Decidability and Complexity of Some Finitely-valued Dynamic Logics

2021 ◽  
Author(s):  
Igor Sedlár
Keyword(s):  
Transitive Closure ◽  
Description Logics ◽  
Dynamic Logic ◽  
Time Algorithm ◽  
Action Execution ◽  
Propositional Dynamic Logic ◽  
Relational Models ◽  
Reasoning About Actions ◽  
Decidability And Complexity ◽  
Dynamic Logics

Propositional Dynamic Logic, PDL, is a well known modal logic formalizing reasoning about complex actions. We study many-valued generalizations of PDL based on relational models where satisfaction of formulas in states and accessibility between states via action execution are both seen as graded notions, evaluated in a finite Łukasiewicz chain. For each n>1, the logic PDŁn is obtained using the n-element Łukasiewicz chain, PDL being equivalent to PDŁ2. These finitely-valued dynamic logics can be applied in formalizing reasoning about actions specified by graded predicates, reasoning about costs of actions, and as a framework for certain graded description logics with transitive closure of roles. Generalizing techniques used in the case of PDL we obtain completeness and decidability results for all PDŁn. A generalization of Pratt's exponential-time algorithm for checking validity of formulas is given and EXPTIME-hardness of each PDŁn validity problem is established by embedding PDL into PDŁn.

2-Exp Time lower bounds for propositional dynamic logics with intersection

2005 ◽  
Vol 70 (4) ◽  
pp. 1072-1086 ◽  
Author(s):  
Martin Lange ◽  
Carsten Lutz
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Open Problem ◽  
Dynamic Logic ◽  
Propositional Dynamic Logic ◽  
Tree Structures ◽  
Exponential Time ◽  
Double Exponential ◽  
Test Operator ◽  
Dynamic Logics

AbstractIn 1984. Danecki proved that satisfiability in IPDL, i.e., Propositional Dynamic Logic (PDL) extended with an intersection operator on programs, is decidabie in deterministic double exponential time. Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004. the first author established ExpSpace-hardness. In this paper, we finally close the gap and prove that IPDL is hard for 2-ExpTime. thus 2-ExpTime-complete. We then sharpen our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures.

scholarly journals Propositional Dynamic Logic for Planning

2020 ◽  
Author(s):  
Mario Folhadela Benevides ◽  
Anna Moreira De Oliveira
Keyword(s):  
Computational Complexity ◽  
Dynamic Logic ◽  
Planning Problem ◽  
Propositional Dynamic Logic ◽  
Reasoning About Actions ◽  
Future Work ◽  
New Framework

This paper presents an on going work on Propositional Dynamic Logic PDL in which atomic programs are STRIPS actions. We think that this new framework is appropriate to reasoning about actions and plans when dealing with planning problem. Unlike, PDL atomic programs, STRIPS actions have pre-conditions and post-conditions. We propose a novel operator of action composition that takes in account the features of STRIPS actions. We propose an axiomatization and prove its soundness. Completeness, decidability and computational complexity are left as future work.

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

2019 ◽  
Vol 29 (8) ◽  
pp. 1289-1310
Author(s):  
Linh Anh Nguyen
Keyword(s):  
Description Logics ◽  
Dynamic Logic ◽  
Expressive Power ◽  
Propositional Dynamic Logic ◽  
Test Operator ◽  
Local Reflexivity ◽  
Horn Fragment ◽  
Logical Formalism

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 $\mathcal{ALC}_{reg}$.

scholarly journals A layered rule-based architecture for approximate knowledge fusion?

2010 ◽  
Vol 7 (3) ◽  
pp. 617-642 ◽  
Author(s):  
Barbara Dunin-Kęplicz ◽  
Anh Nguyen ◽  
Andrzej Szałas
Keyword(s):  
Real World ◽  
Description Logics ◽  
Dynamic Logic ◽  
Deductive Database ◽  
General Mechanism ◽  
Multi Agent Systems ◽  
Propositional Dynamic Logic ◽  
Real World Data ◽  
Knowledge Fusion ◽  
Database Technology

In this paper we present a framework for fusing approximate knowledge obtained from various distributed, heterogenous knowledge sources. This issue is substantial in modeling multi-agent systems, where a group of loosely coupled heterogeneous agents cooperate in achieving a common goal. In paper [5] we have focused on defining general mechanism for knowledge fusion. Next, the techniques ensuring tractability of fusing knowledge expressed as a Horn subset of propositional dynamic logic were developed in [13,16]. Propositional logics may seem too weak to be useful in real-world applications. On the other hand, propositional languages may be viewed as sublanguages of first-order logics which serve as a natural tool to define concepts in the spirit of description logics [2]. These notions may be further used to define various ontologies, like e.g. those applicable in the Semantic Web. Taking this step, we propose a framework, in which our Horn subset of dynamic logic is combined with deductive database technology. This synthesis is formally implemented in the framework of HSPDL architecture. The resulting knowledge fusion rules are naturally applicable to real-world data.

scholarly journals A Comparison of Solvers for Propositional Dynamic Logic

10.29007/63hq ◽  
2018 ◽  
Author(s):  
Ullrich Hustadt ◽  
Renate A. Schmidt
Keyword(s):  
Dynamic Logic ◽  
Propositional Dynamic Logic ◽  
Dynamic Logics

Calculi for propositional dynamic logics have been investigated since the introduction of this logic in the late seventies. Only in recent years have practical procedures been suggested and implemented. In this paper, we compare three such systems, namely, the Tableau Workbench by Abate, Gore, and Widmann (2009), the pdlProver system by Gore and Widmann (2009), and the MLSolver system by Friedmann and Lange (2009).

scholarly journals Reasoning about actions with Temporal Answer Sets

2012 ◽  
Vol 13 (2) ◽  
pp. 201-225 ◽  
Author(s):  
LAURA GIORDANO ◽  
ALBERTO MARTELLI ◽  
DANIELE THESEIDER DUPRÉ
Keyword(s):  
Temporal Logic ◽  
Linear Time ◽  
Dynamic Logic ◽  
Bounded Model Checking ◽  
Propositional Dynamic Logic ◽  
Reasoning About Actions ◽  
Action Language ◽  
Linear Time Temporal Logic ◽  
Answer Sets ◽  
Answer Set

AbstractIn this paper, we combine Answer Set Programming (ASP) with Dynamic Linear Time Temporal Logic (DLTL) to define a temporal logic programming language for reasoning about complex actions and infinite computations. DLTL extends propositional temporal logic of linear time with regular programs of propositional dynamic logic, which are used for indexing temporal modalities. The action language allows general DLTL formulas to be included in domain descriptions to constrain the space of possible extensions. We introduce a notion of Temporal Answer Set for domain descriptions, based on the usual notion of Answer Set. Also, we provide a translation of domain descriptions into standard ASP and use Bounded Model Checking (BMC) techniques for the verification of DLTL constraints.

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