Space Efficiency of Propositional Knowledge Representation Formalisms

We investigate the space efficiency of a Propositional Knowledge Representation (PKR) formalism. Intuitively, the space efficiency of a formalism F in representing a certain piece of knowledge A, is the size of the shortest formula of F that represents A. In this paper we assume that knowledge is either a set of propositional interpretations (models) or a set of propositional formulae (theorems). We provide a formal way of talking about the relative ability of PKR formalisms to compactly represent a set of models or a set of theorems. We introduce two new compactness measures, the corresponding classes, and show that the relative space efficiency of a PKR formalism in representing models/theorems is directly related to such classes. In particular, we consider formalisms for nonmonotonic reasoning, such as circumscription and default logic, as well as belief revision operators and the stable model semantics for logic programs with negation. One interesting result is that formalisms with the same time complexity do not necessarily belong to the same space efficiency class

Redundancy in Logic I: CNF Propositional Formulae

A knowledge base is redundant if it contains parts that can be inferred from the rest of it. We study the problem of checking whether a CNF formula (a set of clauses) is redundant, that is, it contains clauses that can be derived from the other ones. Any CNF formula can be made irredundant by deleting some of its clauses: what results is an irredundant equivalent subset (I.E.S.) We study the complexity of some related problems: verification, checking existence of a I.E.S. with a given size, checking necessary and possible presence of clauses in I.E.S.'s, and uniqueness. We also consider the problem of redundancy with different definitions of equivalence.Comment: Extended and revised version of a paper that has been presented at ECAI 200

On the Role of Canonicity in Bottom-up Knowledge Compilation

We consider the problem of bottom-up compilation of knowledge bases, which is usually predicated on the existence of a polytime function for combining compilations using Boolean operators (usually called an Apply function). While such a polytime Apply function is known to exist for certain languages (e.g., OBDDs) and not exist for others (e.g., DNNF), its existence for certain languages remains unknown. Among the latter is the recently introduced language of Sentential Decision Diagrams (SDDs), for which a polytime Apply function exists for unreduced SDDs, but remains unknown for reduced ones (i.e. canonical SDDs). We resolve this open question in this paper and consider some of its theoretical and practical implications. Some of the findings we report question the common wisdom on the relationship between bottom-up compilation, language canonicity and the complexity of the Apply function

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

Algebraic model counting

Weighted model counting (WMC) is a well-known inference task on knowledge bases, and the basis for some of the most efficient techniques for probabilistic inference in graphical models. We introduce algebraic model counting (AMC), a generalization of WMC to a semiring structure that provides a unified view on a range of tasks and existing results. We show that AMC generalizes many well-known tasks in a variety of domains such as probabilistic inference, soft constraints and network and database analysis. Furthermore, we investigate AMC from a knowledge compilation perspective and show that all AMC tasks can be evaluated using sd-DNNF circuits, which are strictly more succinct, and thus more efficient to evaluate, than direct representations of sets of models. We identify further characteristics of AMC instances that allow for evaluation on even more succinct circuits

The Language of Search

This paper is concerned with a class of algorithms that perform exhaustive search on propositional knowledge bases. We show that each of these algorithms defines and generates a propositional language. Specifically, we show that the trace of a search can be interpreted as a combinational circuit, and a search algorithm then defines a propositional language consisting of circuits that are generated across all possible executions of the algorithm. In particular, we show that several versions of exhaustive DPLL search correspond to such well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF. By thus mapping search algorithms to propositional languages, we provide a uniform and practical framework in which successful search techniques can be harnessed for compilation of knowledge into various languages of interest, and a new methodology whereby the power and limitations of search algorithms can be understood by looking up the tractability and succinctness of the corresponding propositional languages

Studying strategies and types of players:Experiments, logics and cognitive models

How do people reason about their opponent in turn-taking games? Often, people do not make the decisions that game theory would prescribe. We present a logic that can play a key role in understanding how people make their decisions, by delineating all plausible reasoning strategies in a systematic manner. This in turn makes it possible to construct a corresponding set of computational models in a cognitive architecture. These models can be run and fitted to the participants’ data in terms of decisions, response times, and answers to questions. We validate these claims on the basis of an earlier game-theoretic experiment about the turn-taking game “Marble Drop with Surprising Opponent”, in which the opponent often starts with a seemingly irrational move. We explore two ways of segregating the participants into reasonable “player types”. The first way is based on latent class analysis, which divides the players into three classes according to their first decisions in the game: Random players, Learners, and Expected players, who make decisions consistent with forward induction. The second way is based on participants’ answers to a question about their opponent, classified according to levels of theory of mind: zero-order, first-order and second-order. It turns out that increasing levels of decisions and theory of mind both correspond to increasing success as measured by monetary awards and increasing decision times. Next, we use the logical language to express different kinds of strategies that people apply when reasoning about their opponent and making decisions in turn-taking games, as well as the ‘reasoning types’ reflected in their behavior. Then, we translate the logical formulas into computational cognitive models in the PRIMs architecture. Finally, we run two of the resulting models, corresponding to the strategy of only being interested in one’s own payoff and to the myopic strategy, in which one can only look ahead to a limited number of nodes. It turns out that the participant data fit to the own-payoff strategy, not the myopic one. The article closes the circle from experiments via logic and cognitive modelling back to predictions about new experiments

Learning design – making practice explicit

New technologies have immense potential for learning, but the sheer variety possible also creates challenges for learners in terms of navigating through an increasingly complex digital landscape and for teachers in terms of how to design and support learning interventions. How can learners and teachers make informed decisions about what technologies to use in the design and support of learning activities? This presentation will consider this question and present a new methodology for design – 'learning design', which aims to shift the creation and support of learning from what has traditionally been an implicit, belief-based practice to one that is explicit and design based. Learning design research at the Open University, UK has included the development of a set of conceptual design views, a tool for visualising designs (CompendiumLD) and a social networking site, for sharing and discussing learning and teaching ideas and designs (Cloudworks). An overview of this work will be provided, along with a discussion of the perceived benefits of this new approach to educational design