# Custom college term papers

I will focus on the two related logics that are the most mature and best studied: TPTL and hybrid logic. At first glance, these seem to be peripheral issues, but they turn out to be important. These seemingly subtle semantic changes have substantial ramifications on the theoret- 30 3. For example, custom college term papers consider the problem of model-checking a path: evaluating whether a formula holds for a single path. Alur and Henzinger show that this problem is EXPSPACE-complete for TPTL, further noting that it would be undecidable if TPTL were modified to allow more powerful atomic propositions, such as addition over terms. Our path constraint language, on the other hand, goes so far as to allow atomic propositions made up of arbitrary predicates over arbitrary terms (just like FOL), but then we regain decidability by studying finite sequences rather than infinite sequences. Unsurprisingly, this completely changes the problem of model-checking a path. What is perhaps surprising is that the problem is still hard and interesting. As I will show, the problem of model-checking an evolution path constraint on a finite path is hard (PSPACE-complete).

Although hybrid logic can be used to express constraints over finite, linear paths, it does not seem to be done often, at least not often enough that anyone has bothered to study the theoretical properties of that case. And again, although naively we might assume this case to be trivial, boring, or a straightforward specialization of the more general case, in fact interesting and surprising results emerge from these restrictions. Hybrid logic is quite different in character and focus from our constraint logic. In particular, although the general idea of nominals (propositions that are true in only one state and hence uniquely identify that state) is quite central to hybrid logic, custom college term papers the J, binder that corresponds to our variable-binding operator is not.

This is not to say that the theory of j in hybrid logic is underdeveloped—on the contrary, some remarkable results about it have been published—but the focus of hybrid logic is different from ours. Not having the satisfaction operator imposes some unexpected challenges. Of course, nominals themselves are also somewhat different from bound variables in our language. This is an important semantic difference, although the effect is similar. This can be easily stated as a model-checking problem. In general, model checking is the problem of checking whether a specific 31 3 Theoretical results on evolution path constraint verification formula is true of a specific Kripke structure. More specifically, model checking is usually used to verify that a state transition system has some property.

For others, such as LTL, it is hard—PSPACE-complete, in fact. Moreover, the solution to the model-checking problem for LTL is intellectually rather challenging too, involving an intricate tableau construction. Fortunately, we are not terribly interested in this form of the problem. Instead, we are interested in model-checking a single, particular path—not verifying a formula over an entire state transition system. Likewise, all our paths are finite—and in most cases probably rather short, since they must be comprehensible to the human architects reasoning about them.

This simple dynamic-programming algorithm is no longer adequate, because we also need to keep track of variable valuations.

So it is clear that model-checking our path constraint language will be harder than model-checking Unite LTL paths. There are such valuations, which is 0(m fc ) for custom college term papers fixed m.

Note that k, the number of rigid variables, is asymptotically proportional to the length of the formula, (. This gives us a strong hint that we have departed the realm of polynomial-time algorithms. However, we can model-check a path constraint in polynomial space, because we only need to work with one valuation at a time. In fact, even the naive recursive algorithm is polynomial-space: Theorem 1. There is a polynomial-space algorithm for model-checking a path constraint on a single path. Since every recursive call is of a strict subformula, the stack depth of this algorithm never exceeds (. Each execution of Check (excluding the recursion) uses only 0(1) space. Thus, the space complexity of the algorithm is Off).

The way to prove that a problem is PSPACE-hard is to prove that the quantified-Boolean-formula (QBF) problem reduces to it. In fact QBF is a 33 3 Theoretical results on evolution path constraint verification direct generalization of SAT. A quantified Boolean formula is just what it sounds like: a formula such as 3xVy3z(z A x v y) (2) where each variable is interpreted as Boolean and is quantified. Any QBF can be converted to prenex normal form in polynomial time, so from now on we will assume prenex normal form, as is typical in QBF proofs. SAT can be viewed as the special case of QBF where all the quantifiers are existential.

There is a fairly obvious transformation from QBF to our model-checking problem, but unfortunately this obvious reduction does not work. The obvious reduction is to change the Vs into Ds and the 3s into Os, add variable bindings after each quantifier, then check the formula on a path of length 2, where the first state represents falsehood and the second represents truth. Thus the path constraint is not equivalent to formula (2).

Flowever, it is temptingly close, and it is easy to imagine language extensions that would make the proof work. Proving that model-checking a path constraint is PSPACE-hard is more challenging. Instead of a simple two-state model, we must build a 2k-state model, where Jc is the number of quantifiers. The problem of model-checking a path constraint on a single path is PSPACE-complete.

To show it is PSPACE-hard, we exhibit a polynomial-time reduction of QBF.

We translate this QBF into a path constraint model-checking problem as follows. A graphical representation of the path constraint language model that we construct in the reduction of a QBF. Note that we can construct this model in linear time. We now construct the path constraint that corresponds to the QBF.

At the end of the formula, we add Ic closing parentheses. It is custom college term papers larger than the original QBF, but only by a constant factor. Let Xi denote y with the first i quantifiers removed. And custom college term papers of course hybrid logic with J, has the same problem as our path constraint language. There are practical reasons not to be overly concerned about the PSPACE-hardness of checking path constraints.