Extension Of Propositional Dynamic Logic Research Articles

Exploring Non-Regular Extensions of Propositional Dynamic Logic with Description-Logics Features

We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.

Smart Sensorization Using Propositional Dynamic Logic.

The current high energy prices pose a serious challenge, especially in the domestic economy. In this respect, one of the main problems is obtaining domestic hot water. For this reason, this article develops a heating system applied to a conventional water tank in such a way as to minimize the necessary energy supply by converting it, under certain circumstances, into atmospheric. For this purpose, the domotic system has been equipped with sensors that automate the pressurization of the compartment and solenoid valves that regulate the external water supply. This design, to which different level sensors are applied, sends the information in real time to an artificial intelligence system, by means of deductive control, which recognizes the states of the system. This work shows the introduction of an extension of propositional dynamic logic in the field of energy efficiency. Thanks to this formalism, a qualitative control of the program variables is achieved by incorporating qualitative reasoning tools. On the other hand, it solves preventive maintenance systems through the early detection of faults in the installation. This research has led to the patenting of an intelligent domestic hot water system that considerably reduces energy consumption by setting disjointed heating intervals that, powered by renewable or non-renewable sources, are controlled by a propositional dynamic logic.

The Influence of the Test Operator on the Expressive Power of PDL-like Logics

Abstract Berman and Paterson proved that test-free propositional dynamic logic (PDL) is weaker than PDL. One would raise questions: does a similar result also hold for extensions of PDL? For example, is test-free converse-PDL (CPDL) weaker than CPDL? In what circumstances the test operator can be eliminated without reducing the expressive power of a PDL-based logical formalism? These problems have not yet been studied. As the description logics $\mathcal{ALC}_{trans}$ and $\mathcal{ALC}_{reg}$ are, respectively, variants of test-free PDL and PDL, there is a concept of $\mathcal{ALC}_{reg}$ that is not equivalent to any concept of $\mathcal{ALC}_{trans}$. Generalizing this, we prove that there is a concept of $\mathcal{ALC}_{reg}$ that is not equivalent to any concept of the logic that extends $\mathcal{ALC}_{trans}$ with inverse roles, nominals, qualified number restrictions, the universal role and local reflexivity of roles. We also provide some results for the case with RBoxes and TBoxes. One of them states that tests can be eliminated from TBoxes of the deterministic Horn fragment of $\mathcal{ALC}_{reg}$.

Extending Propositional Dynamic Logic for Petri Nets

Propositional Dynamic Logic (PDL) is a multi-modal logic used for specifying and reasoning on sequential programs. Petri Nets is a widely used formalism to specify and analyze concurrent programs with a very intuitive graphical representation. Petri-PDL is an extension of PDL, whose programs are Petri Nets, which combines the advantages of both formalisms using a compositional and structural approach to deal with Petri Nets.In this work we present an extension of Petri-PDL to deal with Stochastic Petri Nets, the DS3 logic. This system is an alternative to model performance evaluation in a compositional and structural approach. We discuss about its soundness, decidability and completeness regarding our semantics and present a proof of EXPTime-completeness of its SAT problem. A usage example is presented.

A logic framework for reasoning with movement based on fuzzy qualitative representation

We present a logic approach to reason with moving objects under fuzzy qualitative representation. This way, we can deal both with qualitative and quantitative information, and consequently, to obtain more accurate results. The proposed logic system is introduced as an extension of Propositional Dynamic Logic: this choice, on the one hand, simplifies the theoretical study concerning soundness, completeness and decidability; on the other hand, provides the possibility of constructing complex relations from simpler ones and the use of a language very close to programming languages.

Separation of Test-Free Propositional Dynamic Logics over Context-Free Languages

For a class L of languages let PDL[L] be an extension of Propositional Dynamic Logic which allows programs to be in a language of L rather than just to be regular. If L contains a non-regular language, PDL[L] can express non-regular properties, in contrast to pure PDL. For regular, visibly pushdown and deterministic context-free languages, the separation of the respective PDLs can be proven by automata-theoretic techniques. However, these techniques introduce non-determinism on the automata side. As non-determinism is also the difference between DCFL and CFL, these techniques seem to be inappropriate to separate PDL[DCFL] from PDL[CFL]. Nevertheless, this separation is shown but for programs without test operators.

A logic for reasoning about responsibility

One way to allocate tasks to agents is by ascribing them obligations. From obligations to be, agents are able to infer what are the forbidden, permitted and obligatory actions they may perform, by using the well-known Meyer’s reduction from obligations to be to obligations to do. However, we show through an example that this method is not completely adequate to guide agents’ decisions. We then propose a solution using, instead of obligations, the concept of ‘responsibility’. To formalise responsibility we use a multiagent extension of propositional dynamic logic as framework, and then we define some basic concepts, such as ‘agent ability’, also briefly discussing the problem of uniform strategies and a possible solution. In the last part, we show that our framework can be used in the specification of normative multiagent systems, by presenting an extensive running example.

A note on an extension of PDL

Recently visibly pushdown automata have been extended to so-called k-phase multi-stack visibly pushdown automata ( k-MVPAs). On the occasion of introducing k-MVPAs, it has been asked whether the extension of Propositional Dynamic Logic with k-MVPAs still leads to a decidable logic. This question is answered negatively here.

PDL with negation of atomic programs

Propositional dynamic logic (PDL) is one of the most successful variants of modal logic. To make it even more useful for applications, many extensions of PDL have been considered in the literature. A very natural and useful such extension is with negation of programs. Unfortunately, as long-known, reasoning with the resulting logic is undecidable. In this paper, we consider the extension of PDL with negation of atomic programs, only. We argue that this logic is still useful, e.g. in the context of description logics, and prove that satisfiability is decidable and EXPTIME-complete using an approach based on Büchi tree automata.

The complexity of PDL with interleaving

To provide a logic for reasoning about concurrently executing programs, Abrahamson has defined an extension of propositional dynamic logic (PDL) by allowing interleaving as an operator for combining programs, in addition to the regular PDL operators union, concatenation, and star. We show that the satisfiability problem for interleaving PDL is complete for deterministic double-exponential time, and that this problem requires time double-exponential in cn log n for some positive constant c. Moreover, this lower bound holds even when restricted to formulas where each program appearing in the formula has the form a 1¦a 2¦ … ¦a k where ¦ denotes the interleaving operator and where a 1, …, a k are regular programs, i.e., programs built from atomic programs using only the regular operators. Another consequence of the method used to prove this result is that the equivalence problem for regular expressions with interleaving requires space 2 cn log n and that this lower bound holds even to decide whether (E 1¦E 2¦ … ¦E k) ∪ F ≡ ∑ ∗ where E 1, …, E k , F are ordinary regular expressions; this improves a previous result of the authors. Moreover, the same lower bound holds for the containment problem for expressions of the form E 1¦E 2¦ … ¦E k .

∏ 11-universality of some propositional logics of concurrent programs

It is proved that validity problems for two variants of propositional dynamic logic (PDL) connected with concurrent programming are highly undecidable (∏ 1 1-universal). These variants are an extension of PDL by asynchronous programming constructs shuffle and iterated shuffle and a variant of PDL with a partial commutativity relation on primitive programs. In both cases propositional variables are not used.

Deciding Properties of Nonregular Programs

Extensions of propositional dynamic logic (PDL) with nonregular programs are considered. Three classes of nonregular languages are defined, and for each of them it is shown that for any language L in the class, PDL, with L added to the set of regular programs as a new program, is decidable. The first class consists of the languages accepted by pushdown automata that act only on the basis of their input symbol, except when determining whether they reject or continue. The second class (which contains even noncontext-free languages) consists of the languages accepted by deterministic stack machines, but which have a unique new symbol prefixing each word. The third class represents a certain delicate combination of these, and, in particular, it serves to prove the 1983 conjecture that PDL with the addition of the language $\{ {a^i b^i c^i |i \geqslant 0} \}$ is decidable.

Propositional dynamic logic with local assignment

We propose an extension of Propositional Dynamic Logic which allows a new kind of program terms—local assignments to propositional variables. They are very close to the known array assignments in Dynamic Logic beacuse they allow us to change the truth values of predicate variables. In this logic, many notions, like equivalence of programs, looping and finitely branching, are expressible on a propositional level. In fact, we show that the resulting logic is equivalent in expressive power to first-order logic augmented by a device to express transitive closures. In other words, it is (modulo extra predicate symbols) equivalent to first-order dynamic logic. Not suprisingly, therefore, the validity problem for this extension is Π 1 1-complete.

Looping vs. repeating in dynamic logic

Two extensions of propositional dynamic logic for dealing with infinite computations, LPDL and RPDL, are compared in expressive power. The first is obtained by adding the assertion loop (α) for any program α, meaning “α contains an infinite computation,≓ and the second by adding repeat (α), meaning “α can be repeated indefinitely.≓ While repeat can be used to encode loop , and hence LPDL ⩽ RPDL, it is shown here that the converse fails. Thus LPDL < RPDL. The proof is surprisingly nontrivial, especially in the presence of tests. The significance of the result is dicussed and is put in perspective with other known results for LPDL, RPDL, and their first-order counterparts. Some open questions are posed.

Propositional dynamic logic of looping and converse is elementarily decidable

Propositional dynamic logic is a formal system for reasoning about the before—after behavior of regular program schemes. An extension of propositional dynamic logic which includes both an infinite looping construct and a converse or backtracking construct is considered and it is proved that the satisfiability problem for this logic is elementarily decidable. In order to establish this result, deterministic two-way automata on infinite trees are defined, and it is shown how they can be simulated by nondeterministic one-way automata. The satisfiability problem for propositional dynamic logic of looping and converse is then reduced to the emptiness problem for these two-way automata.