Machine learning models that automate decision-making are increasingly being used in consequential areas such as loan approvals, pretrial bail approval, hiring, and many more. Unfortunately, most of these models are black-boxes, i.e., they are unable to reveal how they reach these prediction decisions. A need for transparency demands justification for such predictions. An affected individual might also desire explanations to understand why a decision was made. Ethical and legal considerations may further require informing the individual of changes in the input attribute that could be made to produce a desirable outcome. This paper focuses on the latter problem of automatically generating counterfactual explanations. We propose a framework Counterfactual Generation with s(CASP) (CFGS) that utilizes answer set programming (ASP) and the s(CASP) goal-directed ASP system to automatically generate counterfactual explanations from rules generated by rule-based machine learning (RBML) algorithms. In our framework, we show how counterfactual explanations are computed and justified by imagining worlds where some or all factual assumptions are altered/changed. More importantly, we show how we can navigate between these worlds, namely, go from our original world/scenario where we obtain an undesired outcome to the imagined world/scenario where we obtain a desired/favourable outcome.

Mathematical reasoning poses a significant challenge for language models due to its complex and structured nature. In this paper, we introduce DeepSeekMath 7B, which continues pre-training DeepSeek-Coder-Base-v1.5 7B with 120B math-related tokens sourced from Common Crawl, together with natural language and code data. DeepSeekMath 7B has achieved an impressive score of 51.7% on the competition-level MATH benchmark without relying on external toolkits and voting techniques, approaching the performance level of Gemini-Ultra and GPT-4. Self-consistency over 64 samples from DeepSeekMath 7B achieves 60.9% on MATH. The mathematical reasoning capability of DeepSeekMath is attributed to two key factors: First, we harness the significant potential of publicly available web data through a meticulously engineered data selection pipeline. Second, we introduce Group Relative Policy Optimization (GRPO), a variant of Proximal Policy Optimization (PPO), that enhances mathematical reasoning abilities while concurrently optimizing the memory usage of PPO.

Answer Set Programming (ASP) is a generic problem modeling and solving framework with a strong focus on knowledge representation and a rapid growth of industrial applications. So far, the study of complexity resulted in characterizing hardness and determining their sources, fine-grained insights in the form of dichotomy-style results, as well as detailed parameterized complexity landscapes. Unfortunately, for the well-known parameter treewidth disjunctive programs require double-exponential runtime under reasonable complexity assumptions. This quickly becomes out of reach. We deal with the classification of structural parameters for disjunctive ASP on the program's rule structure (incidence graph). First, we provide a polynomial kernel to obtain single-exponential runtime in terms of vertex cover size, despite subset-minimization being not represented in the program's structure. Then we turn our attention to strictly better structural parameters between vertex cover size and treewidth. Here, we provide double-exponential lower bounds for the most prominent parameters in that range: treedepth, feedback vertex size, and cliquewidth. Based on this, we argue that unfortunately our options beyond vertex cover size are limited. Our results provide an in-depth hardness study, relying on a novel reduction from normal to disjunctive programs, trading the increase of complexity for an exponential parameter compression.

Representing time is crucial for cyber-physical systems and has been studied extensively in the Situation Calculus. The most commonly used approach represents time by adding a real-valued fluent $\mathit{time}(a)$ that attaches a time point to each action and consequently to each situation. We show that in this approach, checking whether there is a reachable situation that satisfies a given formula is undecidable, even if the domain of discourse is restricted to a finite set of objects. We present an alternative approach based on well-established results from timed automata theory by introducing clocks as real-valued fluents with restricted successor state axioms and comparison operators. %that only allow comparisons against fixed rationals. With this restriction, we can show that the reachability problem for finite-domain basic action theories is decidable. Finally, we apply our results on Golog program realization by presenting a decidable procedure for determining an action sequence that is a successful execution of a given program.

Metamodeling is a general approach to expressing knowledge about classes and properties in an ontology. It is a desirable modeling feature in multiple applications that simplifies the extension and reuse of ontologies. Nevertheless, allowing metamodeling without restrictions is problematic for several reasons, mainly due to undecidability issues. Practical languages, therefore, forbid classes to occur as instances of other classes or treat such occurrences as semantically different objects. Specifically, meta-querying in SPARQL under the Direct Semantic Entailment Regime (DSER) uses the latter approach, thereby effectively not supporting meta-queries. However, several extensions enabling different metamodeling features have been proposed over the last decade. This paper deals with the Metamodeling Semantics (MS) over OWL 2 QL and the Metamodeling Semantic Entailment Regime (MSER), as proposed in Lenzerini et al. (2015) and Lenzerini et al. (2020); Cima et al. (2017). A reduction from OWL 2 QL to Datalog for meta-querying was proposed in Cima et al. (2017). In this paper, we experiment with various logic programming tools that support Datalog querying to determine their suitability as back-ends to MSER query answering. These tools stem from different logic programming paradigms (Prolog, pure Datalog, Answer Set Programming, Hybrid Knowledge Bases). Our work shows that the Datalog approach to MSER querying is practical also for sizeable ontologies with limited resources (time and memory). This paper significantly extends Qureshi & Faber (2021) by a more detailed experimental analysis and more background. Under consideration in Theory and Practice of Logic Programming (TPLP).

We introduce SymbolicAI, a versatile and modular framework employing a logic-based approach to concept learning and flow management in generative processes. SymbolicAI enables the seamless integration of generative models with a diverse range of solvers by treating large language models (LLMs) as semantic parsers that execute tasks based on both natural and formal language instructions, thus bridging the gap between symbolic reasoning and generative AI. We leverage probabilistic programming principles to tackle complex tasks, and utilize differentiable and classical programming paradigms with their respective strengths. The framework introduces a set of polymorphic, compositional, and self-referential operations for data stream manipulation, aligning LLM outputs with user objectives. As a result, we can transition between the capabilities of various foundation models endowed with zero- and few-shot learning capabilities and specialized, fine-tuned models or solvers proficient in addressing specific problems. In turn, the framework facilitates the creation and evaluation of explainable computational graphs. We conclude by introducing a quality measure and its empirical score for evaluating these computational graphs, and propose a benchmark that compares various state-of-the-art LLMs across a set of complex workflows. We refer to the empirical score as the "Vector Embedding for Relational Trajectory Evaluation through Cross-similarity", or VERTEX score for short. The framework codebase and benchmark are linked below.

In this paper we propose a general framework to integrate supervised and unsupervised examples with background knowledge expressed by a collection of first-order logic clauses into kernel machines. In particular, we consider a multi-task learning scheme where multiple predicates defined on a set of objects are to be jointly learned from examples, enforcing a set of FOL constraints on the admissible configurations of their values. The predicates are defined on the feature spaces, in which the input objects are represented, and can be either known a priori or approximated by an appropriate kernel-based learner. A general approach is presented to convert the FOL clauses into a continuous implementation that can deal with the outputs computed by the kernel-based predicates. The learning problem is formulated as a semi-supervised task that requires the optimization in the primal of a loss function that combines a fitting loss measure on the supervised examples, a regularization term, and a penalty term that enforces the constraints on both the supervised and unsupervised examples. Unfortunately, the penalty term is not convex and it can hinder the optimization process. However, it is possible to avoid poor solutions by using a two stage learning schema, in which the supervised examples are learned first and then the constraints are enforced.

Despite the great success of neural visual generative models in recent years, integrating them with strong symbolic reasoning systems remains a challenging task. There are two levels of symbol grounding problems among the core challenges: the first is symbol assignment, i.e. mapping latent factors of neural visual generators to semantic-meaningful symbolic factors from the reasoning systems by learning from limited labeled data. The second is rule learning, i.e. learning new rules that govern the generative process to enhance the symbolic reasoning systems. To deal with these two problems, we propose a neurosymbolic learning approach, Abductive visual Generation (AbdGen), for integrating logic programming systems with neural visual generative models based on the abductive learning framework. To achieve reliable and efficient symbol grounding, the quantized abduction method is introduced for generating abduction proposals by the nearest-neighbor lookup within semantic codebooks. To achieve precise rule learning, the contrastive meta-abduction method is proposed to eliminate wrong rules with positive cases and avoid less informative rules with negative cases simultaneously. Experimental results show that compared to the baseline approaches, AbdGen requires significantly less labeled data for symbol assignment. Furthermore, AbdGen can effectively learn underlying logical generative rules from data, which is out of the capability of existing approaches.

Agda is a dependently-typed programming language and a proof assistant, pivotal in proof formalization and programming language theory. This paper extends the Agda ecosystem into machine learning territory, and, vice versa, makes Agda-related resources available to machine learning practitioners. We introduce and release a novel dataset of Agda program-proofs that is elaborate and extensive enough to support various machine learning applications -- the first of its kind. Leveraging the dataset's ultra-high resolution, detailing proof states at the sub-type level, we propose a novel neural architecture targeted at faithfully representing dependently-typed programs on the basis of structural rather than nominal principles. We instantiate and evaluate our architecture in a premise selection setup, where it achieves strong initial results.

Neuro-Symbolic AI (NeSy) holds promise to ensure the safe deployment of AI systems, as interpretable symbolic techniques provide formal behaviour guarantees. The challenge is how to effectively integrate neural and symbolic computation, to enable learning and reasoning from raw data. Existing pipelines that train the neural and symbolic components sequentially require extensive labelling, whereas end-to-end approaches are limited in terms of scalability, due to the combinatorial explosion in the symbol grounding problem. In this paper, we leverage the implicit knowledge within foundation models to enhance the performance in NeSy tasks, whilst reducing the amount of data labelling and manual engineering. We introduce a new architecture, called NeSyGPT, which fine-tunes a vision-language foundation model to extract symbolic features from raw data, before learning a highly expressive answer set program to solve a downstream task. Our comprehensive evaluation demonstrates that NeSyGPT has superior accuracy over various baselines, and can scale to complex NeSy tasks. Finally, we highlight the effective use of a large language model to generate the programmatic interface between the neural and symbolic components, significantly reducing the amount of manual engineering required.