Conference Paper

Pruning Search Space for Weighted First Order Horn Clause Satisfiability

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Many SRL models pose logical inference as weighted satisfiability solving. Performing logical inference after completely grounding clauses with all possible constants is computationally expensive and approaches such as LazySAT [8] utilize the sparseness of the domain to deal with this. Here, we investigate the efficiency of restricting the Knowledge Base (Σ) to the set of first order horn clauses. We propose an algorithm that prunes the search space for satisfiability in horn clauses and prove that the optimal solution is guaranteed to exist in the pruned space. The approach finds a model, if it exists, in polynomial time; otherwise it finds an interpretation that is most likely given the weights. We provide experimental evidence that our approach reduces the size of search space substantially.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

Article
The Indian Institute of Technology (IIT) Bombay has a history of research and development in the area of databases, dating back to the early 1980s. D. B. Phatak and N. L. Sarda were among the first faculty members at IIT Bombay to work in the area of database systems. The number of PhD students increased from around 1 or 2 enrolled at a time in the early 1990s, to about 12 to 15 students at a time in recent years. While this number is much better than earlier, and is increasing rapidly, it is still small by most standards. However, the master's and bachelor's students have compensated for the shortage of PhD students, and have made very significant contributions to the research efforts, with well over three fourths of the papers having such students as coauthors. Graph data models are ubiquitous in semistructured search. Modeling a data graph as an electrical network, or equivalently, as a Markovian 'random surfer' process, is widely used in applications that need to characterize some notion of graph proximity.
Article
Full-text available
We introduce a greedy local search procedure called GSAT for solving propositional satisfiability problems. Our experiments show that this procedure can be used to solve hard, randomly generated problems that are an order of magnitude larger than those that can be handled by more traditional approaches such as the Davis-Putnam procedure or resolution. We also show that GSAT can solve structured satisfiability problems quickly. In particular, we solve encodings of graph coloring problems, N-queens, and Boolean induction. General application strategies and limitations of the ap-proach are also discussed. GSAT is best viewed as a model-finding procedure. Its good performance suggests that it may be advan-tageous to reformulate reasoning tasks that have tra-ditionally been viewed as theorem-proving problems as model-finding tasks.
Conference Paper
Full-text available
As exact inference for first-order probabilistic graphical models at the propositional level can be formidably expensive, there is an ongoing effort to design efficient lifted inference algorithms for such models. This paper discusses directed first-order models that require an aggregation operator when a parent random variable is parameterized by logical variables that are not present in a child random vari- able. We introduce a new data structure, aggrega- tion parfactors, to describe aggregation in directed first-order models. We show how to extend Milch et al.'s C-FOVE algorithm to perform lifted infer- ence in the presence of aggregation parfactors. We also show that there are cases where the polynomial time complexity (in the domain size of logical vari- ables) of the C-FOVE algorithm can be reduced to logarithmic time complexity using aggregation par- factors.
Article
Full-text available
In this paper we introduce MINIMAXSAT, a new Max-SAT solver that is built on top of MIN- ISAT+. It incorporates the best current SAT and Max-SAT techniques. It can handle hard clauses (clauses of mandatory satisfaction as in SAT), soft clauses (clauses whose falsification is penal- ized by a cost as in Max-SAT) as well as pseudo-boolean objective functions and constraints. Its main features are: learning and backjumping on hard clauses; resolution-based and substraction- based lower bounding; and lazy propagation with the two-watched literal scheme. Our empirical evaluation comparing a wide set of solving alternatives on a broad set of optimization benchmarks indicates that the performance of MINIMAXSAT is usually close to the best specialized alternative and, in some cases, even better.
Conference Paper
Full-text available
We introduce a greedy local search procedure called GSAT for solving propositional satisfiability problems. Our experiments show that this procedure can be used to solve hard, randomly generated problems that are an order of magnitude larger than those that can be handled by more traditional approaches such as the Davis-Putnam procedure or resolution. We also show that GSAT can solve structured satisfiability problems quickly. In particular, we solve encodings of graph coloring problems, N-queens, and Boolean induction. General application strategies and limitations of the approach are also discussed. GSAT is best viewed as a model-finding procedure. Its good performance suggests that it may be advantageous to reformulate reasoning tasks that have traditionally been viewed as theorem-proving problems as model-finding tasks. Introduction The property of NP-hardness is traditionally taken to be the barrier separating tasks that can be solved computationally with realistic resources fr...
Article
Full-text available
It has recently been shown that local search is surprisingly good at finding satisfying assignments for certain classes of CNF formulas (Selman et al. 1992). In this paper we demonstrate that the power of local search for satisfiability testing can be further enhanced by employing a new strategy, called "mixed random walk", for escaping from local minima. We present a detailed comparison of this strategy with simulated annealing, and show that mixed random walk is the superior strategy on several classes of computationally difficult problem instances. We also present results demonstrating the effectiveness of local search with walk for solving circuit synthesis and diagnosis problems. Finally, we show that mixed random walk improves upon the results of Hansen and Jaumard on MAX-SAT problems. 1 Introduction Local search algorithms have been successfully applied to many optimization problems. Hansen and Jaumard (1990) describe experiments using local search for MAX-SAT, i.e...
Conference Paper
Many machine learning applications require a combination of probability and rst-order logic. Markov logic networks (MLNs) accomplish this by attaching weights to rst-order clauses, and viewing these as templates for features of Markov networks. Model parameters (i.e., clause weights) can be learned by maximizing the likelihood of a relational database, but this can be quite costly and lead to suboptimal results for any given prediction task. In this paper we pro- pose a discriminative approach to training MLNs, one which optimizes the conditional likelihood of the query predicates given the evidence ones, rather than the joint likelihood of all predicates. We extend Collins's (2002) voted perceptron algorithm for HMMs to MLNs by replacing the Viterbi algo- rithm with a weighted satisability solver. Experiments on entity resolution and link prediction tasks show the advan- tages of this approach compared to generative MLN training, as well as compared to purely probabilistic and purely logical approaches.
Conference Paper
Propositionalization of a first-order theory followed by sat- isfiability testing has proved to be a remarkably efficient approach to inference in relational domains such as plan- ning (Kautz & Selman 1996) and verification (Jackson 2000). More recently, weighted satisfiability solvers have been used successfully for MPE inference in statistical relational learn- ers (Singla & Domingos 2005). However, fully instantiating a finite first-order theory requires memory on the order of the number of constants raised to the arity of the clauses, which significantly limits the size of domains it can be applied to. In this paper we propose LazySAT, a variation of the Walk- SAT solver that avoids this blowup by taking advantage of the extreme sparseness that is typical of relational domains (i.e., only a small fraction of ground atoms are true, and most clauses are trivially satisfied). Experiments on entity reso- lution and planning problems show that LazySAT reduces memory usage by orders of magnitude compared to Walk- SAT, while taking comparable time to run and producing the same solutions.
Conference Paper
Markov logic networks (MLNs) are a statistical relational learning model that consists of a set of weighted first-order clauses and provides a way of softening first-order logic. Several machine learning problems have been successfully addressed by treating MLNs as a "programming language" where a set of features expressed in first-order logic is manually engineered by the designer and then weights for these features are learned from the data. Inference over the learned model is an important step in this process both because several weight-learning algorithms involve performing inference multiple times during training and because inference is used to evaluate and use the final model. "Programming" with an MLN would therefore be significantly facilitated by speeding up inference, thus providing the ability to quickly observe the performance of new hand-coded features. This paper presents a meta-inference algorithm that can speed up any of the available inference techniques by first clustering the query literals and then performing full inference for only one representative from each cluster. Our approach to clustering the literals does not depend on the weights of the clauses in the model. Thus, when learning weights for a fixed set of clauses, the clustering step incurs only a one-time up-front cost.
Conference Paper
Statistical-relational reasoning has received much attention due to its ability to robustly model com- plex relationships. A key challenge is tractable inference, especially in domains involving many objects, due to the combinatorics involved. One can accelerate inference by using approximation techniques, "lazy" algorithms, etc. We consider Markov Logic Networks (MLNs), which involve counting how often logical formulae are satisfied. We propose a preprocessing algorithm that can substantially reduce the effective size of MLNs by rapidly counting how often the evidence satis- fies each formula, regardless of the truth values of the query literals. This is a general preprocess- ing method that loses no information and can be used for any MLN inference algorithm. We eval- uate our algorithm empirically in three real-world domains, greatly reducing the work needed during subsequent inference. Such reduction might even allow exact inference to be performed when sam- pling methods would be otherwise necessary.
Article
We study the problem of learning to infer hidden-state sequences of processes whose states and observations are propositionally or relationally factored. Unfortunately, standard exact inference techniques such as Viterbi and graphical model inference exhibit exponential complexity for these processes. The main motivation behind our work is to identify a restricted space of models, which facilitate efficient inference, yet are expressive enough to remain useful in many applications. In particular, we present the penalty-logic simple-transition model, which utilizes a very simple-transition structure where the transition cost between any two states is constant. While not appropriate for all complex processes, we argue that it is often rich enough in many applications of interest, and when it is applicable there can be inference and learning advantages compared to more general models. In particular, we show that sequential inference for this model, that is, finding a minimum-cost state sequence, efficiently reduces to a single-state minimization (SSM) problem. We then show how to define atemporal-cost models in terms of penalty logic, or weighted logical constraints, and how to use this representation for practically efficient SSM computation. We present a method for learning the weights of our model from labeled training data based on Perceptron updates. Finally, we give experiments in both propositional and relational video-interpretation domains showing advantages compared to more general models.
Article
We present a new branch and bound algorithm for weighted Max-SAT, called Lazy which incorporates original data structures and inference rules, as well as a lower bound of better quality. We provide experimental evidence that our solver is very competitive and outperforms some of the best performing Max-SAT and weighted Max-SAT solvers on a wide range of instances.
Article
The programming of a proof procedure is discussed in connection with trial runs and possible improvements.