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"First-order logic."
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Recommender systems based on neuro-symbolic knowledge graph embeddings encoding first-order logic rules
In this paper, we present a knowledge-aware recommendation model based on neuro-symbolic graph embeddings that encode first-order logic rules. Our approach is based on the intuition that is the basis of neuro-symbolic AI systems: to combine deep learning and symbolic reasoning in one single model, in order to take the best out of both the paradigms. To this end, we start from a knowledge graph (KG) encoding information about users, ratings, and descriptive properties of the items and we design a model that combines background knowledge encoded in logical rules mined from the KG with explicit knowledge encoded in the triples of the KG itself to obtain a more precise representation of users and items. Specifically, our model is based on the combination of: (i) a rule learner that extracts first-order logic rules based on the information encoded in the knowledge graph; (ii) a graph embedding module, that jointly learns a vector space representation of users and items based on the triples encoded in the knowledge graph and the rules previously extracted; (iii) a recommendation module that uses the embeddings to feed a deep learning architecture that provides users with top-k recommendations. In the experimental section, we evaluate the effectiveness of our strategy on three datasets, and the results show that the combination of knowledge graph embeddings and first-order logic rules led to an improvement in the predictive accuracy and in the novelty of the recommendations. Moreover, our approach overcomes several competitive baselines, thus confirming the validity of our intuitions.
Journal Article
A Canonical Model for Constant Domain Basic First-Order Logic
I build a canonical model for constant domain basic first-order logic (BQLCD), the constant domain first-order extension of Visser's basic propositional logic, and use the canonical model to verify that BQLCD satisfies the disjunction and existence properties.
Journal Article
Undecidability of First-Order Modal and Intuitionistic Logics with Two Variables and One Monadic Predicate Letter
We prove that the positive fragment of first-order intuitionistic logic in the language with two individual variables and a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable. This holds true regardless of whether we consider semantics with expanding or constant domains. We then generalise this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of the weak law of the excluded middle and QBL and QFL are first-order counterparts of Visser's basic and formal logics, respectively. We also show that, for most \"natural\" first-order modal logics, the two-variable fragment with a single monadic predicate letter, without functional symbols, constants, and equality, is undecidable, regardless of whether we consider semantics with expanding or constant domains. These include all sublogics of QKTB, QGL, and QGrz—among them, QK, QT, QKB, QD, QK4, and QS4.
Journal Article
A Unified Approach to Structural Limits and Limits of Graphs with Bounded Tree-Depth
In this paper we introduce a general framework for the study of limits of relational structures and graphs in particular, which is
based on a combination of model theory and (functional) analysis. We show how the various approaches to graph limits fit to this
framework and that they naturally appear as “tractable cases” of a general theory. As an outcome of this, we provide extensions of known
results. We believe that this puts these into a broader context. The second part of the paper is devoted to the study of sparse
structures. First, we consider limits of structures with bounded diameter connected components and we prove that in this case the
convergence can be “almost” studied component-wise. We also propose the structure of limit objects for convergent sequences of sparse
structures. Eventually, we consider the specific case of limits of colored rooted trees with bounded height and of graphs with bounded
tree-depth, motivated by their role as “elementary bricks” these graphs play in decompositions of sparse graphs, and give an explicit
construction of a limit object in this case. This limit object is a graph built on a standard probability space with the property that
every first-order definable set of tuples is measurable. This is an example of the general concept of
eBook
An Intuitionistically Complete System of Basic Intuitionistic Conditional Logic
We introduce a basic intuitionistic conditional logic
IntCK
that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that
IntCK
stands in a very natural relation to other similar logics, like the basic classical conditional logic
CK
and the basic intuitionistic modal logic
IK
. As for the basic intuitionistic conditional logic
ICK
proposed in Weiss (
Journal of Philosophical Logic
,
48
, 447–469,
2019
),
IntCK
extends its language with a diamond-like conditional modality
◊
→
, but its (
◊
→
)-free fragment is also a proper extension of
ICK
. We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.
Journal Article
Markov logic networks
Issue Title: Special Issue: Multi-Relational Data Mining and Statistical Relational Learning We propose a simple approach to combining first-order logic and probabilistic graphical models in a single representation. A Markov logic network (MLN) is a first-order knowledge base with a weight attached to each formula (or clause). Together with a set of constants representing objects in the domain, it specifies a ground Markov network containing one feature for each possible grounding of a first-order formula in the KB, with the corresponding weight. Inference in MLNs is performed by MCMC over the minimal subset of the ground network required for answering the query. Weights are efficiently learned from relational databases by iteratively optimizing a pseudo-likelihood measure. Optionally, additional clauses are learned using inductive logic programming techniques. Experiments with a real-world database and knowledge base in a university domain illustrate the promise of this approach.[PUBLICATION ABSTRACT]
Journal Article
THE FLUTED FRAGMENT REVISITED
We study the fluted fragment, a decidable fragment of first-order logic with an unbounded number of variables, motivated by the work of W. V. Quine.We show that the satisfiability problem for this fragment has nonelementary complexity, thus refuting an earlier published claim by W. C. Purdy that it is in NExpTime. More precisely, we consider 𝓕𝓛
m
, the intersection of the fluted fragment and the m-variable fragment of first-order logic, for all m ≥ 1. We show that, for m ≥ 2, this subfragment forces ⌊m/2⌋-tuply exponentially large models, and that its satisfiability problem is ⌊m/2⌋-NExpTime-hard. We further establish that, for m ≥ 3, any satisfiable 𝓕𝓛
m
-formula has a model of at most (m − 2)-tuply exponential size, whence the satisfiability (= finite satisfiability) problem for this fragment is in (m − 2)-NExpTime. Together with other, known, complexity results, this provides tight complexity bounds for 𝓕𝓛
m
for all m ≤ 4.
Journal Article
Developing a Novel Ontology for Cybersecurity in Internet of Medical Things-Enabled Remote Patient Monitoring
IoT has seen remarkable growth, particularly in healthcare, leading to the rise of IoMT. IoMT integrates medical devices for real-time data analysis and transmission but faces challenges in data security and interoperability. This research identifies a significant gap in the existing literature regarding a comprehensive ontology for vulnerabilities in medical IoT devices. This paper proposes a fundamental domain ontology named MIoT (Medical Internet of Things) ontology, focusing on cybersecurity in IoMT (Internet of Medical Things), particularly in remote patient monitoring settings. This research will refer to similar-looking acronyms, IoMT and MIoT ontology. It is important to distinguish between the two. IoMT is a collection of various medical devices and their applications within the research domain. On the other hand, MIoT ontology refers to the proposed ontology that defines various concepts, roles, and individuals. MIoT ontology utilizes the knowledge engineering methodology outlined in Ontology Development 101, along with the structured life cycle, and establishes semantic interoperability among medical devices to secure IoMT assets from vulnerabilities and cyberattacks. By defining key concepts and relationships, it becomes easier to understand and analyze the complex network of information within the IoMT. The MIoT ontology captures essential key terms and security-related entities for future extensions. A conceptual model is derived from the MIoT ontology and validated through a case study. Furthermore, this paper outlines a roadmap for future research, highlighting potential impacts on security automation in healthcare applications.
Journal Article
Explainable models via compression of tree ensembles
Ensemble models (bagging and gradient-boosting) of relational decision trees have proved to be some of the most effective learning methods in the area of probabilistic logic models (PLMs). While effective, they lose one of the most important benefits of PLMs—interpretability. In this paper we consider the problem of compressing a large set of learned trees into a single explainable model. To this effect, we propose CoTE—Compression of Tree Ensembles—that produces a single small decision list as a compressed representation. CoTE first converts the trees to decision lists and then performs the combination and compression with the aid of the original training set. An experimental evaluation demonstrates the effectiveness of CoTE in several benchmark relational data sets.
Journal Article