Source-linked AI summary
Logic Tensor Networks: Deep Learning and Logical Reasoning from Data and Knowledge
Luciano Serafini, Artur d'Avila Garcez
TL;DR
The paper addresses integrating relational learning from real-valued data with first-order reasoning. It proposes Logic Tensor Networks, which implement Real Logic with tensor networks. In a knowledge-completion example, background knowledge supports learning additional facts, though comparison with other neural-symbolic approaches remains future work.
Problem
The paper seeks a computationally adequate way for an agent to infer relational knowledge and predict object properties or classes from partial relations and quantitative attributes.
Source: §1 · p. 2Method
Logic Tensor Networks combine tensor-network learning with first-order many-valued logic, representing data and constraints through real-valued vectors.
Source: §1 · p. 2Results
With background knowledge, LTN predicts additional facts, and all generic background axioms except the smoking-causes-cancer axiom have satisfiability above 90%; that axiom reaches 77%.
Source: §5 · p. 9Takeaways & Limitations
The paper presents approximate satisfiability as learning over real-valued vectors, integrating relational knowledge constraints with data-driven approaches.
Source: §7 · p. 11Experimental evaluation and comparison with other neural-symbolic approaches, including newer developments, remain desirable.
Source: §6 · p. 11
Abstract
from arXiv · showhide
We propose Logic Tensor Networks: a uniform framework for integrating automatic learning and reasoning. A logic formalism called Real Logic is defined on a first-order language whereby formulas have truth-value in the interval [0,1] and semantics defined concretely on the domain of real numbers. Logical constants are interpreted as feature vectors of real numbers. Real Logic promotes a well-founded integration of deductive reasoning on a knowledge-base and efficient data-driven relational machine learning. We show how Real Logic can be implemented in deep Tensor Neural Networks with the use of Google's tensorflow primitives. The paper concludes with experiments applying Logic Tensor Networks on a simple but representative example of knowledge completion.
1 Introduction
The paper addresses how an agent can jointly learn from real-valued object features and infer relational knowledge. It proposes Logic Tensor Networks, using a vector-based representation to integrate first-order reasoning and tensor-network learning.
1 Introduction
- LTN integrates tensor-network learning with first-order many-valued logic, representing data and logical constraints with real-valued vectors.The framework is intended to let reasoning improve learning and new data revise constraints.Source: §1 · p. 2
- An agent must infer new relations among objects and predict their numerical properties or classes from partial relational knowledge and quantitative attributes.Source: §1 · p. 2
- Classes and relations can constrain one another: a relation may support inferring another object's class, while contrary examples may revise that conclusion.Source: §1 · p. 2
- The framework combines open-domain first-order inference with relational multi-class learning using tensor networks.The paper contrasts this expected capability with related literature.Source: §1 · p. 2
- In practice, first-order reasoning with function symbols is approximated through iterative deepening of clause depth.Source: §1 · p. 2
- The proposed vector-based representation is presented as adequate for integrating learning and reasoning and mapping symbols to real-world manifestations.Source: §1 · p. 3
2 Real Logic
Real Logic gives first-order language a concrete semantics over real numbers. Groundings map constants and functions to real-valued representations, while clauses receive truth values in [0,1].
2 Real Logic
- The language signature contains constant, functional, and predicate symbols, and sentences are assumed to be in prenex conjunctive, skolemised normal form.For example, ∀x(A(x) → ∃yR(x, y)) becomes ¬A(x) ∨ R(x, f(x)).Source: §2 · p. 3
- Real Logic interprets sentences of a first-order language with concrete real-valued semantics rather than standard abstract FOL semantics.Source: §2 · p. 3
- A grounding maps each closed term to an n-tuple of real numbers representing its object's numeric features, and each clause to a truth value in [0,1].Higher truth values represent greater confidence in the clause.Source: §2 · p. 3
- Grounding extends compositionally: functions act on grounded terms, predicates yield real-valued truth, and negation maps truth to 1 minus that value.Source: §2 · p. 4
- Disjunction combines the truth values of its clauses with an s-norm, whose choice specifies an instance of Real Logic.Listed choices include Lukasiewicz, product, and Gödel s-norms.Source: §2 · p. 4
- A document example grounds objects as bag-of-words vectors, concatenation as vector addition, and similarity as cosine similarity.Source: §2 · p. 4
3 Learning as approximate satisfiability
Learning is framed as finding an extension of a partial grounding that best satisfies a finite set of clause instantiations. Because searching all groundings and instantiations is impractical, the approach restricts both.
3 Learning as approximate satisfiability
- A grounded theory pairs clauses with confidence intervals and a partial grounding; it is satisfiable if an extending grounding meets every interval for every clause instantiation.Source: §3 · p. 5
- Checking satisfiability over all possible groundings and clause instantiations is impractical, so both the grounding family and instantiation set are restricted.For clauses containing function symbols, instantiations may be limited by depth.Source: §3 · p. 5
- The loss is zero when a clause’s truth value lies within its prescribed interval, and otherwise measures its distance from that interval.The intended grounding should capture correlations between quantitative attributes and relational properties.Source: §3 · p. 5
- Approximate satisfiability seeks a grounding extension that minimizes error across a finite subset of clause instantiations.The grounding must extend the given partial grounding and belong to a specified family of grounding functions.Source: §3 · pp. 5–6
4 Implementing Real Logic in Tensor Networks
Real Logic is instantiated with tensor transformations for functions and neural tensor networks for predicates. A neural network computes clause truth by grounding literals and combining them with an s-norm.
4 Implementing Real Logic in Tensor Networks
- Function symbols are interpreted as linear transformations of their argument vectors.The transformation uses a matrix and bias vector.Source: §4 · p. 6
- Predicate grounding generalizes a neural tensor network, mapping argument vectors to truth values in [0, 1].The predicate network uses a tensor, matrix, bias vector, and sigmoid.Source: §4 · p. 6
- Clause truth is computed by grounding its literals and combining them with the selected s-norm.The illustrated network represents ¬P(x, y) → A(y), with G(x) = v, G(y) = u, and k = 2.Source: §4 · p. 6, Figure 1 · p. 7
- The network parameters are learned by minimizing loss, equivalently maximizing satisfiability of the clause.Source: §4 · p. 6
5 An Example of Knowledge Completion
The friends-and-smokers example tests whether LTN can complete a knowledge base with missing facts and unknown truth degrees. With background knowledge, it learns more facts, while fitting an inconsistent smoking–cancer rule less closely than the other axioms.
5 An Example of Knowledge Completion
- The example uses 14 people in two groups, with smoking facts known in both and cancer facts known only in the first group.Friendship knowledge is complete within each group only when symmetry is assumed.Source: §5 · p. 7
- The task is to estimate truth degrees for listed facts, assign truth values to missing facts, and find groundings for constants.The setup assumes listed facts are true initially; the second experiment adds general background knowledge to factual knowledge.Source: §5 · pp. 7–8
- Without background knowledge, LTN fits the factual KB and infers additional friendship and cancer facts by exploiting similarities between learned groundings.Examples include F(c, b), F(g, b), and ¬F(b, a).Source: §5 · p. 8, Table 1 · p. 9
- The authors implement LTN as a Python library using TensorFlow and report both experiments’ results in Table 1.The table caption notes that Kexp1 has satisfiability approximately 1.Source: §5 · pp. 7–8, Table 1 · p. 9
- With background knowledge, LTN predicts C(i) and C(n), infers the symmetric friendship m–i, and satisfies all generic axioms above 90% except smoking-causes-cancer at 77%.The smoking–cancer axiom conflicts with data in which f and g smoke but do not have cancer.Source: §5 · p. 9, Table 1 · p. 9
6 Related work
Related work positions Real Logic among approaches that embed logical knowledge in real-valued spaces and combine logic with learning. Its distinctions include open-domain first-order reasoning, general grounding semantics, and Skolemized existential quantifiers that can generate prediction features; broader comparative evaluation remains desirable.
6 Related work
- Real Logic shares geometric interpretations of terms with, but generalizes its semantics for functions and predicates; is representable as a single-layer LTN.Source: §6 · pp. 9–10
- Unlike MLNs, Real Logic determines complex-formula truth through fuzzy reasoning and learns relations between object features by error minimization, while operating on open domains.Source: §6 · p. 10
- LTNs share open-domain scope with BLOG but use tensor networks for computational efficiency without requiring a probabilistic interpretation.Source: §6 · p. 10
- Unlike, Real Logic grounds functional symbols, uses a more general atomic-formula model, avoids the closed-world assumption for quantifiers, and does not fix a t-norm.Source: §6 · p. 10
- LTN existential quantifiers are Skolemized rather than reduced to finite disjunctions, allowing satisfaction by new individuals and generation of feature vectors for prediction.For ∀x∃yR(x, y), the grounded Skolem function maps an input vector to features of a typical related individual.Source: §6 · pp. 10–11
- Comparative evaluation against newer neural-symbolic approaches is identified as desirable, so the paper does not establish such comparisons.Source: §6 · p. 11
7 Conclusion and future work
The conclusion presents Real Logic as a uniform framework for learning and reasoning, implemented in deep tensor networks as Logic Tensor Networks. It reports applications to knowledge completion and data prediction, while identifying large-scale experiments and benchmark comparisons as future work.
7 Conclusion and future work
- Approximate satisfiability is framed as learning with knowledge and data mapped onto real-valued vectors, integrating relational constraints with data-driven approaches.Source: §7 · p. 11
- Real Logic is implemented in deep tensor networks called Logic Tensor Networks and applied to knowledge completion and data prediction.Source: §7 · p. 11
- Future work includes releasing the TensorFlow implementation and conducting large-scale experiments and relational-learning benchmark comparisons.Source: §7 · p. 11