Knowledge Compilation Map Research Articles

An extended knowledge compilation map for conditional preference statements-based and generalized additive utilities-based languages

An extended knowledge compilation map for conditional preference statements-based and generalized additive utilities-based languages

A Compiler for Weak Decomposable Negation Normal Form

This paper integrates weak decomposable negation normal form (wDNNF) circuits, introduced by Akshay et al. in 2018, into the knowledge compilation map. This circuit type generalises decomposable negation normal form (DNNF) circuits in such a way that they allow a restricted form of sharing variables among the inputs of a conjunction node. We show that wDNNF circuits have the same properties as DNNF circuits regarding the queries and transformations presented in the knowledge compilation map, whilst being strictly more succinct than DNNF circuits (that is, they can represent Boolean functions compactly). We also present and evaluate a knowledge compiler, called Bella, for converting CNF formulae into wDNNF circuits. Our experiments demonstrate that wDNNF circuits are suitable for configuration instances.

On the role of logical separability in knowledge compilation

Knowledge compilation is an alternative solution to address demanding reasoning tasks with high complexity via converting knowledge bases into a suitable target language. The notion of logical separability, proposed by Levesque, offers a general explanation for the tractability of clausal entailment for two remarkable languages: decomposable negation normal form and prime implicates. It is interesting to explore what role logical separability plays in problem tractability. In this paper, we apply the notion of logical separability to a number of reasoning problems within the context of propositional logic: satisfiability checking (CO), clausal entailment checking (CE), model counting (CT), model enumeration (ME) and forgetting (FO), as well as their dual tasks, contributing to several recursive procedures. We provide the corresponding logical separability based properties: CO-logical separability, CE-logical separability, CT-logical separability, ME-logical separability and their duals. Based on these properties, we then identify four novel normal forms: CO-LSNNF, CE-LSNNF, CT-LSNNF and ME-LSNNF, as well as their dual languages. We show that each of them is the necessary and sufficient condition under which the corresponding procedure is correct. We finally integrate the above normal forms into the knowledge compilation map.

Knowledge Compilation Meets Logical Separability

Knowledge compilation is an alternative solution to address demanding reasoning tasks with high complexity via converting knowledge bases into a suitable target language. Interestingly, the notion of logical separability, proposed by Levesque, offers a general explanation for the tractability of clausal entailment for two remarkable languages: decomposable negation normal form and prime implicates. It is interesting to explore what role logical separability on earth plays in problem tractability. In this paper, we apply the notion of logical separability in three reasoning problems within the context of propositional logic: satisfiability check (CO), clausal entailment check (CE) and model counting (CT), contributing to three corresponding polytime procedures. We provide three logical separability based properties: CO- logical separability, CE-logical separability and CT-logical separability. We then identify three novel normal forms: CO-LSNNF, CE-LSNNF and CT-LSNNF based on the above properties. Besides, we show that every normal form is the necessary and sufficient condition under which the corresponding procedure is correct. We finally integrate the above four normal forms into the knowledge compilation map.

Backdoor Decomposable Monotone Circuits and Propagation Complete Encodings

We describe a compilation language of backdoor decomposable monotone circuits (BDMCs) which generalizes several concepts appearing in the literature, e.g. DNNFs and backdoor trees. A C-BDMC sentence is a monotone circuit which satisfies decomposability property (such as in DNNF) in which the inputs (or leaves) are associated with CNF encodings from a given base class C. We consider the class of propagation complete (PC) encodings as a base class and we show that PC-BDMCs are polynomially equivalent to PC encodings. Additionally, we use this to determine the properties of PC-BDMCs and PC encodings with respect to the knowledge compilation map including the list of efficient operations on the languages.

Properties of Switch-List Representations of Boolean Functions

In this paper, we focus on a less usual way to represent Boolean functions, namely on representations by switch-lists, which are closely related to interval representations. Given a truth table representation of a Boolean function f the switch-list representation of f is a list of Boolean vectors from the truth table which have a different function value than the preceding Boolean vector in the truth table. The main aim of this paper is to include this type of representation in the Knowledge Compilation Map by Darwiche and Marquis and to argue that switch-lists may in certain situations constitute a reasonable choice for a target language in knowledge compilation. First, we compare switch-list representations with a number of standard representations (such as CNF, DNF, and OBDD) with respect to their relative succinctness. As a by-product of this analysis, we also give a short proof of a longstanding open question proposed by Darwiche and Marquis, namely the incomparability of MODS (models) and PI (prime implicates) representations. Next, using the succinctness result between switch-lists and OBDDs, we develop a polynomial time compilation algorithm from switch-lists to OBDDs. Finally, we analyze which standard transformations and queries (those considered by Darwiche and Marquis) can be performed in polynomial time with respect to the size of the input if the input knowledge is represented by a switch-list. We show that this collection is very broad and the combination of polynomial time transformations and queries is quite unique. Some of the queries can be answered directly using the switch-list input, others require a compilation of the input to OBDD representations which are then used to answer the queries.

Complementary Knowledge Compilation Using the Hyper Extension Rule

We introduce the concept of Complementary formula (COMF), which is a new and non-equivalent way for Knowledge compilation (KC). Based on the Hyper extension rule (HER) which is an expansion of Extension rule (ER), we design a compilation algorithm which can formula compile each Conjunctive normal form (CNF) formula to complementary Fully complementary connected diagram (c-FCCD), named as C2C (CNF formula to cFCCD). Theoretically, c-FCCD is a kind of complementary formulae of the input formulae and can support all queries and partial transformations in KC map. Experimentally, C2C is competitive with the EPCCL compilers KCER, C2E, UKCHER, DKCHER and IKCHER.

Augmenting Negation Normal Form With Irrelevant Variables

Irrelevant variables are always omitted in knowledge compilation languages since their assignments do not change the satisfiability of sentences. In order to identify new knowledge compilation languages and reduce the scale of compiling result of d-DNNF, we augment NNF with irrelevant variables in this paper. The NNF PNI , NNF PI , and NNF NI are proposed based on different combinations of positive literals, negative literals, and irrelevant variables. Each sentence in NNF, NNF PI , NNF NI , or NNF PNI can be translated to an equivalent sentence in another language in polynomial time. We also define d-DNNF NI , d-DNNF PI , and d-DNNF PNI based on decomposability and determinism in NNF, which are subclasses of NNF PI , NNF NI , and NNF PNI , respectively. A number of querying and transforming methods for d-DNNF PNI , d-DNNF PI , and d-DNNF PI are designed to solve relevant reasoning problems in knowledge compilation map. Overall, d-DNNF PI and d-DNNF NI do not reduce the tractability of d-DNNF, so we propose a compressing method for d-DNNF based on d-DNNF PI and d-DNNF NI . The experimentally, the compiling results of the d-DNNF PI and d-DNNF NI are better with respect to d-DNNF for most instances, and our compressing method is significantly effective for all instances.

On the Relative Succinctness of Sentential Decision Diagrams

Sentential decision diagrams (SDDs) introduced by Darwiche in 2011 are a promising representation language for propositional knowledge bases. The relative succinctness of representation languages is an important subject in knowledge compilation. The aim of the paper is to identify which kind of Boolean functions can be represented by SDDs of small size with respect to the number of variables the functions are defined on. For this reason the sets of Boolean functions representable by different representation languages in polynomial size are investigated and SDDs are compared with representation languages from the classical knowledge compilation map of Darwiche and Marquis. Ordered binary decision diagrams (OBDDs) which are a popular data structure for Boolean functions are one of them. SDDs are more general than OBDDs by definition but only recently, a Boolean function was presented with polynomial SDD size but exponential OBDD size. This result is strengthened in several ways. The main result is that a function can be represented by SDDs of small size if the function and its negation have small restricted nondeterministic OBDD representations. Moreover, for important Boolean functions called storage access function polynomial-size SDDs are presented. As a side effect an open problem about the relative succinctness between SDDs and free binary decision diagrams which are more general than OBDDs is answered.

Compile!

This paper is concerned with knowledge compilation (KC), a family of approaches developed in AI for more than twenty years. Knowledge compilation consists in pre-processing some pieces of the available information in order to improve the computational efficiency (especially, the time complexity) of some tasks. In this paper, the focus is laid on three KC topics which gave rise to many works: the development of knowledge compilation techniques for the clausal entailment problem in propositional logic, the concept of compilability and the notion of knowledge compilation map. The three topics, as well as an overview of the main results from the literature, are presented. Some recent research lines are also discussed.

Disjunctive closures for knowledge compilation

The knowledge compilation (KC) map [1] can be viewed as a multi-criteria evaluation of a number of target classes of representations for propositional KC. Using this map, the choice of a class for a given application can be made, considering both the space efficiency of it (i.e., its ability to represent information using little space), and its time efficiency, i.e., the queries and transformations which can be achieved in polynomial time, among those of interest for the application under consideration. When no class of propositional representations offers all the transformations one would expect, some of them can be left implicit. This is the key idea underlying the concept of closure introduced here: instead of performing computationally expensive transformations, one just remembers that they have to be done. In this paper, we investigate the disjunctive closure principles, i.e., disjunction, existential quantification, and their combinations. We provide several characterization results concerning the corresponding closures. We also extend the KC map with new propositional languages obtained as disjunctive closures of several incomplete propositional languages, including the well-known KROM (the CNF formulae containing only binary clauses), HORN (the CNF formulae containing only Horn clauses), and AFF (the affine language, which is the set of conjunctions of XOR-clauses). Each introduced language is evaluated along the lines of the KC map.

A Knowledge Compilation Map for Ordered Real-Valued Decision Diagrams

Valued decision diagrams (VDDs) are data structures that represent functions mapping variable-value assignments to non-negative real numbers. They prove useful to compile cost functions, utility functions, or probability distributions. While the complexity of some queries (notably optimization) and transformations (notably conditioning) on VDD languages has been known for some time, there remain many significant queries and transformations, such as the various kinds of cuts, marginalizations, and combinations, the complexity of which has not been identified so far. This paper contributes to filling this gap and completing previous results about the time and space efficiency of VDD languages, thus leading to a knowledge compilation map for real-valued functions. Our results show that many tasks that are hard on valued CSPs are actually tractable on VDDs.

Reduced ordered binary decision diagram with implied literals: a new knowledge compilation approach

Knowledge compilation is an approach to tackle the computational intractability of general reasoning problems. According to this approach, knowledge bases are converted off-line into a target compilation language which is tractable for on-line querying. Reduced ordered binary decision diagram (ROBDD) is one of the most influential target languages. We generalize ROBDD by associating some implied literals in each node and the new language is called reduced ordered binary decision diagram with implied literals (ROBDD-L). Then we discuss a kind of subsets of ROBDD-L called ROBDD-i with precisely i implied literals (0 \leq i \leq \infty). In particular, ROBDD-0 is isomorphic to ROBDD; ROBDD-\infty requires that each node should be associated by the implied literals as many as possible. We show that ROBDD-i has uniqueness over some specific variables order, and ROBDD-\infty is the most succinct subset in ROBDD-L and can meet most of the querying requirements involved in the knowledge compilation map. Finally, we propose an ROBDD-i compilation algorithm for any i and a ROBDD-\infty compilation algorithm. Based on them, we implement a ROBDD-L package called BDDjLu and then get some conclusions from preliminary experimental results: ROBDD-\infty is obviously smaller than ROBDD for all benchmarks; ROBDD-\infty is smaller than the d-DNNF the benchmarks whose compilation results are relatively small; it seems that it is better to transform ROBDDs-\infty into FBDDs and ROBDDs rather than straight compile the benchmarks.

Knowledge Compilation in the Modal Logic S5

In this paper, we study the knowledge compilation task for propositional epistemic logic S5. We first extend many of the queries and transformations considered in the classical knowledge compilation map to S5. We then show that the notion of disjunctive normal form (DNF) can be profitably extended to the epistemic case; we prove that the DNF fragment of S5, when appropriately defined, satisfies essentially the same queries and transformations as its classical counterpart.

A Knowledge Compilation Map

We propose a perspective on knowledge compilation which calls for analyzing different compilation approaches according to two key dimensions: the succinctness of the target compilation language, and the class of queries and transformations that the language supports in polytime. We then provide a knowledge compilation map, which analyzes a large number of existing target compilation languages according to their succinctness and their polytime transformations and queries. We argue that such analysis is necessary for placing new compilation approaches within the context of existing ones. We also go beyond classical, flat target compilation languages based on CNF and DNF, and consider a richer, nested class based on directed acyclic graphs (such as OBDDs), which we show to include a relatively large number of target compilation languages.