Determinism and non-determinism in PDL

In classical deterministic planning, solutions to planning tasks are simply sequences of actions, but that is not sufficient for contingent plans in non-deterministic environments. Contingent plans are often expressed through policies that map states to actions. An alternative is to specify contingent plans as programs, e.g. in the syntax of Propositional Dynamic Logic (PDL). PDL is a logic for reasoning about programs with sequential composition, test and non-deterministic choice. However, as we show in the paper, none of the existing PDL modalities directly captures the notion of a solution to a planning task under non-determinism. We add a new modality to star-free PDL correctly capturing this notion. We prove the appropriateness of the new modality by showing how to translate back and forth between policies and PDL programs under the new modality. More precisely, we show how a policy solution to a planning task gives rise to a program solution expressed via the new modality, and vice versa. We also provide an axiomatisation of our PDL extension through reduction axioms into standard star-free PDL.

Propositional dynamic logic of nonregular programs

Deterministic process logic is elementary

It is an open problem whether definability in Propositional Dynamic Logic (PDL) on forests is decidable. Based on an algebraic characterization by Boja\'nczyk, et. al.,(2012) in terms of forest algebras, Straubing (2013) described an approach to PDL based on a k-fold iterated distributive law. A proof that all languages satisfying such a k-fold iterated distributive law are in PDL would settle decidability of PDL. We solve this problem in the case k=2: All languages recognized by forest algebras satisfying a 2-fold iterated distributive law are in PDL. Furthermore, we show that this class is decidable. This provides a novel nontrivial decidable subclass of PDL, and demonstrates the viability of the proposed approach to deciding PDL in general.

In its many guises and variations, propositional dynamic logic (PDL) plays an important role in various areas of computer science such as databases, artificial intelligence, and computer linguistics. One relevant and powerful variation is ICPDL, the extension of PDL with intersection and converse. Although ICPDL has several interesting applications, its computational properties have never been investigated. In this paper, we prove that ICPDL is decidable by developing a translation to the monadic second order logic of infinite trees. Our result has applications in information logic, description logic, and epistemic logic. In particular, we solve a long-standing open problem in information logic. Another virtue of our approach is that it provides a decidability proof that is more transparent than existing ones for PDL with intersection (but without converse).

The completeness proofs for Propositional Dynamic Logic (PDL) in the literature are non-constructive and usually presented in an informal manner. We obtain a formal and constructive completeness proof for Converse PDL by recasting a completeness proof by Kozen and Parikh into our constructive setting. We base our proof on a Pratt-style decision method for satisfiability constructing finite models for satisfiable formulas and pruning refutations for unsatisfiable formulas. Completeness of Segerberg's axiomatization of PDL is then obtained by translating pruning refutations to derivations in the Hilbert system. We first treat PDL without converse and then extend the proofs to Converse PDL. All results are formalized in Coq/Ssreflect.

Propositional dynamic logic with local assignment

We introduce dynamic algebras and show how they can be used to give an algebraic interpretation to propositional dynamic logic (PDL). Dynamic algebras include all Kripke models, the standard interpretation of PDL. We give a simple algebraic condition on *-free dynamic algebras that is necessary and sufficient for representation by *-free Kripke models. In the presence of*, the condition is sufficient for representation by a nonstandard Kripke model. This result leads to a duality between certain topological Kripke models and dynamic algebras analogous to the duality between Boolean algebras and their Stone spaces.

1 - Structures, Languages and Translations: the Structural Approach to Feature Logic

Agent-based systems have gained immense popularity in the last decade or so. Reasoning methods such as planning and decision-making under uncertainty applied to agents and multi-agent systems is an important area of research in agent-based systems. Most works in planning in uncertain environments incorporate sensing actions. We wish to address planning problems when there are no such sensing actions. For this, we introduce the notion of ‘try actions’ and develop a logic PDL_try that is an extension to propositional dynamic logic (PDL). The logic also allows specification of the implicit knowledge of an agent. We illustrate planning in our logic with a new adaptation of the popular blocks world domain. Example programs (plans) are discussed in detail for single agent and two agent situations.

Dynamic linear time temporal logic

In this paper we study automatic composition synthesis of e-Services, based on automated reasoning. We represent the behavior of an e-Service in terms of a deterministic transition system (or a finite state machine), in which for each action the role of the e-Service, either as initiator or as servant, is highlighted. In this setting we present an algorithm based on satisfiability in a variant of Propositional Dynamic Logic that solves the automatic composition problem. Specifically, given (i) a possibly incomplete specification of the sequences of actions that a client would like to realize, and (ii) aset of available e-Services, our technique synthesizes a composite e-Service that (i) uses only the available e-Services and (ii) interacts with the client “in accordance” to the given specification. We also study the computational complexity of the proposed algorithm.

In this paper we discuss an effective technique for automatic service composition and we present the prototype software that implements it. In particular, we characterize the behavior of a service in terms of a finite state machine. In this setting we discuss a technique based on satisfiability in a variant of Propositional Dynamic Logic that solves the automatic composition problem. Specifically, given (i) a client specification of his desired service, i.e., the service he would like to interact with, and (ii) a set of available services, our technique synthesizes the orchestration schema of a composite service that uses only the available services and fully realizes the client specification. The developed system is an open-source software tool, called \(\mathcal {ESC}\) (e-service composer), that implements our composition technique starting from services, each of them described in terms of a WSDL specification and a behavioral description expressed in any language that can capture finite state machines.KeywordsService CompositionDescription LogicComposite ServiceBehavioral DescriptionService Orient ComputingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dynamic logic is a powerful framework for reasoning about imperative programs. An extension with a concurrent operator, called concurrent propositional dynamic logic (CPDL) [20], was introduced to formalise programs running in parallel. In a different direction, other authors proposed a systematic method for generating multi-valued propositional dynamic logics to reason about weighted programs [15]. This paper presents the first step of combining these two frameworks to introduce uncertainty in concurrent computations. In the proposed framework, a weight is assigned to each branch of the parallel execution, resulting in a (possible) asymmetric parallelism, inherent to the fuzzy programming paradigm [2, 23]. By adopting such an approach, a family of logics is obtained, called multi-valued concurrent propositional dynamic logics (\(\mathcal {GCDL}(\mathbf {A})\)), parametric on an action lattice \(\mathbf {A}\) specifying a notion of “weight” assigned to program execution. Additionally, the validity of some axioms of CPDL is discussed in the new family of generated logics.

Description logics provide powerful languages for representing and reasoning about knowledge of static application domains. The main strength of description logics is that they offer considerable expressive power going far beyond propositional logic, while reasoning is still decidable. There is a demand to bring the power and character of description logics into the description and reasoning of dynamic application domains which are characterized by actions. In this paper, based on a combination of the propositional dynamic logic PDL, a family of description logics and an action formalism constructed over description logics, we propose a family of dynamic description logics DDL(X @) for representing and reasoning about actions, where X represents well-studied description logics ranging from the to the , and X @ denotes the extension of X with the @ constructor. The representation power of DDL(X @) is reflected in four aspects. Firstly, the static knowledge of application domains is represented as RBoxes and acyclic TBoxes of the description logic X. Secondly, the states of the world and the pre-conditions of atomic actions are described by ABox assertions of the description logic X @, and the post-conditions of atomic actions are described by primitive literals of X @. Thirdly, starting with atomic actions and ABox assertions of X @, complex actions are constructed with regular program constructors of PDL, so that various control structures on actions such as the "Sequence", "Choice", "Any-Order", "Iterate", "If-Then-Else", "Repeat-While" and "Repeat-Until" can be represented. Finally, both atomic actions and complex actions are used as modal operators for the construction of formulas, so that many properties on actions can be explicitly stated by formulas. A tableau-algorithm is provided for deciding the satisfiability of DDL(X @)-formulas; based on this algorithm, reasoning tasks such as the realizability, executability and projection of actions can be effectively carried out. As a result, DDL(X @) not only offers considerable expressive power going beyond many action formalisms which are propositional, but also provides decidable reasoning services for actions described by it.