Gaoyan Xie

A solvable class of quadratic Diophantine equations with applications to verification of infinite-state systems

A k-system consists of k quadratic Diophantine equations over nonnegative integer variables s1, ..., sm, t1, ..., tn of the form: Σ 1≤j≤l B1j(t1, ..., tn)A1j(s1, ..., sm) = C1(s1, ..., sm) Σ 1≤j≤l Bkj(t1, ..., tn)Akj(s1, ..., sm) = Ck(s1, ..., sm) where l, n, m are positive integers, the Bs are nonnegative linear polynomials over t1, ..., tn (i.e., they are of the form b0+b1t1+...+bntn, where each bi is a nonnegative integer), and the As and Cs are nonnegative linear polynomials over s1, ..., sm. We show that it is decidable to determine, given any 2-system, whether it has a solution in s1, ..., sm, t1, ..., tn, and give applications of this result to some interesting problems in verification of infinite-state systems. The general problem is undecidable; in fact, there is a fixed k > 2 for which the k-system problem is undecidable. However, certain special cases are decidable and these, too, have applications to verification.

On model-checking of p systems

Membrane computing is a branch of molecular computing that aims to develop models and paradigms that are biologically motivated. It identifies an unconventional computing model, namely a P system, from natural phenomena of cell evolutions and chemical reactions. Because of the nature of maximal parallelism inherent in the model, P systems have a great potential for implementing massively concurrent systems in an efficient way that would allow us to solve currently intractable problems. In this paper, we look at various models of P systems and investigate their model-checking problems. We identify what is decidable (or undecidable) about model-checking these systems under extended logic formalisms of CTL. We also report on some experiments on whether existing conservative (symbolic) model-checking techniques can be practically applied to handle P systems with a reasonable size.

Dense Counter Machines and Verification Problems

We generalize the traditional definition of a multicounter machine (where the counters, which can only assume nonnegative integer values, can be incremented/decremented by 1 and tested for zero) by allowing the machine the additional ability to increment/decrement each counter C i by a nondeterministically chosen fractional amount δ i between 0 and 1 (δ i may be different at each step). Further at each step, the δ i ’s of some counters can be linearly related in that they can be integral multiples of the same fractional δ (e.g., δ i = 3δ, δ3 = 6δ). We show that, under some restrictions on counter behavior, the binary reachability set of such a machine is definable in the additive theory of the reals and integers. There are applications of this result in verification, and we give an example in the paper. We also extend the notion of “semilinear language” to “dense semilinear language” and show its connection to a restricted class of dense multicounter automata.

An automata-theoretic approach for model-checking systems with unspecified components

This paper introduces a new approach for the verification of systems with unspecified components. In our approach, some model-checking problems concerning a component-based system are first reduced to the emptiness problem of an oracle finite automaton, which is then solved by testing the unspecified components on-the-fly with test-cases generated automatically from the oracle finite automaton. The generated test-cases are of bounded length, and with a properly chosen bound, a complete and sound solution is immediate. Particularly, the whole verification process can be carried out in an automatic way. In the paper, a symbolic algorithm is given for generating test-cases and performing the testings, and an example is drawn from an TinyOS application to illustrate our approach.

Real-Counter automata and their decision problems

We introduce real-counter automata, which are two-way finite automata augmented with counters that take real values. In contrast to traditional word automata that accept sequences of symbols, real-counter automata accept real words that are bounded and closed real intervals delimited by a finite number of markers. We study the membership and emptiness problems for one-way/two-way real-counter automata as well as those automata further augmented with other unbounded storage devices such as integer-counters and pushdown stacks.

Linear reachability problems and minimal solutions to linear Diophantine equation systems

The linear reachability problem for finite state transition systems is to decide whether there is an execution path in a given finite state transition system such that the counts of labels on the path satisfy a given linear constraint. Using some known results on minimal solutions (in nonnegative integers) for linear Diophantine equation systems, we present new time complexity bounds for the problem. In contrast to the previously known results, the bounds obtained in this paper are polynomial in the size of the transition system in consideration, when the linear constraint is fixed. The bounds are also used to establish a worst-case time complexity result for the linear reachability problem for timed automata.

Testability of oracle automata

An oracle finite automaton (OFA) is a finite/Buchi automaton augmented with a finite number of unbounded, one-way, and writable query tapes. By each transition, an OFA can read an input symbol, append a symbol to the end of a query tape, erase the content of a query tape, or query an oracle with the content of a query tape (called a query string). Here, an oracleO is a language in some language class

New complexity results for some linear counting problems using minimal solutions to linear diophantine equations

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Real-Counter Automata and Their Decision Problems (Extended Abstract)

(all oracles in the OFA must be in the same language class

New complexity results for some linear counting problems using minimal solutions to linear diophantine equations: (Extended abstract)

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