Intuitively, locally finiteness means that the refinement operator is computable, completeness means we can generate, by refinement of a, any element of G related to a given element g 1 by the order relation, and properness means that a refinement operator does not generate elements which are equivalent to the element being refined. When a refinement operator is locally finite, complete and proper, we say that it is ideal. Notice that all the subsumption relations presented above satisfy the reflexive 2 and transitive 3 properties. Therefore, the pair (G,), where G is the set of all DLGs given a set of labels L, and is any of the subsumption relations defined above is a quasi-ordered set. Thus, this opens the door to defining refinement operators for DLGs. Intuitively, a downward refinement operator for DLGs will generate refinements of a given DLG by either adding vertices, edges, or by making some of the labels more specific, thus making the graph more specific. In the following subsections, we will introduce a collection of refinement operators for connected DLGs, and discuss their theoretical properties. A summary of these operators is shown in Table 1, where we show that under the object-identity constraint, all the refinement operators presented in this document are ideal. If we do not impose object-identity, then the operators are locally complete and complete, but not proper.

Steen's (2018) Hintikka set properties for Church's type theory based on primitive equality are reduced to the Hintikka set properties of Benzm\"uller, Brown and Kohlhase (2004) which are based on the logical connectives negation, disjunction and universal quantification.

To a multi-agent logic of knowledge or belief we can add public announcements to model publicly observed information change, or action models to model information change that is differently observed by different agents, but also modalities representing quantification over such information change, such as quantifiers over announcements or quantifiers over actions models. Such additions may result in more complex or undecidable logics, and create a very open landscape of relative expressivity. The survey [88] of such logics focused on open problems. Some such open problems have since then been resolved, and yet others have come to the fore. In this updated survey we review what is known about such logics with quantification over information change, including digressions into what are known as relation changing modal(but often not epistemic) logics. Again we focus on open problems.

What kind of programming language would be most appropriate to serve the needs of integrative, multi-paradigm, multi-software-system approaches to AGI? This question is broached via exploring the more particular question of how to create a more scalable and usable version of the "Atomese" programming language that forms a key component of the OpenCog AGI design (an "Atomese 2.0") . It is tentatively proposed that the core of Atomese 2.0 should be a very flexible framework of rewriting rules for rewriting a metagraph (where the rules themselves are represented within the same metagraph, and some of the intermediate data created and used during the rule-interpretation process may be represented in the same metagraph). This framework should support concurrent rewriting of the metagraph according to rules that are labeled with various sorts of uncertainty-quantifications, and that are labeled with various sorts of types associated with various type systems. A gradual typing approach should be used to enable mixture of rules and other metagraph nodes/links associated with various type systems, and untyped metagraph nodes/links not associated with any type system. This must be done in a way that allows reasonable efficiency and scalability, including in concurrent and distributed processing contexts, in the case where a large percentage of of processing time is occupied with evaluating static pattern-matching queries on specific subgraphs of a large metagraph (including a rich variety of queries such as matches against nodes representing variables, and matches against whole subgraphs, etc.).

The Sentential Decision Diagram (SDD) is a tractable representation of Boolean functions that subsumes the famous Ordered Binary Decision Diagram (OBDD) as a strict subset. SDDs are attracting much attention because they are more succinct than OBDDs, as well as having canonical forms and supporting many useful queries and transformations such as model counting and Apply operation. In this paper, we propose a more succinct variant of SDD named Variable Shift SDD (VS-SDD). The key idea is to create a unique representation for Boolean functions that are equivalent under a specific variable substitution. We show that VS-SDDs are never larger than SDDs and there are cases in which the size of a VS-SDD is exponentially smaller than that of an SDD. Moreover, despite such succinctness, we show that numerous basic operations that are supported in polytime with SDD are also supported in polytime with VS-SDD. Experiments confirm that VS-SDDs are significantly more succinct than SDDs when applied to classical planning instances, where inherent symmetry exists.

Probabilistic Computation Tree Logic (PCTL) is frequently used to formally specify control objectives such as probabilistic reachability and safety. In this work, we focus on model checking PCTL specifications statistically on Markov Decision Processes (MDPs) by sampling, e.g., checking whether there exists a feasible policy such that the probability of reaching certain goal states is greater than a threshold. We use reinforcement learning to search for such a feasible policy for PCTL specifications, and then develop a statistical model checking (SMC) method with provable guarantees on its error. Specifically, we first use upper-confidence-bound (UCB) based Q-learning to design an SMC algorithm for bounded-time PCTL specifications, and then extend this algorithm to unbounded-time specifications by identifying a proper truncation time by checking the PCTL specification and its negation at the same time. Finally, we evaluate the proposed method on case studies.

Given a graph rewrite system, a graph G is a quine graph if it has a non-void maximal collection of non-conflicting matches of left patterns of graphs rewrites, such that after the parallel application of the rewrites we obtain a graph isomorphic with G. Such graphs exhibit a metabolism, they can multiply or they can die, when reduced by a random rewriting algorithm. These are introductory notes to the pages of artificial chemistry experiments with chemlambda, lambda calculus or interaction combinators, available from the entry page https://chemlambda.github.io/index.html . The experiments are bundled into pages, all of them based on a library of programs, on a database which contains hundreds of graphs and on a database of about 150 pages of text comments and a collection of more than 200 animations, most of them which can be re-done live, via the programs. There are links to public repositories of other contributors to these experiments, with versions of these programs in python, haskell, awk or javascript.

Commonsense reasoning has long been considered as one of the holy grails of artificial intelligence. Most of the recent progress in the field has been achieved by novel machine learning algorithms for natural language processing. However, without incorporating logical reasoning, these algorithms remain arguably shallow. With some notable exceptions, developers of practical automated logic-based reasoners have mostly avoided focusing on the problem. The paper argues that the methods and algorithms used by existing automated reasoners for classical first-order logic can be extended towards commonsense reasoning. Instead of devising new specialized logics we propose a framework of extensions to the mainstream resolution-based search methods to make these capable of performing search tasks for practical commonsense reasoning with reasonable efficiency. The proposed extensions mostly rely on operating on ordinary proof trees and are devised to handle commonsense knowledge bases containing inconsistencies, default rules, taxonomies, topics, relevance, confidence and similarity measures. We claim that machine learning is best suited for the construction of commonsense knowledge bases while the extended logic-based methods would be well-suited for actually answering queries from these knowledge bases.

Neuro-symbolic and statistical relational artificial intelligence both integrate frameworks for learning with logical reasoning. This survey identifies several parallels across seven different dimensions between these two fields. These cannot only be used to characterize and position neuro-symbolic artificial intelligence approaches but also to identify a number of directions for further research.

Program synthesis is the task of automatically generating a program consistent with a given specification. A natural way to specify programs is to provide examples of desired input-output behavior, and many current program synthesis approaches have achieved impressive results after training on randomly generated input-output examples. However, recent work has discovered that some of these approaches generalize poorly to data distributions different from that of the randomly generated examples. We show that this problem applies to other state-of-the-art approaches as well and that current methods to counteract this problem are insufficient. We then propose a new, adversarial approach to control the bias of synthetic data distributions and show that it outperforms current approaches.