A continuous-time Markov chain (CTMC) execution is a continuous class of probability distributions over states. This paper proposes a probabilistic linear-time temporal logic, namely continuous-time linear logic (CLL), to reason about the probability distribution execution of CTMCs. We define the syntax of CLL on the space of probability distributions. The syntax of CLL includes multiphase timed until formulas, and the semantics of CLL allows time reset to study relatively temporal properties. We derive a corresponding model-checking algorithm for CLL formulas. The correctness of the model-checking algorithm depends on Schanuel's conjecture, a central open problem in transcendental number theory. Furthermore, we provide a running example of CTMCs to illustrate our method.
We present a logical framework for the verification of relational properties in imperative programs. Our work is motivated by relational properties which come from security applications and often require reasoning about formulas with quantifier-alternations. Our framework reduces verification of relational properties of imperative programs to a validity problem into trace logic, an expressive instance of first-order predicate logic. Trace logic draws its expressiveness from its syntax, which allows expressing properties over computation traces. Its axiomatization supports fine-grained reasonin…
There is a scalability gap between probabilistic and non-probabilistic verification. Probabilistic model checking tools are based either on explicit engines or on (Multi-Terminal) Binary Decision Diagrams. These structures are complemented in areas of non-probabilistic verification by more scalable techniques, such as IC3. We present a symbolic probabilistic model checking algorithm based on IC3-like incremental construction of inductive clauses to partition the state space, interleaved with incremental construction of a system of linear inequalities. This paper compares our implementation to …
We consider Hoare-style verification for the graph programming language GP 2. In previous work, graph properties were specified by so-called E-conditions which extend nested graph conditions. However, this type of assertions is not easy to comprehend by programmers that are used to formal specifications in standard first-order logic. In this paper, we present an approach to verify GP 2 programs with a standard first-order logic. We show how to construct a strongest liberal postcondition with respect to a rule schema and a precondition. We then extend this construction to obtain strongest liber…
We consider Hoare-style verification for the graph programming language GP 2. In previous work, graph properties were specified by so-called E-conditions which extend nested graph conditions. However, this type of assertions is not easy to comprehend by programmers that are used to formal specifications in standard first-order logic. In this paper, we present an approach to verify GP 2 programs with a standard first-order logic. We show how to construct a strongest liberal postcondition with respect to a rule schema and a precondition. We then extend this construction to obtain strongest liber…
In this paper, we integrate separation logic with Propositional Projection Temporal Logic (PPTL) to obtain a two-dimensional logic, namely PPTL$^{\tiny\mbox{SL}}$. The spatial dimension is realized by a decidable fragment of separation logic which can be used to describe linked lists, and the temporal dimension is expressed by PPTL. We show that PPTL and PPTL$^{\tiny\mbox{SL}}$ are closely related in their syntax structures. That is, for any PPTL$^{\tiny\mbox{SL}}$ formula in a restricted form, there exists an "isomorphic" PPTL formula. The "isomorphic" PPTL formulas can be obtained by first a…
The Internet, as it stands today, is highly vulnerable to attacks. However, little has been done to understand and verify the formal security guarantees of proposed secure inter-domain routing protocols, such as Secure BGP (S-BGP). In this paper, we develop a sound program logic for SANDLog-a declarative specification language for secure routing protocols for verifying properties of these protocols. We prove invariant properties of SANDLog programs that run in an adversarial environment. As a step towards automated verification, we implement a verification condition generator (VCGen) to automa…
This paper presents a formalisation of pGCL in Isabelle/HOL. Using a shallow embedding, we demonstrate close integration with existing automation support. We demonstrate the facility with which the model can be extended to incorporate existing results, including those of the L4.verified project. We motivate the applicability of the formalism to the mechanical verification of probabilistic security properties, including the effectiveness of side-channel countermeasures in real systems.
Decentralized techniques are becoming crucial and ubiquitous with the rapid advancement of distributed ledger technologies such as the blockchain. Numerous decentralized systems have been developed to address security and privacy issues with great dependability and reliability via these techniques. Meanwhile, formalization and verification of the decentralized systems is the key to ensuring correctness of the design and security properties of the implementation. In this paper, we propose a novel method of formalizing and verifying decentralized systems with a kind of extended concurrent separa…
We introduce a method of verifying termination of logic programs with respect to concrete queries (instead of abstract query patterns). A necessary and sufficient condition is established and an algorithm for automatic verification is developed. In contrast to existing query pattern-based approaches, our method has the following features: (1) It applies to all general logic programs with non-floundering queries. (2) It is very easy to automate because it does not need to search for a level mapping or a model, nor does it need to compute an interargument relation based on additional mode or typ…
We present a framework in Isabelle for verifying asymptotic time complexity of imperative programs. We build upon an extension of Imperative HOL and its separation logic to include running time. In addition to the basic arguments, our framework is able to handle advanced techniques for time complexity analysis, such as the use of the Akra-Bazzi theorem and amortized analysis. Various automation is built and incorporated into the auto2 prover to reason about separation logic with time credits, and to derive asymptotic behavior of functions. As case studies, we verify the asymptotic time complex…
The Complexity of Verifying Strategic Abilities in Succinct Multiagent Systems
Griffith University（格里菲斯大学）；CNRS, LIPN, Sorbonne Paris North University（法国国家科学研究中心，LIPN，索邦巴黎北大学）；University of Naples Federico II（那不勒斯费德里科二世大学）；University of New South Wales（新南威尔士大学）
The interplay between process behaviour and spatial aspects of computation has become more and more relevant in Computer Science, especially in the field of collective adaptive systems, but also, more generally, when dealing with systems distributed in physical space. Traditional verification techniques are well suited to analyse the temporal evolution of programs; properties of space are typically not explicitly taken into account. We propose a methodology to verify properties depending upon physical space. We define an appropriate logic, stemming from the tradition of topological interpretat…
Probabilistic team semantics is a framework for logical analysis of probabilistic dependencies. Our focus is on the axiomatizability, complexity, and expressivity of probabilistic inclusion logic and its extensions. We identify a natural fragment of existential second-order logic with additive real arithmetic that captures exactly the expressivity of probabilistic inclusion logic. We furthermore relate these formalisms to linear programming, and doing so obtain PTIME data complexity for the logics. Moreover, on finite structures, we show that the full existential second-order logic with additi…
Manna and Waldinger's theory of substitutions and unification has been verified using the Cambridge LCF theorem prover. A proof of the monotonicity of substitution is presented in detail, as an example of interaction with LCF. Translating the theory into LCF's domain-theoretic logic is largely straightforward. Well-founded induction on a complex ordering is translated into nested structural inductions. Correctness of unification is expressed using predicates for such properties as idempotence and most-generality. The verification is presented as a series of lemmas. The LCF proofs are compared …
In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under …
Larsen and Skou characterized probabilistic bisimilarity over reactive probabilistic systems with a logic including true, negation, conjunction, and a diamond modality decorated with a probabilistic lower bound. Later on, Desharnais, Edalat, and Panangaden showed that negation is not necessary to characterize the same equivalence. In this paper, we prove that the logical characterization holds also when conjunction is replaced by disjunction, with negation still being not necessary. To this end, we introduce reactive probabilistic trees, a fully abstract model for reactive probabilistic system…
The fuzzy modality `probably` is interpreted over probabilistic type spaces by taking expected truth values. The arising probabilistic fuzzy description logic is invariant under probabilistic bisimilarity; more informatively, it is non-expansive wrt. a suitable notion of behavioural distance. In the present paper, we provide a characterization of the expressive power of this logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we…
In this paper, we propose a new logic for expressing and reasoning about probabilistic hyperproperties. Hyperproperties characterize the relation between different independent executions of a system. Probabilistic hyperproperties express quantitative dependencies between such executions. The standard temporal logics for probabilistic systems, i.e., PCTL and PCTL* can refer only to a single path at a time and, hence, cannot express many probabilistic hyperproperties of interest. The logic proposed in this paper, \HyperPCTL, adds explicit and simultaneous quantification over multiple traces to P…
Probabilistic justification logic is a modal logic with two kind of modalities: probability measures and explicit justification terms. We present a tableau procedure that can be used to decide the satisfiability problem for this logic in polynomial space. We show that this upper complexity bound is tight.
We study discrete probabilistic programs with potentially unbounded looping behaviors over an infinite state space. We present, to the best of our knowledge, the first decidability result for the problem of determining whether such a program generates exactly a specified distribution over its outputs (provided the program terminates almost surely). The class of distributions that can be specified in our formalism consists of standard distributions (geometric, uniform, etc.) and finite convolutions thereof. Our method relies on representing these (possibly infinite-support) distributions as pro…
Lossy channel systems (LCSs) are systems of finite state automata that communicate via unreliable unbounded fifo channels. In order to circumvent the undecidability of model checking for nondeterministic LCSs, probabilistic models have been introduced, where it can be decided whether a linear-time property holds almost surely. However, such fully probabilistic systems are not a faithful model of nondeterministic protocols. We study a hybrid model for LCSs where losses of messages are seen as faults occurring with some given probability, and where the internal behavior of the system remains non…
Probabilistic timed automata (PTAs) are timed automata (TAs) extended with discrete probability distributions.They serve as a mathematical model for a wide range of applications that involve both stochastic and timed behaviours. In this work, we consider the problem of model-checking linear \emph{dense-time} properties over {PTAs}. In particular, we study linear dense-time properties that can be encoded by TAs with infinite acceptance criterion.First, we show that the problem of model-checking PTAs against deterministic-TA specifications can be solved through a product construction. Based on t…
In this paper, we propose Probabilistic discrete-time Projection Temporal Logic (PrPTL), which extends Projection Temporal Logic (PTL) with probability. To this end, some useful formulas are derived and some logic laws are given. Further, we define Time Normal Form (TNF) for PrPTL as the standard form and prove that any PrPTL formulas can be rewritten to TNF. According to the TNF, we construct the time normal form graph which can be used for the probabilistic model checking on PrPTL.
The probability theory is a well-studied branch of mathematics, in order to carry out formal reasoning about probability. Thus, it is important to have a logic, both for computation of probabilities and for reasoning about probabilities, with a well-defined syntax and semantics. Both current approaches, based on Nilsson's probability structures/logics, and on linear inequalities in order to reason about probabilities, have some weak points. In this paper we have presented the complete revision of both approaches. We have shown that the full embedding of Nilsson'probabilistic structure into pro…
In this paper we extend the predicate logic introduced in [Beauquier et al. 2002] in order to deal with Semi-Markov Processes. We prove that with respect to qualitative probabilistic properties, model checking is decidable for this logic applied to Semi-Markov Processes. Furthermore we apply our logic to Probabilistic Timed Automata considering classical and urgent semantics, and considering also predicates on clock. We prove that results on Semi Markov Processes hold also for Probabilistic Timed Automata for both the two semantics considered. Moreover, we prove that results for Markov Process…
The logic PJ is a probabilistic logic defined by adding (non-iterated) probability operators to the basic justification logic J. In this paper we establish upper and lower bounds for the complexity of the derivability problem in the logic PJ. The main result of the paper is that the complex- ity of the derivability problem in PJ remains the same as the complexity of the derivability problem in the underlying logic J, which is Πp2-complete. This implies hat the probability operators do not increase the complex- ity of the logic, although they arguably enrich the expressiveness of the language.
The paper presents probabilistic extensions of interval temporal logic (ITL) and duration calculus (DC) with infinite intervals and complete Hilbert-style proof systems for them. The completeness results are a strong completeness theorem for the system of probabilistic ITL with respect to an abstract semantics and a relative completeness theorem for the system of probabilistic DC with respect to real-time semantics. The proposed systems subsume probabilistic real-time DC as known from the literature. A correspondence between the proposed systems and a system of probabilistic interval temporal …
We present verification methods for logic programs with delay declarations. The verified properties are termination and freedom from errors related to built-ins. Concerning termination, we present two approaches. The first approach tries to eliminate the well-known problem of speculative output bindings. The second approach is based on identifying the predicates for which the textual position of an atom using this predicate is irrelevant with respect to termination. Three features are distinctive of this work: it allows for predicates to be used in several modes; it shows that block declaratio…
Over the last two decades, there has been an extensive study on logical formalisms for specifying and verifying real-time systems. Temporal logics have been an important research subject within this direction. Although numerous logics have been introduced for the formal specification of real-time and complex systems, an up to date comprehensive analysis of these logics does not exist in the literature. In this paper we analyse real-time and probabilistic temporal logics which have been widely used in this field. We extrapolate the notions of decidability, axiomatizability, expressiveness, mode…
Probabilistic Soft Logic has been proposed and used in several applications as an efficient way to deal with inconsistency, uncertainty and relational representation. In several applications, this approach has led to an adequate description of the corresponding human reasoning. In this paper, we provide a theoretical explanation for one of the semi-heuristic choices made in this approach: namely, we explain the choice of the corresponding conjunction operations. Our explanation leads to a more general family of operations which may be used in future applications of probabilistic soft logic.
We present a formulation of the problem of probabilistic model checking as one of query evaluation over probabilistic logic programs. To the best of our knowledge, our formulation is the first of its kind, and it covers a rich class of probabilistic models and probabilistic temporal logics. The inference algorithms of existing probabilistic logic-programming systems are well defined only for queries with a finite number of explanations. This restriction prohibits the encoding of probabilistic model checkers, where explanations correspond to executions of the system being model checked. To over…
The semantics of probabilistic languages has been extensively studied, but specification languages for their properties have received little attention. This paper introduces the probabilistic dynamic logic pDL, a specification logic for programs in the probabilistic guarded command language (pGCL) of McIver and Morgan. The proposed logic pDL can express both first-order state properties and probabilistic reachability properties, addressing both the non-deterministic and probabilistic choice operators of pGCL. In order to precisely explain the meaning of specifications, we formally define the s…
There are numerous applications where we have to deal with temporal uncertainty associated with events. The Temporal Probabilistic (TP) Logic Programs should provide support for valid-time indeterminacy of events, by proposing the concept of an indeterminate instant, that is, an interval of time-points (event's time-window) with an associated, lower and upper, probability distribution. In particular, we propose the new semantics, for the TP Logic Programs of Dekhtyar and Subrahmanian. Our semantics, based on the possible world semantics is a generalization of the possible world semantics for (…
Generalized Probabilistic Logic (GPL) is a temporal logic, based on the modal mu-calculus, for specifying properties of reactive probabilistic systems. We explore XPL, an extension to GPL allowing the semantics of nondeterminism present in Markov decision processes (MDPs). XPL is expressive enough that a number of independently studied problems--- such as termination of Recursive MDPs (RMDPs), PCTL* model checking of MDPs, and reachability for Branching MDPs--- can all be cast as model checking over XPL. Termination of multi-exit RMDPs is undecidable; thus, model checking in XPL is undecidable…
Probabilistic specifications are fast gaining ground as a tool for statistical modeling of probabilistic systems. One of the main goals of formal methods in this domain is to ensure that specific behavior is present or absent in the system, up to a certain confidence threshold, regardless of the way it operates amid uncertain information. This paper presents a rewriting logic semantics for a probabilistic extension of Event-B, a proof-based formal method for discrete systems modeling. The proposed semantics adequately captures the three sources of probabilistic behavior, namely, probabilistic …
Digitization provides a sound and complete method to reduce the problem of verifying whether a real-time system satisfies a property under dense-time semantics to whether the same real-time system satisfies the property over discrete-time. This is a brief overview of digitization of real-time models and logics covering known results, value, limitations, and alternatives.
Dynamically typed object-oriented languages enable programmers to write elegant, reusable and extensible programs. However, with the current methodology for program verification, the absence of static type information creates significant overhead. Our proposal is two-fold: First, we propose a layer of abstraction hiding the complexity of dynamic typing when provided with sufficient type information. Since this essentially creates the illusion of verifying a statically-typed program, the effort required is equivalent to the statically-typed case. Second, we show how the required type informatio…
The purpose of this paper is to develop further the main concepts of Phenomena Dynamic Logic (P-DL) and Cognitive Dynamic Logic (C-DL), presented in the previous paper. The specific character of these logics is in matching vagueness or fuzziness of similarity measures to the uncertainty of models. These logics are based on the following fundamental notions: generality relation, uncertainty relation, simplicity relation, similarity maximization problem with empirical content and enhancement (learning) operator. We develop these notions in terms of logic and probability and developed a Probabili…
Probabilistic Computation Tree Logic (PCTL) and Continuous Stochastic Logic (CSL) are often used to describe specifications of probabilistic properties for discrete time and continuous time, respectively. In PCTL and CSL, the possibility of executions satisfying some temporal properties can be quantitatively represented by the probabilistic extension of the path quantifiers in their basic Computation Tree Logic (CTL), however, path formulae of them are expressed via the same operators in CTL. For this reason, both of them cannot represent formulae with quantitative temporal properties, such as…
A recent line of research has developed around logics of belief based on evidence. The approach of Bílková et al understands belief as based on information confirmed by a reliable source. We propose a finer analysis of how belief can be based on information, where the confirmation comes from multiple possibly conflicting sources and is of a probabilistic nature. We use Belnap-Dunn logic and its probabilistic extensions to account for potentially contradictory information on which belief is grounded. We combine it with an extension of Lukasiewicz logic, or a bilattice logic, within a two-layer …
