In the study of reasoning in neural networks, recent efforts have sought to improve consistency and coherence of sequence models, leading to important developments in the area of neuro-symbolic AI. In symbolic AI, the concepts of consistency and coherence can be defined and verified formally, but for neural networks these definitions are lacking. The provision of such formal definitions is crucial to offer a common basis for the quantitative evaluation and systematic comparison of connectionist, neuro-symbolic and transfer learning approaches. In this paper, we introduce formal definitions of consistency and coherence for neural systems. To illustrate the usefulness of our definitions, we propose a new dynamic relation-decoder model built around the principles of consistency and coherence. We compare our results with several existing relation-decoders using a partial transfer learning task based on a novel data set introduced in this paper. Our experiments show that relation-decoders that maintain consistency over unobserved regions of representation space retaincoherence across domains, whilst achieving better transfer learning performance.

Assessing the reasoning ability of Large Language Models (LLMs) over data remains an open and pressing research question. Compared with LLMs, human reasoning can derive corresponding modifications to the output based on certain kinds of changes to the input. This reasoning pattern, which relies on abstract rules that govern relationships between changes of data, has not been comprehensively described or evaluated in LLMs. In this paper, we formally define this reasoning pattern as the Derivation Relation (DR) and introduce the concept of Derivation Capability (DC), i.e. applying DR by making the corresponding modification to the output whenever the input takes certain changes. To assess DC, a systematically constructed evaluation framework named DEVAL is proposed and used to evaluate five popular LLMs and one Large Reasoning Model in seven mainstream tasks. The evaluation results show that mainstream LLMs, such as GPT-4o and Claude3.5, exhibit moderate DR recognition capabilities but reveal significant drop-offs on applying DR effectively in problem-solving scenarios. To improve this, we propose a novel prompt engineering approach called Derivation Prompting (DP). It achieves an average improvement of 15.2% in DC for all tested LLMs, outperforming commonly used prompt engineering techniques.

We thank the reviewers for their valuable suggestions. Please find our answers ( A) for each reviewer ( R) below. A: We will add the following definition: VCDpX, Hq " max | X In particular, we will clarify that the teacher knows the learner's preference This is the protocol used in existing teaching models for both the batch settings (e.g., as in RTD / PBTD (line 221). We will further add a discussion on this in the updated version of the paper. R4: Discussion on the "presumably increased complexity of sequential learners" We will add a discussion in the updated version of the paper.

One reason could be the number of models used for the violin plot.

To prove Theorem 2, we need several auxiliary lemmas. Lemma 2. Given estimated parameters null θ and null F, for any bidding policy π, we have R ( π; null θ, null F) R ( π; θ, null F) E Recall, by definition of R (π; null θ, null F) in Eq. (6), for any null θ and null F, R (π; null θ, null F) = E Lemma 3. F or any fixed bidding strategy π, we have null null R ( π; θ, null F Given the MDP re-formulation in Subsection 3.1, we have for any null,b and h = 2, 3,,H, null Given the above two auxiliary lemmas, we are ready to prove Theorem 2. Proof of Theorem 2. For notation simplicity, let OPT( null CR Then we bound the above terms separately in the following. In this section, we enumerate several useful technical lemmas used in this paper. Finally, we describe the well-known simulation lemma. Then we complete the proof.

We would be happy to add more based on further reviewer suggestions.

This trade-off is posed as a general problem of utility maximization under local differential privacy.