In recent years, G\"odel's ontological proof and variations of it were formalized and analyzed with automated tools in various ways. We supplement these analyses with a modeling in an automated environment based on first-order logic extended by predicate quantification. Formula macros are used to structure complex formulas and tasks. The analysis is presented as a generated type-set document where informal explanations are interspersed with pretty-printed formulas and outputs of reasoners for first-order theorem proving and second-order quantifier elimination. Previously unnoticed or obscured aspects and details of G\"odel's proof become apparent. Practical application possibilities of second-order quantifier elimination are shown and the encountered elimination tasks may serve as benchmarks.

The relevance of polynomial formula classes to deductive efficiency motivated their search, and currently, a great number of such classes is known. Nonetheless, they have been exclusively sought in the setting of clausal form and propositional logic, which is of course expressively limiting for real applications. As a consequence, a first polynomial propositional class in non-clausal (NC) form has recently been proposed. Along these lines and towards making NC tractability applicable beyond propositional logic, firstly, we define the Regular many-valued Horn Non-Clausal class, or RH, obtained by suitably amalgamating both regular classes: Horn and NC. Secondly, we demonstrate that the relationship between (1) RH and the regular Horn class is that syntactically RH subsumes the Horn class but that both classes are equivalent semantically; and between (2) RH and the regular non-clausal class is that RH contains all NC formulas whose clausal form is Horn. Thirdly, we define Regular Non-Clausal Unit-Resolution, or RUR-NC , and prove both that it is complete for RH and that checks its satisfiability in polynomial time. The latter fact shows that our intended goal is reached since RH is many-valued, non-clausal and tractable. As RH and RUR-NC are, both, basic in the DPLL scheme, the most efficient in propositional logic, and can be extended to some other non-classical logics, we argue that they pave the way for efficient non-clausal DPLL-based approximate reasoning.

The ability to recognise and make analogies is often used as a measure or test of human intelligence. The ability to solve Bongard problems is an example of such a test. It has also been postulated that the ability to rapidly construct novel abstractions is critical to being able to solve analogical problems. Given an image, the ability to construct a program that would generate that image is one form of abstraction, as exemplified in the Dreamcoder project. In this paper, we present a preliminary examination of whether programs constructed by Dreamcoder can be used for analogical reasoning to solve certain Bongard problems. We use Dreamcoder to discover programs that generate the images in a Bongard problem and represent each of these as a sequence of state transitions. We decorate the states using positional information in an automated manner and then encode the resulting sequence into logical facts in Prolog. We use inductive logic programming (ILP), to learn an (interpretable) theory for the abstract concept involved in an instance of a Bongard problem. Experiments on synthetically created Bongard problems for concepts such as 'above/below' and 'clockwise/counterclockwise' demonstrate that our end-to-end system can solve such problems. We study the importance and completeness of each component of our approach, highlighting its current limitations and pointing to directions for improvement in our formulation as well as in elements of any Dreamcoder-like program synthesis system used for such an approach.

Reasoning is an essential part of human intelligence and thus has been a long-standing goal in artificial intelligence research. With the recent success of deep learning, incorporating reasoning with deep learning systems, i.e., neuro-symbolic AI has become a major field of interest. We propose the Neuro-Symbolic Forward Reasoner (NSFR), a new approach for reasoning tasks taking advantage of differentiable forward-chaining using first-order logic. The key idea is to combine differentiable forward-chaining reasoning with object-centric (deep) learning. Differentiable forward-chaining reasoning computes logical entailments smoothly, i.e., it deduces new facts from given facts and rules in a differentiable manner. The object-centric learning approach factorizes raw inputs into representations in terms of objects. Thus, it allows us to provide a consistent framework to perform the forward-chaining inference from raw inputs. NSFR factorizes the raw inputs into the object-centric representations, converts them into probabilistic ground atoms, and finally performs differentiable forward-chaining inference using weighted rules for inference. Our comprehensive experimental evaluations on object-centric reasoning data sets, 2D Kandinsky patterns and 3D CLEVR-Hans, and a variety of tasks show the effectiveness and advantage of our approach.

We present an approach for representing abstract argumentation frameworks based on an encoding into classical higher-order logic. This provides a uniform framework for computer-assisted assessment of abstract argumentation frameworks using interactive and automated reasoning tools. This enables the formal analysis and verification of meta-theoretical properties as well as the flexible generation of extensions and labellings with respect to well-known argumentation semantics.

Given a Boolean formula $\varphi$ over the set of variables $X$ and a projection set $\mathcal{P} \subseteq X$, a subset of variables $\mathcal{I}$ is independent support of $\mathcal{P}$ if two solutions agree on $\mathcal{I}$, then they also agree on $\mathcal{P}$. The notion of independent support is related to the classical notion of definability dating back to 1901, and have been studied over the decades. Recently, the computational problem of determining independent support for a given formula has attained importance owing to the crucial importance of independent support for hashing-based counting and sampling techniques. In this paper, we design an efficient and scalable independent support computation technique that can handle formulas arising from real-world benchmarks. Our algorithmic framework, called Arjun, employs implicit and explicit definability notions, and is based on a tight integration of gate-identification techniques and assumption-based framework. We demonstrate that augmenting the state of the art model counter ApproxMC4 and sampler UniGen3 with Arjun leads to significant performance improvements. In particular, ApproxMC4 augmented with Arjun counts 387 more benchmarks out of 1896 while UniGen3 augmented with Arjun samples 319 more benchmarks within the same time limit.

Autonomous cyber-physical systems (CPS) can improve safety and efficiency for safety-critical applications, but require rigorous testing before deployment. The complexity of these systems often precludes the use of formal verification and real-world testing can be too dangerous during development. Therefore, simulation-based techniques have been developed that treat the system under test as a black box operating in a simulated environment. Safety validation tasks include finding disturbances in the environment that cause the system to fail (falsification), finding the most-likely failure, and estimating the probability that the system fails. Motivated by the prevalence of safety-critical artificial intelligence, this work provides a survey of state-of-the-art safety validation techniques for CPS with a focus on applied algorithms and their modifications for the safety validation problem. We present and discuss algorithms in the domains of optimization, path planning, reinforcement learning, and importance sampling. Problem decomposition techniques are presented to help scale algorithms to large state spaces, which are common for CPS. A brief overview of safety-critical applications is given, including autonomous vehicles and aircraft collision avoidance systems. Finally, we present a survey of existing academic and commercially available safety validation tools.

Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by introducing and studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification, and identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former, leading to a refinement of quantified Boolean logic with literal quantification as its primitive.

We report on the design and development of the CASPR system, a socialbot designed to compete in the Amazon Alexa Socialbot Challenge 4. CASPR's distinguishing characteristic is that it will use automated commonsense reasoning to truly "understand" dialogs, allowing it to converse like a human. Three main requirements of a socialbot are that it should be able to "understand" users' utterances, possess a strategy for holding a conversation, and be able to learn new knowledge. We developed techniques such as conversational knowledge template (CKT) to approximate commonsense reasoning needed to hold a conversation on specific topics. We present the philosophy behind CASPR's design as well as details of its implementation. We also report on CASPR's performance as well as discuss lessons learned.

We propose a new formalization of legal competences, and in particular for the Hohfeldian categories of power and immunity, through a deontic reinterpretation of dynamic epistemic logic. We argue that this logic explicitly captures the norm-changing character of legal competences while providing a sophisticated reduction of the latter to static normative positions. The logic is completely axiomatizable, and we apply it to a concrete case in German contract law to illustrate that it can capture the distinction between legal ability and legal permissibility.