In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all t…
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48 results for “second order logic”
We propose a unified framework to address a family of classical mixed-integer optimization problems with logically constrained decision variables, including network design, facility location, unit commitment, sparse portfolio selection, binary quadratic optimization, sparse principal analysis and sparse learning proble…
Polyhedral semantics for intermediate logics; Nerve Criterion ensures completeness.
problem Characterize polyhedrally-complete intermediate logics.
method Developed Nerve Criterion to characterize polyhedrally-complete logics combinatorially.
result Nerve Criterion provides a necessary and sufficient condition for polyhedrally-completeness.
This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significa…
Knowledge graph reasoning, which aims at predicting the missing facts through reasoning with the observed facts, is critical to many applications. Such a problem has been widely explored by traditional logic rule-based approaches and recent knowledge graph embedding methods. A principled logic rule-based approach is th…
The artificial neural network is a popular framework in machine learning. To empower individual neurons, we recently suggested that the current type of neurons could be upgraded to 2nd order counterparts, in which the linear operation between inputs to a neuron and the associated weights is replaced with a nonlinear qu…
Uniform first order interpretation of the second order theory of countable groups of homeomorphismsmath.GR
Uniform interpretation of group theory in manifold homeomorphisms.
problem Understanding group properties in manifold homeomorphisms.
method First order theory interpretation of second order group theory.
result Many group theory problems encoded in homeomorphism groups.
Markov logic networks (MLNs) reconcile two opposing schools in machine learning and artificial intelligence: causal networks, which account for uncertainty extremely well, and first-order logic, which allows for formal deduction. An MLN is essentially a first-order logic template to generate Markov networks. Inference …
We introduce neural Markov logic networks (NMLNs), a statistical relational learning system that borrows ideas from Markov logic. Like Markov logic networks (MLNs), NMLNs are an exponential-family model for modelling distributions over possible worlds, but unlike MLNs, they do not rely on explicitly specified first-ord…
Deep learning methods capable of handling relational data have proliferated over the last years. In contrast to traditional relational learning methods that leverage first-order logic for representing such data, these deep learning methods aim at re-representing symbolic relational data in Euclidean spaces. They offer …
Combining deep neural networks with structured logic rules is desirable to harness flexibility and reduce uninterpretability of the neural models. We propose a general framework capable of enhancing various types of neural networks (e.g., CNNs and RNNs) with declarative first-order logic rules. Specifically, we develop…
A fundamental challenge in developing high-impact machine learning technologies is balancing the need to model rich, structured domains with the ability to scale to big data. Many important problem areas are both richly structured and large scale, from social and biological networks, to knowledge graphs and the Web, to…
Study logical generalization in GNNs using a new benchmark.
problem Understanding how GNNs adapt to new logical tasks.
method Developed GraphLog benchmark suite for logical tasks, evaluated GNNs in supervised, pretraining, and continual learning settings.
result Logical diversity during training affects GNNs' ability to generalize.
Statistical relational frameworks such as Markov logic networks and probabilistic soft logic (PSL) encode model structure with weighted first-order logical clauses. Learning these clauses from data is referred to as structure learning. Structure learning alleviates the manual cost of specifying models. However, this be…
Paper formalizes analogy between data sets and models using Hoare logic.
problem Lack of formal criteria for transferring machine learning models between data domains.
method Formalization of analogy using first-order logic and Hoare logic, rigorous theorem proving.
result Rigorous formalization of analogy in knowledge transfer between machine learning models.
RWTNs improve NTN performance in SRL tasks.
problem Improving NTN performance in SRL tasks.
method Randomly weighted, untrained tensors in NTN encoder network.
result RWTNs outperform LTNs in SII tasks.
We present a heuristic based algorithm to induce \textit{nonmonotonic} logic programs that will explain the behavior of XGBoost trained classifiers. We use the technique based on the LIME approach to locally select the most important features contributing to the classification decision. Then, in order to explain the mo…
Learning low-dimensional embeddings of knowledge graphs is a powerful approach used to predict unobserved or missing edges between entities. However, an open challenge in this area is developing techniques that can go beyond simple edge prediction and handle more complex logical queries, which might involve multiple un…
Study on topological order on fractal geometries, proving no-go theorem and fault-tolerant gates.
problem Investigating topological order on fractal geometries embedded in n dimensions.
method Using quantum error-correcting codes and systolic geometry to diagnose topological order.
result Proves no-go theorem for topological order on 2D fractals, survival on higher dimensions, and construction of fault-tolerant gates.
Transformers improve logical reasoning on longer proofs but struggle with length.
problem Understanding systematic generalization in neural proof generation.
method Soft theorem-proving using Transformer models, evaluating logical consistency and inference accuracy.
result Transformers improve generalization with longer proofs but have difficulty with length.
A key feature of inductive logic programming (ILP) is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs. Specifically, we extend meta-interpretive learning (MIL) to support learning higher…
Combining logic and probability has been a long stand- ing goal of AI research. Markov Logic Networks (MLNs) achieve this by attaching weights to formulas in first-order logic, and can be seen as templates for constructing features for ground Markov networks. Most techniques for learning weights of MLNs are domain-size…
Today, the dominant paradigm for training neural networks involves minimizing task loss on a large dataset. Using world knowledge to inform a model, and yet retain the ability to perform end-to-end training remains an open question. In this paper, we present a novel framework for introducing declarative knowledge to ne…
We propose Trusted Neural Network (TNN) models, which are deep neural network models that satisfy safety constraints critical to the application domain. We investigate different mechanisms for incorporating rule-based knowledge in the form of first-order logic constraints into a TNN model, where rules that encode safet…
AlphaLogics mines market logic to generate interpretable alpha factors.
problem Complex, opaque alpha factors from factor mining overlook market logic.
method Market Logic Mining, Factor Generation and Optimization, Market Logic Generation and Optimization.
result AlphaLogics improves predictive metrics and risk-adjusted returns over baselines.
Our interest in this paper is in the construction of symbolic explanations for predictions made by a deep neural network. We will focus attention on deep relational machines (DRMs, first proposed by H. Lodhi). A DRM is a deep network in which the input layer consists of Boolean-valued functions (features) that are defi…
The paper explores how neural networks learn logical functions and their generalization error.
problem Learning logical functions with neural networks and understanding generalization error.
method Gradient descent on neural networks, analyzing noise-stability and Boolean influence.
result Gradient descent on certain neural architectures tends to favor low-degree representations, impacting generalization error.
Determining when two knots are equivalent (more precisely isotopic) is a fundamental problem in topology. Here we formulate this problem in terms of Predicate Calculus, using the formulation of knots in terms of braids and some basic topological results. Concretely, Knot theory is formulated in terms of a language with…
Unified model learns concepts across domains like left and right.
problem Limited generalization of language concepts in inference-only models.
method Logic-Enhanced Foundation Model (LEFT) with a differentiable, domain-independent program executor.
result LEFT flexibly learns and reasons with concepts across 2D images, 3D scenes, human motions, and robotic manipulation.
Explaining neural network computation in terms of probabilistic/fuzzy logical operations has attracted much attention due to its simplicity and high interpretability. Different choices of logical operators such as AND, OR and XOR give rise to another dimension for network optimization, and in this paper, we study the o…
HyperFair integrates fairness in recommender systems using probabilistic soft logic.
problem Ensuring fairness in recommender systems across diverse domains.
method Soft fairness constraints integrated as regularization in a joint inference objective function.
result HyperFair improves fairness of predictions from black-box models and hybrid systems.
Recent years have witnessed the great success of deep neural networks in many research areas. The fundamental idea behind the design of most neural networks is to learn similarity patterns from data for prediction and inference, which lacks the ability of logical reasoning. However, the concrete ability of logical reas…
Boolean logic used for neural network training and inference, with convergence analysis.
problem Discrete optimization in neural networks with Boolean logic.
method Boolean logic backpropagation with convergence analysis.
result First convergence analysis for Boolean logic in neural networks.
Paper proposes a novel RRL framework that learns from images and incorporates expert knowledge.
problem Lack of effective methods to incorporate expert background knowledge and learn from non-relational data in RRL.
method Differentiable Inductive Logic Programming (ILP) for learning relational information from images and incorporating expert knowledge.
result Efficacy demonstrated on various environments and datasets, showing improved learning and generalization.
Neural Logic Reasoning integrates deep learning and symbolic logic for better prediction tasks.
problem Lack of cognitive reasoning in deep neural networks limits their ability to solve complex prediction tasks.
method Proposes Logic-Integrated Neural Network (LINN) that learns logical operations and conducts propositional logical reasoning.
result LINN significantly outperforms state-of-the-art recommendation models in Top-K recommendation.
Combines neural networks and logic circuits for interpretable, accurate, and cost-effective learning.
problem Lack of generalizability and interpretability in neural networks and high hardware cost in logic circuits.
method Trains a neural network, then translates it to random forests, and finally to AND-Inverter logic.
result The pipeline maintains greater accuracy and minimizes logic complexity.
This paper introduces a new reward shaping method for average-reward reinforcement learning.
problem Speeding up convergence to an optimal policy in average-reward reinforcement learning tasks.
method Developed a temporal logic-based approach to automatically generate reward shaping functions.
result The optimal policy can be recovered using the proposed reward shaping framework.
Real artificial intelligence always has been focused on by many machine learning researchers, especially in the area of deep learning. However deep neural network is hard to be understood and explained, and sometimes, even metaphysics. The reason is, we believe that: the network is essentially a perceptual model. There…
The purpose of this paper is to discuss how topology and geometry provide, in many instances, the connective tissue that enables logical comprehension. We illustrate this theme with many examples including Venn diagrams, knot diagrams, knot-logical diagrams and an arrow of reference that elucidates self-reference and G…
This study proposes a logic architecture for the high-speed and power efficiently training of a gradient boosting decision tree model of binary classification. We implemented the proposed logic architecture on an FPGA and compared training time and power efficiency with three general GBDT software libraries using CPU a…
Kernel for STL formulae enables machine learning in temporal logic.
problem Lack of a kernel for STL formulae.
method Define a kernel for STL formulae and embed them into a Hilbert space.
result Kernel-based machine learning algorithms can now be applied to STL formulae.
Transformers learn to predict temporal logic solutions from classical solver outputs.
problem Training neural networks on logic problem solutions for verification.
method Training a Transformer on generated training data from classical solvers, focusing on one solution per formula.
result Transformers can predict correct solutions to temporal logic problems, even to unseen benchmarks.
Many software analysis methods have come to rely on machine learning approaches. Code segmentation - the process of decomposing source code into meaningful blocks - can augment these methods by featurizing code, reducing noise, and limiting the problem space. Traditionally, code segmentation has been done using syntact…
We propose the Neural Logic Machine (NLM), a neural-symbolic architecture for both inductive learning and logic reasoning. NLMs exploit the power of both neural networks---as function approximators, and logic programming---as a symbolic processor for objects with properties, relations, logic connectives, and quantifier…
The logic of uncertainty as a logic of experience and chance and the co~event-based Bayes' theoremq-fin.GN
The logic of uncertainty is not the logic of experience and as well as it is not the logic of chance. It is the logic of experience and chance. Experience and chance are two inseparable poles. These are two dual reflections of one essence, which is called co~event. The theory of experience and chance is the theory of c…
Deep learning is very effective at jointly learning feature representations and classification models, especially when dealing with high dimensional input patterns. Probabilistic logic reasoning, on the other hand, is capable to take consistent and robust decisions in complex environments. The integration of deep learn…
We describe the concept of logical scaffolds, which can be used to improve the quality of software that relies on AI components. We explain how some of the existing ideas on runtime monitors for perception systems can be seen as a specific instance of logical scaffolds. Furthermore, we describe how logical scaffolds ma…
Foundations for Restraining Bolts: Reinforcement Learning with LTLf/LDLf restraining specificationscs.LG
In this work we investigate on the concept of "restraining bolt", envisioned in Science Fiction. Specifically we introduce a novel problem in AI. We have two distinct sets of features extracted from the world, one by the agent and one by the authority imposing restraining specifications (the "restraining bolt"). The tw…