Runtime Verification is a lightweight formal verification technique. It is used to verify at runtime whether the system under analysis behaves as expected. The expected behaviour is usually formally specified by means of properties, which are used to automatically synthesise monitors. A monitor is a device that, given a sequence of events representing a system execution, returns a verdict symbolising the satisfaction or violation of the formal property. Properties that can (resp. cannot) be verified at runtime by a monitor are called monitorable and non-monitorable, respectively. In this paper, we revise the notion of monitorability from a practical perspective, where we show how non-monitorable properties can still be used to generate partial monitors, which can partially check the properties. Finally, we present the implications both from a theoretical and practical perspectives.

We consider the problem of automatically constructing computer programs from input-output examples. We investigate how to augment probabilistic and neural program synthesis methods with new search algorithms, proposing a framework called distribution-based search. Within this framework, we introduce two new search algorithms: Heap Search, an enumerative method, and SQRT Sampling, a probabilistic method. We prove certain optimality guarantees for both methods, show how they integrate with probabilistic and neural techniques, and demonstrate how they can operate at scale across parallel compute environments. Collectively these findings offer theoretical and applied studies of search algorithms for program synthesis that integrate with recent developments in machine-learned program synthesizers.

Goal-directed evaluation of Answer Set Programs is gaining traction thanks to its amenability to create AI systems that can, due to the evaluation mechanism used, generate explanations and justifications. s(CASP) is one of these systems and has been already used to write reasoning systems in several fields. It provides enhanced expressiveness w.r.t. other ASP systems due to its ability to use constraints, data structures, and unbound variables natively. However, the performance of existing s(CASP) implementations is not on par with other ASP systems: model consistency is checked once models have been generated, in keeping with the generate-and-test paradigm. In this work, we present a variation of the top-down evaluation strategy, termed Dynamic Consistency Checking, which interleaves model generation and consistency checking. This makes it possible to determine when a literal is not compatible with the denials associated to the global constraints in the program, prune the current execution branch, and choose a different alternative. This strategy is specially (but not exclusively) relevant in problems with a high combinatorial component. We have experimentally observed speedups of up to 90x w.r.t. the standard versions of s(CASP).

Autonomous systems are highly complex and present unique challenges for the application of formal methods. Autonomous systems act without human intervention, and are often embedded in a robotic system, so that they can interact with the real world. As such, they exhibit the properties of safety-critical, cyber-physical, hybrid, and real-time systems. This EPTCS volume contains the proceedings for the third workshop on Formal Methods for Autonomous Systems (FMAS 2021), which was held virtually on the 21st and 22nd of October 2021. Like the previous workshop, FMAS 2021 was an online, stand-alone event, as an adaptation to the ongoing COVID-19 restrictions. Despite the challenges this brought, we were determined to build on the success of the previous two FMAS workshops. The goal of FMAS is to bring together leading researchers who are tackling the unique challenges of autonomous systems using formal methods, to present recent and ongoing work. We are interested in the use of formal methods to specify, model, or verify autonomous and/or robotic systems; in whole or in part. We are also interested in successful industrial applications and potential future directions for this emerging application of formal methods.

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.