Formal series are a natural extension of formal languages by associating each word with a value called a coefficient or a weight. Among them, recognizable series and algebraic series can be regarded as extensions of regular languages and context-free languages, respectively. The coefficient of a word w can represent quantities such as the cost taken by an operation on w, the probability that w is emitted. One of the possible applications of formal series is the string counting in quantitative analysis of software. In this paper, we define the counting problems for formal series and propose algorithms for the problems. The membership problem for an automaton or a grammar corresponds to the problem of computing the coefficient of a given word in a given series. Accordingly, we define the counting problem for formal series in the following two ways. For a formal series S and a natural number d, we define CC(S,d) to be the sum of the coefficients of all the words of length d in S and SC(S,d) to be the number of words of length d that have non-zero coefficients in S. We show that for a given recognizable series S and a natural number d, CC(S,d) can be computed in O(η log d) time where η is an upper-bound of time needed for a single state-transition matrix operation, and if the state-transition matrices of S are commutative for multiplication, SC(S,d) can be computed in polynomial time of d. We extend the notions to tree series and discuss how to compute them efficiently. Also, we propose an algorithm that computes CC(S,d) in square time of d for an algebraic series S. We show the CPU time of the proposed algorithm for computing CC(S,d) for some context-free grammars as S, one of which represents the syntax of C language. To examine the applicability of the proposed algorithms to string counting for the vulnerability analysis, we also present results on string counting for Kaluza Benchmark.

A program is non-interferent if it leaks no secret information to an observable output. However, non-interference is too strict in many practical cases and quantitative information flow (QIF) has been proposed and studied in depth. Originally, QIF is defined as the average of leakage amount of secret information over all executions of a program. However, a vulnerable program that has executions leaking the whole secret but has the small average leakage could be considered as secure. This counter-intuition raises a need for a new definition of information leakage of a particular run, i.e., dynamic leakage. As discussed in [5], entropy-based definitions do not work well for quantifying information leakage dynamically; Belief-based definition on the other hand is appropriate for deterministic programs, however, it is not appropriate for probabilistic ones.In this paper, we propose new simple notions of dynamic leakage based on entropy which are compatible with existing QIF definitions for deterministic programs, and yet reasonable for probabilistic programs in the sense of [5]. We also investigated the complexity of computing the proposed dynamic leakage for three classes of Boolean programs. We also implemented a tool for QIF calculation using model counting tools for Boolean formulae. Experimental results on popular benchmarks of QIF research show the flexibility of our framework. Finally, we discuss the improvement of performance and scalability of the proposed method as well as an extension to more general cases.