Logical formalisms provide a natural and concise means for specifying and reasoning about preferences. In this paper, we propose lexicographic logic, an extension of classical propositional logic that can express a variety of preferences, most notably lexicographic ones. The proposed logic supports a simple new connective whose semantics can be defined in terms of finite lists of truth values. We demonstrate that, despite the well-known theoretical limitations that pose barriers to the quantitative representation of lexicographic preferences, there exists a subset of the rational numbers over which the proposed new connective can be naturally defined. Lexicographic logic can be used to define in a simple way some well-known preferential operators, like "$A$ and if possible $B$", and "$A$ or failing that $B$". Moreover, many other hierarchical preferential operators can be defined using a systematic approach. We argue that the new logic is an effective formalism for ranking query results according to the satisfaction level of user preferences.

Logical forgetting is removing some variables from a logical formula while preserving its information regarding the others [LLM03]. Seen from a different angle, it is restricting a formula to keep only its information about some of its variables [Del17]. Each of the two interpretations has its own applications. Removing information allow reducing memory requirements [EKI19], simplifying reasoning [DW15, EF07, WSS05] and clarifying the relationship about variables [Del17]. Restricting a formula on some variables allows formalizing the limited knowledge of agents [FHMV95, RHPT14], ensuring privacy [GKLW17] and removing inconsistency [LM10].

A recent work has shown that transformers are able to "reason" with facts and rules in a limited setting where the rules are natural language expressions of conjunctions of conditions implying a conclusion. Since this suggests that transformers may be used for reasoning with knowledge given in natural language, we do a rigorous evaluation of this with respect to a common form of knowledge and its corresponding reasoning -- the reasoning about effects of actions. Reasoning about action and change has been a top focus in the knowledge representation subfield of AI from the early days of AI and more recently it has been a highlight aspect in common sense question answering. We consider four action domains (Blocks World, Logistics, Dock-Worker-Robots and a Generic Domain) in natural language and create QA datasets that involve reasoning about the effects of actions in these domains. We investigate the ability of transformers to (a) learn to reason in these domains and (b) transfer that learning from the generic domains to the other domains.

A sliding puzzle is a combination puzzle where a player slide pieces along certain routes on a board to reach a certain end-configuration. In this paper, we propose a novel measurement of complexity of massive sliding puzzles with paramodulation which is an inference method of automated reasoning. It turned out that by counting the number of clauses yielded with paramodulation, we can evaluate the difficulty of each puzzle. In experiment, we have generated 100 * 8 puzzles which passed the solvability checking by countering inversions. By doing this, we can distinguish the complexity of 8 puzzles with the number of generated with paramodulation. For example, board [2,3,6,1,7,8,5,4, hole] is the easiest with score 3008 and board [6,5,8,7,4,3,2,1, hole] is the most difficult with score 48653. Besides, we have succeeded to obverse several layers of complexity (the number of clauses generated) in 100 puzzles. We can conclude that proposal method can provide a new perspective of paramodulation complexity concerning sliding block puzzles.

Recognition in planning seeks to find agent intentions, goals or activities given a set of observations and a knowledge library (e.g. goal states, plans or domain theories). In this work we introduce the problem of Online Action Recognition. It consists in recognizing, in an open world, the planning action that best explains a partially observable state transition from a knowledge library of first-order STRIPS actions, which is initially empty. We frame this as an optimization problem, and propose two algorithms to address it: Action Unification (AU) and Online Action Recognition through Unification (OARU). The former builds on logic unification and generalizes two input actions using weighted partial MaxSAT. The latter looks for an action within the library that explains an observed transition. If there is such action, it generalizes it making use of AU, building in this way an AU hierarchy. Otherwise, OARU inserts a Trivial Grounded Action (TGA) in the library that explains just that transition. We report results on benchmarks from the International Planning Competition and PDDLGym, where OARU recognizes actions accurately with respect to expert knowledge, and shows real-time performance.

The ability of an agent to comprehend a sentence is tightly connected to the agent's prior experiences and background knowledge. The paper suggests to interpret comprehension as a modality and proposes a complete bimodal logical system that describes an interplay between comprehension and knowledge modalities.

Inductive Logic Programming (ILP) systems learn generalised, interpretable rules in a data-efficient manner utilising existing background knowledge. However, current ILP systems require training examples to be specified in a structured logical format. Neural networks learn from unstructured data, although their learned models may be difficult to interpret and are vulnerable to data perturbations at run-time. This paper introduces a hybrid neural-symbolic learning framework, called NSL, that learns interpretable rules from labelled unstructured data. NSL combines pre-trained neural networks for feature extraction with FastLAS, a state-of-the-art ILP system for rule learning under the answer set semantics. Features extracted by the neural components define the structured context of labelled examples and the confidence of the neural predictions determines the level of noise of the examples. Using the scoring function of FastLAS, NSL searches for short, interpretable rules that generalise over such noisy examples. We evaluate our framework on propositional and first-order classification tasks using the MNIST dataset as raw data. Specifically, we demonstrate that NSL is able to learn robust rules from perturbed MNIST data and achieve comparable or superior accuracy when compared to neural network and random forest baselines whilst being more general and interpretable.

This article contains a proposal to add coinduction to the computational apparatus of natural language understanding. This, we argue, will provide a basis for more realistic, computationally sound, and scalable models of natural language dialogue, syntax and semantics. Given that the bottom up, inductively constructed, semantic and syntactic structures are brittle, and seemingly incapable of adequately representing the meaning of longer sentences or realistic dialogues, natural language understanding is in need of a new foundation. Coinduction, which uses top down constraints, has been successfully used in the design of operating systems and programming languages. Moreover, implicitly it has been present in text mining, machine translation, and in some attempts to model intensionality and modalities, which provides evidence that it works. This article shows high level formalizations of some of such uses. Since coinduction and induction can coexist, they can provide a common language and a conceptual model for research in natural language understanding. In particular, such an opportunity seems to be emerging in research on compositionality. This article shows several examples of the joint appearance of induction and coinduction in natural language processing. We argue that the known individual limitations of induction and coinduction can be overcome in empirical settings by a combination of the the two methods. We see an open problem in providing a theory of their joint use.

We present a model for Bayesian filtering (BF) in discrete dynamic systems where multiple entities (inter)-act, i.e. where the system dynamics is naturally described by a Multiset rewriting system (MRS). Typically, BF in such situations is computationally expensive due to the high number of discrete states that need to be maintained explicitly. We devise a lifted state representation, based on a suitable decomposition of multiset states, such that some factors of the distribution are exchangeable and thus afford an efficient representation. Intuitively, this representation groups together similar entities whose properties follow an exchangeable joint distribution. Subsequently, we introduce a BF algorithm that works directly on lifted states, without resorting to the original, much larger ground representation. This algorithm directly lends itself to approximate versions by limiting the number of explicitly represented lifted states in the posterior. We show empirically that the lifted representation can lead to a factorial reduction in the representational complexity of the distribution, and in the approximate cases can lead to a lower variance of the estimate and a lower estimation error compared to the original, ground representation.

Many computational problems in modern society account to probabilistic reasoning, statistics, and combinatorics. A variety of these real-world questions can be solved by representing the question in (Boolean) formulas and associating the number of models of the formula directly with the answer to the question. Since there has been an increasing interest in practical problem solving for model counting over the last years, the Model Counting (MC) Competition was conceived in fall 2019. The competition aims to foster applications, identify new challenging benchmarks, and to promote new solvers and improve established solvers for the model counting problem and versions thereof. We hope that the results can be a good indicator of the current feasibility of model counting and spark many new applications. In this paper, we report on details of the Model Counting Competition 2020, about carrying out the competition, and the results. The competition encompassed three versions of the model counting problem, which we evaluated in separate tracks. The first track featured the model counting problem (MC), which asks for the number of models of a given Boolean formula. On the second track, we challenged developers to submit programs that solve the weighted model counting problem (WMC). The last track was dedicated to projected model counting (PMC). In total, we received a surprising number of 9 solvers in 34 versions from 8 groups.