federal University of Pernambuco

Walter Cazzola, Università degli Studi di Milano

Cynthia Disenfeld, Technion -- Israel Institute of Technology

Robert Dyer, Bowling Green State University

Erik Ernst, Aarhus University

Matthew Flatt, University of Utah

Michael Haupt, Oracle Labs

Marieke Huisman, University of Twente

Cristina Lopes, University of California Irvine

Tiziana Margaria, University of Potsdam

Linda Northrop, Software Engineering Institute

Nate Nystrom, University of Lugano

Bruno C. d. S. Oliveira, The University of Hong Kong

Awais Rashid, Lancaster University

Henrique Rebêlo, Federal University of Pernambuco

Martin Rinard, Massachusetts Institute of Technology

Norbert Siegmund, University of Passau

Murali Sitaraman, Clemson University

Kevin Sullivan, University of Virginia

External Review Committee

Marsha Chechik, University of Toronto

Steven Edwards, Virginia Tech.

Dan Grossman, University of Washington

Görel Hedin, Lund University

Mira Mezini, TU Darmstadt

Peter Müller, ETH Zurich

Hridesh Rajan, Iowa State University

Clemens Szyperski, Microsoft Research

Éric Tanter, University of Chile

Eelco Visser, Delft University of Technology

For additional information, clarification, or answers to questions please contact the Program Committee Chair.