I have moved again ...

It seems like ages since I last wrote for this blog. Last September, I moved with my advisor to the University of Edinburgh to begin my PhD. It took me longer than I thought to settle in Edinburgh as the atmosphere of the city, ESPECIALLY the gloomy weather, is quite different to that which I am accustomed to. Folklore has it that there are only sixty sunny days in Edinburgh in a year on average. I presume that forty of these days are in the summer. The other peculiarity that you can find in Edinburgh is the wind and rain. If you live outside Edinburgh and think that your city is windy, then I suggest that you come to visit Edinburgh. Actually, you'll find that the combination of wind and rain makes umbrella pretty much useless!

Anyway, apologies to my avid readers that I was silent for so long. Provided there is time and interest, I will blog again on a weekly basis.

Australia qualifies for round of 16

After losing 2 - 0 to Brazil, The Socceroos had to at least draw against Croatia. What a nerve-racking (and controversial) game that was! In the end, Australia managed to draw 2 - 2 against Croatia. Although I am supporting Australia through and through, I think that The Socceroos didn't secure the desired result convincingly. In a way, we were lucky to get a draw! Next: Italy vs. Australia.

Linear Temporal Logic (1)

As their name suggests, temporal logics are logics where truth values of formulas may change over time. The significance of temporal logic in computer science is indisputable, especially in verification of safety-critical reactive systems. Model checkers of many flavors of temporal logics have been developed to the extent that they can quickly verify real-world systems with a huge number of states. Furthermore, in contrast to many other tools and techniques of formal methods which are semi-automatic and relatively expensive, model checking is fully-automatic and quite cheap (excellent free model checkers like SPIN exist). Last year these tools and techniques were awarded the ACM Paris Kanellakis Theory and Practice Award. Amir Pnueli, the researcher who introduced the idea of using temporal logic in verification, was awarded the ACM Turing Award in 1996.

As we mentioned, temporal logic comes in a number of flavors. At present we deal only with Linear Temporal Logic (LTL). It is linear because we evaluate each LTL formula with respect to a vertex-labeled infinite path p₀p₁..., where each vertex p_i in the path corresponds to a point in time. "Present" is p₀. "Future" is any of p_i with i ≥ 0. Each vertex is labeled by a subset of 2^Σ where Σ is a set of atomic propositions. In other words, the label of a vertex indicates which atomic propositions are true in that vertex. Here is a somewhat contrived example. Suppose that Σ = {a,b}. As an example, take the path {a}²⁰⁰⁰{a,b}^ω, which is a path whose first 2000 vertices are {a}-labeled and the rest of the vertices are {a,b}-labeled. An example of an LTL formula is "at some point in the future, 'b' holds", which is true. Another example is "a holds until b holds", which is also true.

More formally, the syntax of LTL formulas can be defined in Backus-Naur form as follows:

φ, ψ := p (p ∈ Σ) | φ ∧ ψ | ¬ φ | X φ | φ U ψ

Given a 2^Σ-labeled (infinite) path w, we write wⁱ to refer to the infinite subpath of w starting from the ith vertex (the initial vertex is the 0th vertex). We may now define the semantics of LTL formulas:

1. w |= p (p ∈ Σ) if the label of the initial vertex of w contains p
   
2. w |= φ ∧ ψ if w |= φ and w |= ψ
   
3. w |= ¬ φ if it is not the case that w |= φ
   
4. w |= X φ if w¹ |= φ
   
5. w |= φ U ψ if there exists j ≥ 0 such that w^j |= ψ and for all 0 ≤ i < j, we have wⁱ |= φ
   

We use F φ to abbreviate "true U φ", which says that φ holds in some future. Note that "future" also includes the present. We use G φ to abbreviate ¬ F ¬ φ, which says that φ must hold at all points in the path, i.e., globally. Finally, note that {¬,∧} captures all other boolean connectives, which allows us to freely use any boolean connectives when we write formulas.

How do we use LTL to verify real world systems? The answer requires that we first define the semantics of LTL formulas with respect to Kripke Structures. A Kripke structure K over a finite alphabet Σ is a tuple (S,E,λ: S -> 2^Σ), where S is a set of vertices (or states)and E a set of directed edges on S. It is normally assumed that each vertex in S has at least one outgoing edge. The function λ defines which propositions in Σ are true in a given state. The reader should convince himself that the notion of Kripke structures can be used to model many practical systems. You can first start with traffic lights, in which case &Sigma = {green,red,yellow}, S consists of three states (each corresponding to the event where the light is green, red, or yellow), and E models how traffic lights work (e.g. green -> yellow -> red). A more interesting example includes Dijkstra's algorithm for the dining philosopher problem.

Now, given a state s in structure K, we may define the semantics of LTL formulas: K,s |= φ if for all directed path w in K starting from s, we have w |= φ. For example, if K models traffic lights, then the formulas G( green -> F red ) and G( green -> X yellow ) have to hold in every state of K. [For example, the first formula says that, it is globally true that if the light is presently green, then at some point in the future it will turn red.]

What we are interested in is an algorithm that solves the following model-checking problem: given a Kripke structure K over Σ, a state s in K, and an LTL formula φ over Σ, decide whether K,s |= φ. The problem is PSPACE-complete. The best known algorithm, which runs in 2^O(|φ|) x |K|, was given by Vardi and Wolper by utilizing automata theory on infinite strings. Although the complexity of the algorithm is exponential in the size of the formula, the algorithm runs quite fast in practice because practical specifications are usually short. We shall talk about this algorithm in the near future.

Australia vs. Japan: 3 - 1

Go Aussie!

Game theoretic characterization of treewidth

There is no doubt that the notion of treewidth introduced by Robertson and Seymour is one of the most important concepts in math and computer science introduced in the past thirty years. Roughly speaking, treewidth of a graph G measures how much G looks like a tree. For example, the treewidths of a tree, of a cycle, and of a clique (with n vertices) are respectively 1, 2, and n.

Some of the readers might not be aware of an intuitive way of understanding the concept of treewidths that is provided by the cops-and-robber games, as pointed out by Seymour and Thomas in their paper "Graph searching and a min-max theorem for treewidth", Journal of Combinatorial Theory B, Vol. 58 (1993). The following presentation follows the second paper cited at the bottom. The k-cops-and-robber games, where k > 1, takes a graph G = (V,E) as a parameter. Each game has two players: the cops and the robber. The game goes as follows:

1. In round 0, the cops choose a subset X₀ of V of size at most k. The robber chooses a vertex v₀ of V - X₀.
   
2. In round i+1, the cops choose a subset X_i+1 of V of size at most k. If possible, the robber chooses a vertex v_i+1 of V - X_i+1 such that there is a (possibly empty) path from v_i to v_i+1 which does not pass through X_i ∩ X_i+1. Otherwise, the cops win the game right there.
   

If this game has no end (i.e. the robber can evade capture), then the robber wins. The cops are said to have a winning strategy on the k-cops-and-robber game on G if they can play in such a way that they will eventually win the game. Otherwise, we say that the robber has a winning strategy on this game.

It is helpful to think of the set X_i+1 - X_i as set of vertices above which the cops are hovering in helicopters in round i+1. [This tells us that cops can move from vertices to vertices unrestrictedly.] Also, the set X_i+1 ∩ X_i can be thought of as the vertices where the cops have landed from helicopters. So, it takes two moves for a cop to be at a particular vertex v (i.e. on the ground): she must first ride a helicopter to v, and second land the helicopter at v. On the other hand, a cop may take off to another vertex (but still is in a helicopter) in just one move. In addition, the robber can see the cops in the helicopter, and may elude capture by running at great speed to another vertex as long as he does not run through a cop on the ground during his escape.

Theorem (Seymour and Thomas): A graph G has treewidth ≤ k-1 iff the cops have a winning strategy in the k-cops-and-robber game on G.

Equivalently, a graph G has treewidth k-1 iff k is the smallest number k' such that the cops have a winning strategy in the k'-cops-and-robber game on G. Let's try this game on a tree. The cops can win the game using 2 cops only. The idea is to gradually shrink the set of vertices the robber can go to in the next round by alternately moving the two cops in a "depth-first traversal" manner. On the other hand, it is clear that the robber player has a winning strategy on a tree if he is chased by only a poor lone cop.

In recent years, some researchers in databases and verification have proposed variants of the cops-and-robber games so as to explain some phenomena in those fields. Let me conclude by mentioning two such papers:

1. Gottlob, Leone, and Scarcello. Robbers, marshals, and guards: game theoretic and logical characterization of hypertree width, JCSS 66, 2003.
   
2. Berwanger, Dawar, Hunter, and Kreutzer. DAG-width and Parity Games, STACS 2006.