scholarly journals The Computational Complexity of Probabilistic Planning

1998 ◽  
Vol 9 ◽  
pp. 1-36 ◽  
Author(s):  
M. L. Littman ◽  
J. Goldsmith ◽  
M. Mundhenk
Keyword(s):  
Computational Complexity ◽  
Efficient Algorithms ◽  
Boolean Satisfiability ◽  
Complexity Classes ◽  
Probabilistic Planning ◽  
Plan Evaluation ◽  
Partially Ordered ◽  
Boolean Satisfiability Problem ◽  
Complete Problem ◽  
Planning Problems

We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought; we examine totally ordered plans, acyclic plans, and looping plans, and partially ordered plans under three natural definitions of plan value. We show that problems of interest are complete for a variety of complexity classes: PL, P, NP, co-NP, PP, NP^PP, co-NP^PP, and PSPACE. In the process of proving that certain planning problems are complete for NP^PP, we introduce a new basic NP^PP-complete problem, E-MAJSAT, which generalizes the standard Boolean satisfiability problem to computations involving probabilistic quantities; our results suggest that the development of good heuristics for E-MAJSAT could be important for the creation of efficient algorithms for a wide variety of problems.

Chapter 34. Stochastic Boolean Satisfiability

2021 ◽  
Author(s):  
Stephen M. Majercik
Keyword(s):  
Artificial Intelligence ◽  
Computational Complexity ◽  
Probabilistic Reasoning ◽  
Broad Class ◽  
Theoretical Perspective ◽  
Boolean Satisfiability ◽  
Probabilistic Planning ◽  
Empirical Results ◽  
Future Work ◽  
Planning Problems

Stochastic satisfiability (SSAT) is an extension of satisfiability (SAT) that merges two important areas of artificial intelligence: logic and probabilistic reasoning. Initially suggested by Papadimitriou, who called it a “game against nature”, SSAT is interesting both from a theoretical perspective–it is complete for PSPACE, an important complexity class–and from a practical perspective–a broad class of probabilistic planning problems can be encoded and solved as SSAT instances. This chapter describes SSAT and its variants, their computational complexity, applications of SSAT, analytical results, algorithms and empirical results, related work, and directions for future work.

Complete problems for fixed-point logics

1995 ◽  
Vol 60 (2) ◽  
pp. 517-527 ◽  
Author(s):  
Martin Grohe
Keyword(s):  
Fixed Point ◽  
Computational Complexity ◽  
Normal Form ◽  
Query Language ◽  
Main Step ◽  
Complexity Classes ◽  
Complete Problem ◽  
Complete Problems ◽  
Database Query Language ◽  
Natural Description

The notion of logical reducibilities is derived from the idea of interpretations between theories. It was used by Lovász and Gács [LG77] and Immerman [Imm87] to give complete problems for certain complexity classes and hence establish new connections between logical definability and computational complexity.However, the notion is also interesting in a purely logical context. For example, it is helpful to establish nonexpressibility results.We say that a class of τ-structures is a >complete problem for a logic under L-reductions if it is definable in [τ] and if every class definable in can be ”translated” into by L-formulae (cf. §4).We prove the following theorem:1.1. Theorem. There are complete problemsfor partial fixed-point logic andfor inductive fixed-point logic under quantifier-free reductions.The main step of the proof is to establish a new normal form for fixed-point formulae (which might be of some interest itself). To obtain this normal form we use theorems of Abiteboul and Vianu [AV91a] that show the equivalence between the fixed-point logics we consider and certain extensions of the database query language Datalog.In [Dah87] Dahlhaus gave a complete problem for least fixed-point logic. Since least fixed-point logic equals inductive fixed-point logic by a well-known result of Gurevich and Shelah [GS86], this already proves one part of our theorem.However, our class gives a natural description of the fixed-point process of an inductive fixed-point formula and hence sheds some light on completely different aspects of the logic than Dahlhaus's construction, which is strongly based on the features of least fixed-point formulae.

scholarly journals NLocalSAT: Boosting Local Search with Solution Prediction

2020 ◽  
Author(s):  
Wenjie Zhang ◽  
Zeyu Sun ◽  
Qihao Zhu ◽  
Ge Li ◽  
Shaowei Cai ◽  
...  
Keyword(s):  
Neural Network ◽  
Local Search ◽  
Prediction Model ◽  
Computer Science ◽  
Boolean Satisfiability ◽  
Stochastic Local Search ◽  
Boolean Satisfiability Problem ◽  
Complete Problem ◽  
Random Manner ◽  
Np Complete

The Boolean satisfiability problem (SAT) is a famous NP-complete problem in computer science. An effective way for solving a satisfiable SAT problem is the stochastic local search (SLS). However, in this method, the initialization is assigned in a random manner, which impacts the effectiveness of SLS solvers. To address this problem, we propose NLocalSAT. NLocalSAT combines SLS with a solution prediction model, which boosts SLS by changing initialization assignments with a neural network. We evaluated NLocalSAT on five SLS solvers (CCAnr, Sparrow, CPSparrow, YalSAT, and probSAT) with instances in the random track of SAT Competition 2018. The experimental results show that solvers with NLocalSAT achieve 27% ~ 62% improvement over the original SLS solvers.

scholarly journals Compilation-Based Approaches to Parallel Planning: An Empirical Comparison

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Kristýna Pantůčková ◽  
Roman Barták
Keyword(s):  
Programming Language ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Automated Planning ◽  
Boolean Satisfiability ◽  
Planning Problem ◽  
Empirical Comparison ◽  
Boolean Satisfiability Problem ◽  
International Planning ◽  
Planning Problems

Automated planning deals with finding a sequence of actions, a plan, to reach a goal. One of the possible approaches to automated planning is a compilation of a planning problem to a Boolean satisfiability problem or to a constraint satisfaction problem, which takes direct advantage of the advancements of satisfiability and constraint satisfaction solvers. This paper provides a comparison of three encodings proposed for the compilation of planning problems: Transition constraints for parallel planning (TCPP), Relaxed relaxed exist-Step encoding and Reinforced Encoding. We implemented the encodings using the programming language Picat 2.8, we suggested certain modifications, and we compared the performance of the encodings on benchmarks from international planning competitions.

Logical Decision Problems and Complexity of Logic Programs

1987 ◽  
Vol 10 (1) ◽  
pp. 1-33
Author(s):  
Egon Börger ◽  
Ulrich Löwen
Keyword(s):  
Fixed Point ◽  
Computational Complexity ◽  
Decision Problem ◽  
Decision Problems ◽  
Complexity Classes ◽  
Logic Programs ◽  
Finite Structures ◽  
Syntactic Restrictions

We survey and give new results on logical characterizations of complexity classes in terms of the computational complexity of decision problems of various classes of logical formulas. There are two main approaches to obtain such results: The first approach yields logical descriptions of complexity classes by semantic restrictions (to e.g. finite structures) together with syntactic enrichment of logic by new expressive means (like e.g. fixed point operators). The second approach characterizes complexity classes by (the decision problem of) classes of formulas determined by purely syntactic restrictions on the formation of formulas.

scholarly journals Incremental FPT Delay

Algorithms ◽  
2020 ◽  
Vol 13 (5) ◽  
pp. 122
Author(s):  
Arne Meier
Keyword(s):  
Computational Complexity ◽  
Complexity Theory ◽  
Complexity Classes ◽  
Computational Complexity Theory ◽  
Function Classes ◽  
Classical Setting ◽  
Classical Function ◽  
Multivalued Functions ◽  
Relationship Of ◽  
The Relationship

In this paper, we study the relationship of parameterized enumeration complexity classes defined by Creignou et al. (MFCS 2013). Specifically, we introduce two hierarchies (IncFPTa and CapIncFPTa) of enumeration complexity classes for incremental fpt-time in terms of exponent slices and show how they interleave. Furthermore, we define several parameterized function classes and, in particular, introduce the parameterized counterpart of the class of nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist, TFNP, known from Megiddo and Papadimitriou (TCS 1991). We show that this class TF(para-NP), the restriction of the function variant of NP to total functions, collapsing to F(FPT), the function variant of FPT, is equivalent to the result that OutputFPT coincides with IncFPT. In addition, these collapses are shown to be equivalent to TFNP = FP, and also equivalent to P equals NP intersected with coNP. Finally, we show that these two collapses are equivalent to the collapse of IncP and OutputP in the classical setting. These results are the first direct connections of collapses in parameterized enumeration complexity to collapses in classical enumeration complexity, parameterized function complexity, classical function complexity, and computational complexity theory.

Chapter 25. Model Counting

2021 ◽  
Author(s):  
Carla P. Gomes ◽  
Ashish Sabharwal ◽  
Bart Selman
Keyword(s):  
Local Search ◽  
Upper Bound ◽  
Probabilistic Inference ◽  
Efficient Algorithms ◽  
Propositional Formula ◽  
Exact Methods ◽  
Complete Problem ◽  
Model Counting ◽  
New Ideas

Model counting, or counting the number of solutions of a propositional formula, generalizes SAT and is the canonical #P-complete problem. Surprisingly, model counting is hard even for some polynomial-time solvable cases like 2-SAT and Horn-SAT. Efficient algorithms for this problem will have a significant impact on many application areas that are inherently beyond SAT, such as bounded-length adversarial and contingency planning, and, perhaps most importantly, general probabilistic inference. Model counting can be solved, in principle and to an extent in practice, by extending the two most successful frameworks for SAT algorithms, namely, DPLL and local search. However, scalability and accuracy pose a substantial challenge. As a result, several new ideas have been introduced in the last few years that go beyond the techniques usually employed in most SAT solvers. These include division into components, caching, compilation into normal forms, exploitation of solution sampling methods, and certain randomized streamlining techniques using special constraints. This chapter discusses these techniques, exploring both exact methods as well as fast estimation approaches, including those that provide probabilistic or statistical guarantees on the quality of the reported lower or upper bound on the model count.

scholarly journals HMCTS-OP: Hierarchical MCTS Based Online Planning in the Asymmetric Adversarial Environment

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 719
Author(s):  
Lina Lu ◽  
Wanpeng Zhang ◽  
Xueqiang Gu ◽  
Xiang Ji ◽  
Jing Chen
Keyword(s):  
Computational Complexity ◽  
State Space ◽  
Planning Horizon ◽  
Search Space ◽  
Monte Carlo Tree Search ◽  
Practical Applications ◽  
Online Planning ◽  
Planning Framework ◽  
Planning Methods ◽  
Planning Problems

The Monte Carlo Tree Search (MCTS) has demonstrated excellent performance in solving many planning problems. However, the state space and the branching factors are huge, and the planning horizon is long in many practical applications, especially in the adversarial environment. It is computationally expensive to cover a sufficient number of rewarded states that are far away from the root in the flat non-hierarchical MCTS. Therefore, the flat non-hierarchical MCTS is inefficient for dealing with planning problems with a long planning horizon, huge state space, and branching factors. In this work, we propose a novel hierarchical MCTS-based online planning method named the HMCTS-OP to tackle this issue. The HMCTS-OP integrates the MAXQ-based task hierarchies and the hierarchical MCTS algorithms into the online planning framework. Specifically, the MAXQ-based task hierarchies reduce the search space and guide the search process. Therefore, the computational complexity is significantly reduced. Moreover, the reduction in the computational complexity enables the MCTS to perform a deeper search to find better action in a limited time. We evaluate the performance of the HMCTS-OP in the domain of online planning in the asymmetric adversarial environment. The experiment results show that the HMCTS-OP outperforms other online planning methods in this domain.

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