We study the almost-sure termination problem for probabilistic programs. First, we show that supermartingales with lower bounds on conditional absolute difference provide a sound approach for the almost-sure termination problem. Moreover, using this approach we can obtain explicit optimal bounds on tail probabilities of non-termination within a given number of steps. Second, we present a new approach based on Central Limit Theorem for the almost-sure termination problem, and show that this approach can establish almost-sure termination of programs which none of the existing approaches can handle. Finally, we discuss algorithmic approaches for the two above methods that lead to automated analysis techniques for almost-sure termination of probabilistic programs.

In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on modularity as their theoretical basis. In non-probabilistic programs, the classical variant rule (V-rule) of Floyd-Hoare logic provides the foundation for modular analysis. Extension of this rule to almost-sure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in [Fioriti and Hermanns 2015]. While the proposed probabilistic variant cautiously addresses the key issue of integrability, we show that the proposed modular rule is still not sound for almost-sure termination of probabilistic programs.Besides establishing unsoundness of the previous rule, our contributions are as follows: First, we present a sound modular rule for almost-sure termination of probabilistic programs. Our approach is based on a novel notion of descent supermartingales. Second, for algorithmic approaches, we consider descent supermartingales that are linear and show that they can be synthesized in polynomial time. Finally, we present experimental results on a variety of benchmarks and several natural examples that model various types of nested while loops in probabilistic programs and demonstrate that our approach is able to efficiently prove their almost-sure termination property.

Abstract. Recent studies reveal that branching bisimilarity is decidable for both nBPP (normed Basic Parallel Processes) and nBPA (normed Basic Process Algebras). These results lead to the question if there are any other models in the hierarchy of PRS (Process Rewrite Systems) whose branching bisimilarity is decidable. It is shown in this paper that the branching bisimilarity for both nOCN (normed One Counter Nets) and nPA (normed Process Algebras) is undecidable. These results essentially imply that the question has a negative answer.

We put forward an exponential-time algorithm for deciding branching bisimilarity on normed BPA (Bacis Process Algebra) systems. The decidability of branching (or weak) bisimilarity on normed BPA was once a long standing open problem which was closed by Yuxi Fu in [1]. The EXPTIME-hardness is an inference of a slight modification of the reduction presented by Richard Mayr [2]. Our result claims that this problem is EXPTIME-complete.Remark 3. In this paper, branching bisimulations in Definition 1 and other bisimulation-like relations in later chapters are forced to be equivalence relations. This technical convention does not affect the notion of branching bisimilarity.The branching bisimilarity is a congruence relation, and it satisfies the following famous lemma.If Γ is realtime, the branching bisimilarity is the same as the strong bisimilarity. In this paper, branching bisimilarity will be abbreviated as bisimilarity. For realtime systems, the term bisimilarity will also be used to indicate strong bisimilarity. III. RELATIVIZED BISIMILARITIES ON NORMED BPA A. RetrospectionIn [1], Yuxi Fu creates the notion of redundant processes, and discover the following Proposition 3, which is crucial to the proof of decidability of bisimilarity for normed BPA.We use Rd(γ) = {X | Xγ ≃ γ} to indicate the set of all constants that is ≃-redundant over γ. Clearly, Rd(γ) ⊆ C G .The following lemma confirms that the redundant processes over γ are completely determined by the redundant constants.The crucial observation in [1] is the following fact.Proposition 3. Assume that Rd(γ 1 ) = Rd(γ 2 ), then αγ 1 ≃ βγ 1 if and only if αγ 2 ≃ βγ 2 .Proposition 3 inspires us to define a relativized version of bisimilarity ≃ R for a given suitable reference set R, which will satisfy the following theorem. Theorem 1. Let γ be a process satisfying Rd(γ) = R. Then α ≃ R β if and only if αγ ≃ βγ. Lemma 10. If γ ≃ R δ and α ≃ β, then αγ ≃ R βδ. In particular, If γ ≃ R δ, then αγ ≃ R αδ.The computation lemma also holds for ≃ R . Lemma 11 (Computation Lemma for C. R-identities and Admissible Reference SetsClearly, R-bisimilarity has the following basic property.Be aware that the converse of Lemma 12 does not hold in general. That is, if X ≃ R ǫ, there is no guarantee that X ∈ R. This basic observation leads to further discussion.By Lemma 12 and Definition 6, R ⊆ Id R ⊆ C G . Moreover, Lemma 13. α ≃ R ǫ if and only if α ∈ (Id R ) * .Below we will demonstrate that, as a reference set, Id R plays an important role. At first we state a useful proposition for relative bisimilarities. It says that ≃ R is monotone.Intuitively, ≃ R is the relative bisimilarity which is induced by regarding the constants in R as ǫ purposely. It is reasonable to expect that X ≃ IdR ǫ if and only if X ∈ Id R . This intuition is confirmed by Proposition 16 and its corollaries.A direct inference of Corollary 18 is the following fact.Lemma 19. X ≃ Id R ǫ if and only if X ∈ Id R . In other words,The properties of ≃-redundant processes (Definition 2 and Proposition 3) in Section III-A...