A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered conjectures are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We give an overview of typical use-cases of QuickSpec, as well as demonstrating how to easily connect it to a theorem prover of the user's choice.

Multi-modal program synthesis refers to the task of synthesizing programs (code) from their specification given in different forms, such as a combination of natural language and examples. Examples provide a precise but incomplete specification, and natural language provides an ambiguous but more "complete" task description. Machine-learned pre-trained models (PTMs) are adept at handling ambiguous natural language, but struggle with generating syntactically and semantically precise code. Program synthesis techniques can generate correct code, often even from incomplete but precise specifications, such as examples, but they are unable to work with the ambiguity of natural languages. We present an approach that combines PTMs with component-based synthesis (CBS): PTMs are used to generate candidates programs from the natural language description of the task, which are then used to guide the CBS procedure to find the program that matches the precise examples-based specification. We use our combination approach to instantiate multi-modal synthesis systems for two programming domains: the domain of regular expressions and the domain of CSS selectors. Our evaluation demonstrates the effectiveness of our domain-agnostic approach in comparison to a state-of-the-art specialized system, and the generality of our approach in providing multi-modal program synthesis from natural language and examples in different programming domains.

We explore different ways of implementing temporal constraints expressed in an extension of Answer Set Programming (ASP) with language constructs from dynamic logic. Foremost, we investigate how automata can be used for enforcing such constraints. The idea is to transform a dynamic constraint into an automaton expressed in terms of a logic program that enforces the satisfaction of the original constraint. What makes this approach attractive is its independence of time stamps and the potential to detect unsatisfiability. On the one hand, we elaborate upon a transformation of dynamic formulas into alternating automata that relies on meta-programming in ASP. This is the first application of reification applied to theory expressions in gringo. On the other hand, we propose two transformations of dynamic formulas into monadic second-order formulas. These can then be used by off-the-shelf tools to construct the corresponding automata. We contrast both approaches empirically with the one of the temporal ASP solver telingo that directly maps dynamic constraints to logic programs. Since this preliminary study is restricted to dynamic formulas in integrity constraints, its implementations and (empirical) results readily apply to conventional linear dynamic logic, too.

Scientists have long aimed to discover meaningful equations which accurately describe data. Machine learning algorithms automate construction of accurate data-driven models, but ensuring that these are consistent with existing knowledge is a challenge. We developed a methodology combining automated theorem proving with symbolic regression, enabling principled derivations of laws of nature. We demonstrate this for Kepler's third law, Einstein's relativistic time dilation, and Langmuir's theory of adsorption, in each case, automatically connecting experimental data with background theory. The combination of logical reasoning with machine learning provides generalizable insights into key aspects of the natural phenomena.

Conditionals are useful for modelling, but are not always sufficiently expressive for capturing information accurately. In this paper we make the case for a form of conditional that is situation-based. These conditionals are more expressive than classical conditionals, are general enough to be used in several application domains, and are able to distinguish, for example, between expectations and counterfactuals. Formally, they are shown to generalise the conditional setting in the style of Kraus, Lehmann, and Magidor. We show that situation-based conditionals can be described in terms of a set of rationality postulates. We then propose an intuitive semantics for these conditionals, and present a representation result which shows that our semantic construction corresponds exactly to the description in terms of postulates. With the semantics in place, we proceed to define a form of entailment for situated conditional knowledge bases, which we refer to as minimal closure. It is reminiscent of and, indeed, inspired by, the version of entailment for propositional conditional knowledge bases known as rational closure. Finally, we proceed to show that it is possible to reduce the computation of minimal closure to a series of propositional entailment and satisfiability checks. While this is also the case for rational closure, it is somewhat surprising that the result carries over to minimal closure.

We describe the design and implementation of a reasoning engine that facilitates the gamification of loop-invariant discovery. Our reasoning engine enables students, computational agents and regular software engineers with no formal methods expertise to collaboratively prove interesting theorems about simple programs using browser-based, online games. Within an hour, players are able to specify and verify properties of programs that are beyond the capabilities of fully-automated tools. The hour limit includes the time for setting up the system, completing a short tutorial explaining game play and reasoning about simple imperative programs. Players are never required to understand formal proofs; they only provide insights by proposing invariants. The reasoning engine is responsible for managing and evaluating the proposed invariants, as well as generating actionable feedback.

We propose Dr.Aid, a logic-based AI framework for automated compliance checking of data governance rules over data-flow graphs. The rules are modelled using a formal language based on situation calculus and are suitable for decentralized contexts with multi-input-multi-output (MIMO) processes. Dr.Aid models data rules and flow rules and checks compliance by reasoning about the propagation, combination, modification and application of data rules over the data flow graphs. Our approach is driven and evaluated by real-world datasets using provenance graphs from data-intensive research.

As robots begin to interact closely with humans, we need to build systems worthy of trust regarding both the safety and the quality of the interaction. To do so, we have to be able to formalize what a "good" interaction is, and we need algorithms that can check that a given system produces good interactions or can even synthesize such systems. To make progress, we must first acknowledge that a human is not another dynamic physical element in the environment, but has beliefs, goals, social norms, desires, and preferences. To address these complexities, we must develop models, specifications, and algorithms that use our knowledge about human behavior to create demonstrably trustworthy systems. In this article, we identified a number of promising research directions and we encourage the HRI and formal methods communities to create strong collaborations to tackle these and other questions toward the goal of trustworthy HRI.

Analogy-making is at the core of human intelligence and creativity with applications to such diverse tasks as commonsense reasoning, learning, language acquisition, and story telling. This paper studies analogical proportions between booleans of the form `$a$ is to $b$ what $c$ is to $d$' called boolean proportions. Technically, we instantiate an abstract algebraic framework of analogical proportions -- recently introduced by the author -- in the boolean domain consisting of the truth values true and false together with boolean functions. It turns out that our notion of boolean proportions has appealing mathematical properties and that it coincides with a prominent model of boolean proportions in the general case. In a broader sense, this paper is a further step towards a theory of analogical reasoning and learning systems with potential applications to fundamental AI-problems like commonsense reasoning and computational learning and creativity.

Abstract argumentation provides us with methods such as gradual and Dung semantics with which to evaluate arguments after potential attacks by other arguments. Some of these methods can take intrinsic strengths of arguments as input, with which to modulate the effects of attacks between arguments. Coming from abstract argumentation, these methods look only at the relations between arguments and not at the structure of the arguments themselves. In structured argumentation the way an argument is constructed, by chaining inference rules starting from premises, is taken into consideration. In this paper we study methods for assigning an argument its intrinsic strength, based on the strengths of the premises and inference rules used to form said argument. We first define a set of principles, which are properties that strength assigning methods might satisfy. We then propose two such methods and analyse which principles they satisfy. Finally, we present a generalised system for creating novel strength assigning methods and speak to the properties of this system regarding the proposed principles.