We elaborate upon the formal foundations of hybrid Answer Set Programming (ASP) and extend its underlying logical framework with aggregate functions over constraint values and variables. This is achieved by introducing the construct of conditional expressions, which allow for considering two alternatives while evaluating constraints. Which alternative is considered is interpretation-dependent and chosen according to an associated condition. We put some emphasis on logic programs with linear constraints and show how common ASP aggregates can be regarded as particular cases of so-called conditional linear constraints. Finally, we introduce a polynomial-size, modular and faithful translation from our framework into regular (condition-free) Constraint ASP, outlining an implementation of conditional aggregates on top of existing hybrid ASP solvers.

This paper studies logics of unknown truths and false beliefs under neighborhood semantics. Intuitively, if p is true but you do not know that p, then you have an unknown truth that p; if p is false but you believe thatp, then you have a false belief thatp, or you are wrong aboutp. The notion of unknown truths is important in philosophy and formal epistemology. For instance, it is related to Verificationism, or'verification thesis' [31]. Verificationism says that all truths can be known. However, from the thesis, the unknown truth of p, formalized p Kp, gives us a consequence that all truths are actually known. In other words, the notion gives rise to a well-known counterexample to Verificationism. This is the so-called Fitch's'paradox of knowability' [13]. 1 To take another example: it gives rise to an important type of Moore sentences, which is essential to Moore's paradox, which says that one cannot claim the paradoxical sentence "p but I do not know it" [23, 18]. It is known that such a Moore sentence is unsuccessful and self-refuting (see, e.g.

We elaborate upon the formal foundations of hybrid Answer Set Programming (ASP) and extend its underlying logical framework with aggregate functions over constraint values and variables. This is achieved by introducing the construct of conditional expressions, which allow for considering two alternatives while evaluating constraints. Which alternative is considered is interpretation-dependent and chosen according to an associated condition. We put some emphasis on logic programs with linear constraints and show how common ASP aggregates can be regarded as particular cases of so-called conditional linear constraints. Finally, we introduce a polynomial-size, modular and faithful translation from our framework into regular (condition-free) Constraint ASP, outlining an implementation of conditional aggregates on top of existing hybrid ASP solvers.

We introduce an implementation of an extension of Answer Set Programming (ASP) with language constructs from dynamic (and temporal) logic that provides an expressive computational framework for modeling dynamic applications. Starting from logical foundations, provided by dynamic and temporal equilibrium logics over finite linear traces, we develop a translation of dynamic formulas into temporal logic programs. This provides us with a normal form result establishing the strong equivalence of formulas in different logics. Our translation relies on the introduction of auxiliary atoms to guarantee polynomial space complexity and to provide an embedding that is doomed to be impossible over the same language. Finally, the reduction of dynamic formulas to temporal logic programs allows us to extend ASP with both approaches in a uniform way and to implement both extensions via temporal ASP solvers such as telingo

Recent research in artificial intelligence and machine learning has largely emphasized general-purpose learning and ever-larger training sets and more and more compute. In contrast, I propose a hybrid, knowledge-driven, reasoning-based approach, centered around cognitive models, that could provide the substrate for a richer, more robust AI than is currently possible.

The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence (AI) in general, since they can serve as an organizing principle by which individuals classify objects, form concepts and make generalizations. While distance functions for propositional representations have been thoroughly studied, work on distance functions for structured representations, such as graphs, frames or logical clauses, has been carried out in different communities and is much less understood. Specifically, a significant amount of work that requires the use of a distance or similarity function for structured representations of data usually employs ad-hoc functions for specific applications. Therefore, the goal of this paper is to provide an overview of this work to identify connections between the work carried out in different areas and point out directions for future work.

We consider the task of automated theorem proving, a key AI task. Deep learning has shown promise for training theorem provers, but there are limited human-written theorems and proofs available for supervised learning. To address this limitation, we propose to learn a neural generator that automatically synthesizes theorems and proofs for the purpose of training a theorem prover. Experiments on real-world tasks demonstrate that synthetic data from our approach improves the theorem prover and advances the state of the art of automated theorem proving in Metamath.

Aiming towards the development of a general clustering theory, we discuss abstract axiomatization for clustering. In this respect, we follow up on the work of Kelinberg, (Kleinberg) that showed an impossibility result for such axiomatization. We argue that an impossibility result is not an inherent feature of clustering, but rather, to a large extent, it is an artifact of the specific formalism used in Kleinberg. As opposed to previous work focusing on clustering functions, we propose to address clustering quality measures as the primitive object to be axiomatized. We show that principles like those formulated in Kleinberg's axioms can be readily expressed in the latter framework without leading to inconsistency.

The task of program synthesis, or automatically generating programs that are consistent with a provided specification, remains a challenging task in artificial intelligence. As in other fields of AI, deep learning-based end-to-end approaches have made great advances in program synthesis. However, more so than other fields such as computer vision, program synthesis provides greater opportunities to explicitly exploit structured information such as execution traces, which contain a superset of the information input/output pairs. While they are highly useful for program synthesis, as execution traces are more difficult to obtain than input/output pairs, we use the insight that we can split the process into two parts: infer the trace from the input/output example, then infer the program from the trace. This simple modification leads to state-of-the-art results in program synthesis in the Karel domain, improving accuracy to 81.3% from the 77.12% of prior work.

We present a neurosymbolic framework for the lifelong learning of algorithmic tasks that mix perception and procedural reasoning. Reusing high-level concepts across domains and learning complex procedures are key challenges in lifelong learning. We show that a program synthesis approach that combines gradient descent with combinatorial search over programs can be a more effective response to these challenges than purely neural methods. Our framework, called HOUDINI, represents neural networks as strongly typed, differentiable functional programs that use symbolic higher-order combinators to compose a library of neural functions. Our learning algorithm consists of: (1) a symbolic program synthesizer that performs a type-directed search over parameterized programs, and decides on the library functions to reuse, and the architectures to combine them, while learning a sequence of tasks; and (2) a neural module that trains these programs using stochastic gradient descent.