ZKP systems have surged attention and held a fundamental role in contemporary cryptography. Zk-SNARK protocols dominate the ZKP usage, often implemented through arithmetic circuit programming paradigm. However, underconstrained or overconstrained circuits may lead to bugs. Underconstrained circuits refer to circuits that lack the necessary constraints, resulting in unexpected solutions in the circuit and causing the verifier to accept a bogus witness. Overconstrained circuits refer to circuits that are constrained excessively, resulting in the circuit lacking necessary solutions and causing the verifier to accept no witness, rendering the circuit meaningless. This paper introduces a novel approach for pinpointing two distinct types of bugs in ZKP circuits. The method involves encoding the arithmetic circuit constraints to polynomial equation systems and solving polynomial equation systems over a finite field by algebraic computation. The classification of verification results is refined, greatly enhancing the expressive power of the system. We proposed a tool, AC4, to represent the implementation of this method. Experiments demonstrate that AC4 represents a substantial 29% increase in the checked ratio compared to prior work. Within a solvable range, the checking time of AC4 has also exhibited noticeable improvement, demonstrating a magnitude increase compared to previous efforts.

Operating effectively in complex environments while complying with specified constraints is crucial for the safe and successful deployment of robots that interact with and operate around people. In this work, we focus on generating long-horizon trajectories that adhere to novel static and temporally-extended constraints/instructions at test time. We propose a data-driven diffusion-based framework, LTLDoG, that modifies the inference steps of the reverse process given an instruction specified using finite linear temporal logic ($\text{LTL}_f$). LTLDoG leverages a satisfaction value function on $\text{LTL}_f$ and guides the sampling steps using its gradient field. This value function can also be trained to generalize to new instructions not observed during training, enabling flexible test-time adaptability. Experiments in robot navigation and manipulation illustrate that the method is able to generate trajectories that satisfy formulae that specify obstacle avoidance and visitation sequences.

This paper addresses the problem of preference learning, which aims to learn user-specific preferences (e.g., "good parking spot", "convenient drop-off location") from visual input. Despite its similarity to learning factual concepts (e.g., "red cube"), preference learning is a fundamentally harder problem due to its subjective nature and the paucity of person-specific training data. We address this problem using a new framework called Synapse, which is a neuro-symbolic approach designed to efficiently learn preferential concepts from limited demonstrations. Synapse represents preferences as neuro-symbolic programs in a domain-specific language (DSL) that operates over images, and leverages a novel combination of visual parsing, large language models, and program synthesis to learn programs representing individual preferences. We evaluate Synapse through extensive experimentation including a user case study focusing on mobility-related concepts in mobile robotics and autonomous driving. Our evaluation demonstrates that Synapse significantly outperforms existing baselines as well as its own ablations. The code and other details can be found on the project website https://amrl.cs.utexas.edu/synapse .

Inductive logic reasoning is a fundamental task in graph analysis, which aims to generalize patterns from data. This task has been extensively studied for traditional graph representations, such as knowledge graphs (KGs), using techniques like inductive logic programming (ILP). Existing ILP methods assume learning from KGs with static facts and binary relations. Beyond KGs, graph structures are widely present in other applications such as procedural instructions, scene graphs, and program executions. While ILP is beneficial for these applications, applying it to those graphs is nontrivial: they are more complex than KGs, which usually involve timestamps and n-ary relations, effectively a type of hypergraph with temporal events. In this work, we propose temporal inductive logic reasoning (TILR), an ILP method that reasons on temporal hypergraphs. To enable hypergraph reasoning, we introduce the multi-start random B-walk, a novel graph traversal method for hypergraphs. By combining it with a path-consistency algorithm, TILR learns logic rules by generalizing from both temporal and relational data. To address the lack of hypergraph benchmarks, we create and release two temporal hypergraph datasets: YouCook2-HG and nuScenes-HG. Experiments on these benchmarks demonstrate that TILR achieves superior reasoning capability over various strong baselines.

Humans can develop new theorems to explore broader and more complex mathematical results. While current generative language models (LMs) have achieved significant improvement in automatically proving theorems, their ability to generate new or reusable theorems is still under-explored. Without the new theorems, current LMs struggle to prove harder theorems that are distant from the given hypotheses with the exponentially growing search space. Therefore, this paper proposes an Automated Theorem Generation (ATG) benchmark that evaluates whether an agent can automatically generate valuable (and possibly brand new) theorems that are applicable for downstream theorem proving as reusable knowledge. Specifically, we construct the ATG benchmark by Figure 1: An example theorem generated by GPT-4 splitting the Metamath library into three sets: (OpenAI, 2023). GPT-4 wrongly refers to the intermediate axioms, library, and problem based on their theorem (A (B A) A A) as proving depth. We conduct extensive experiments ((A B) (A A)). In Step 4, it applies "ax-to investigate whether current LMs can 1" but obtains the wrong expression instead of correct generate theorems in the library and benefit (A (B A)) and can not derive (A A) even the problem theorems proving. The results with the incorrect Steps 4 and 5. demonstrate that high-quality ATG data facilitates models' performances on downstream

Natural language inference (NLI), also known as Recognizing Textual Entailment (RTE), is an important aspect of natural language understanding. Most research now uses machine learning and deep learning to perform this task on specific datasets, meaning their solution is not explainable nor explicit. To address the need for an explainable approach to RTE, we propose a novel pipeline that is based on translating text into an Abstract Meaning Representation (AMR) graph. For this we use a pre-trained AMR parser. We then translate the AMR graph into propositional logic and use a SAT solver for automated reasoning. In text, often commonsense suggests that an entailment (or contradiction) relationship holds between a premise and a claim, but because different wordings are used, this is not identified from their logical representations. To address this, we introduce relaxation methods to allow replacement or forgetting of some propositions. Our experimental results show this pipeline performs well on four RTE datasets.

We propose a modal logic in which counting modalities appear in linear inequalities. We show that each formula can be transformed into an equivalent graph neural network (GNN). We also show that a broad class of GNNs can be transformed efficiently into a formula, thus significantly improving upon the literature about the logical expressiveness of GNNs. We also show that the satisfiability problem is PSPACE-complete. These results bring together the promise of using standard logical methods for reasoning about GNNs and their properties, particularly in applications such as GNN querying, equivalence checking, etc. We prove that such natural problems can be solved in polynomial space.

The goal of inductive program synthesis is for a machine to automatically generate a program from user-supplied examples of the desired behaviour of the program. A key underlying assumption is that humans can provide examples of sufficient quality to teach a concept to a machine. However, as far as we are aware, this assumption lacks both empirical and theoretical support. To address this limitation, we explore the question `Can humans teach machines to code?'. To answer this question, we conduct a study where we ask humans to generate examples for six programming tasks, such as finding the maximum element of a list. We compare the performance of a program synthesis system trained on (i) human-provided examples, (ii) randomly sampled examples, and (iii) expert-provided examples. Our results show that, on most of the tasks, non-expert participants did not provide sufficient examples for a program synthesis system to learn an accurate program. Our results also show that non-experts need to provide more examples than both randomly sampled and expert-provided examples.

Chain-of-Thought (CoT) prompting has emerged as a pivotal technique for augmenting the inferential capabilities of language models during reasoning tasks. Despite its advancements, CoT often grapples with challenges in validating reasoning validity and ensuring informativeness. Addressing these limitations, this paper introduces the Logic Agent (LA), an agent-based framework aimed at enhancing the validity of reasoning processes in Large Language Models (LLMs) through strategic logic rule invocation. Unlike conventional approaches, LA transforms LLMs into logic agents that dynamically apply propositional logic rules, initiating the reasoning process by converting natural language inputs into structured logic forms. The logic agent leverages a comprehensive set of predefined functions to systematically navigate the reasoning process. This methodology not only promotes the structured and coherent generation of reasoning constructs but also significantly improves their interpretability and logical coherence. Through extensive experimentation, we demonstrate LA's capacity to scale effectively across various model sizes, markedly improving the precision of complex reasoning across diverse tasks.

Error Detection and Correction Codes (ECCs) are often used in digital designs to protect data integrity. Especially in safety-critical systems such as automotive electronics, ECCs are widely used and the verification of such complex logic becomes more critical considering the ISO 26262 safety standards. Exhaustive verification of ECC using formal methods has been a challenge given the high number of data bits to protect. As an example, for an ECC of 128 data bits with a possibility to detect up to four-bit errors, the combination of bit errors is given by 128C1 + 128C2 + 128C3 + 128C4 = 1.1 * 10^7. This vast analysis space often leads to bounded proof results. Moreover, the complexity and state-space increase further if the ECC has sequential encoding and decoding stages. To overcome such problems and sign-off the design with confidence within reasonable proof time, we present a pragmatic formal verification approach of complex ECC cores with several complexity reduction techniques and know-how that were learnt during the course of verification. We discuss using the linearity of the syndrome generator as a helper assertion, using the abstract model as glue logic to compare the RTL with the sequential version of the circuit, k-induction-based model checking and using mathematical relations captured as properties to simplify the verification in order to get an unbounded proof result within 24 hours of proof runtime.