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 TENSORFLOWTM primitives. The paper concludes with experiments applying Logic Tensor Networks on a simple but representative example of knowledge completion.
The recent availability of large-scale data combining multiple data modalities, such
as image, text, audio and sensor data, has opened up various research and
commercial opportunities, underpinned by machine learning methods and techniques [
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This paper proposes a framework called Logic Tensor Networks (LTN) which
integrates learning based on tensor networks [
We are interested in providing a computationally adequate approach to
implementing learning and reasoning [
For example, a person may have a grounding into a 4-tuple containing some numerical representation of the person’s name, her height, weight, and number of friends in some social network. Object tuples can participate in a set of relations R = fR1; : : : ; Rkg, with Ri O (Ri), where (Ri) denotes the arity of relation Ri. We presuppose the existence of a latent (unknown) relation between the above numerical properties, i.e. groundings, and partial relational structure R on O. Starting from this partial knowledge, an agent is required to: (i) infer new knowledge about the relational structure on the objects of O; (ii) predict the numerical properties or the class of the objects in O.
Classes and relations are not normally independent. For example, it may be the case that if an object x is of class C, C(x), and it is related to another object y through relation R(x; y) then this other object y should be in the same class C(y). In logic: 8x9y((C(x) ^ R(x; y)) ! C(y)). Whether or not C(y) holds will depend on the application: through reasoning, one may derive C(y) where otherwise there might not have been evidence of C(y) from training examples only; through learning, one may need to revise such a conclusion once examples to the contrary become available. The vectorial representation proposed in this paper permits both reasoning and learning as exemplified above and detailed in the next section.
The above forms of reasoning and learning are integrated in a unifying framework,
implemented within tensor networks, and exemplified in relational domains combining
data and relational knowledge about the objects. It is expected that, through an
adequate integration of numerical properties and relational knowledge, differently from the
immediate related literature [
The main contribution of this paper is two-fold. It introduces a novel framework for the integration of learning and reasoning which can take advantage of the repre1 In practice, FOL reasoning including function symbols is approximated through the usual
iterative deepening of clause depth. sentational power of (multi-valued) first-order logic, and it instantiates the framework using tensor networks into an efficient implementation which shows that the proposed vector-based representation of the logic offers an adequate mapping between symbols and their real-world manifestations, which is appropriate for both rich inference and learning from examples.
The paper is organized as follows. In Section 2, we define Real Logic. In Section 3, we propose the Learning-as-Inference framework. In Section 4, we instantiate the framework by showing how Real Logic can be implemented in deep Tensor Neural Networks leading to Logic Tensor Networks (LTN). Section 5 contains an example of how LTN handles knowledge completion using (possibly inconsistent) data and knowledge from the well-known smokers and friends experiment. Section 6 concludes the paper and discusses directions for future work. 2
We start from a first order language L, whose signature contains a set C of constant
symbols, a set F of functional symbols, and a set P of predicate symbols. The sentences of
L are used to express relational knowledge, e.g. the atomic formula R(o1; o2) states that
objects o1 and o2 are related to each other through binary relation R; 8xy:(R(x; y) !
R(y; x)) states that R is a symmetric relation, where x and y are variables; 9y:R(o1; y)
states that there is an (unknown) object which is related to object o1 through R. For
simplicity, without loss of generality, we assume that all logical sentences of L are in prenex
conjunctive, skolemised normal form [
As for the semantics of L, we deviate from the standard abstract semantics of FOL, and we propose a concrete semantics with sentences interpreted as tuples of real numbers. To emphasise the fact that L is interpreted in a “real” world, we use the term (semantic) grounding, denoted by G, instead of the more standard interpretation2. – G associates an n-tuple of real numbers G(t) to any closed term t of L; intuitively
G(t) is the set of numeric features of the object denoted by t. – G associates a real number in the interval [0; 1] to each clause of L. Intuitively, G( ) represents one’s confidence in the truth of ; the higher the value, the higher the confidence.
A grounding is specified only for the elements of the signature of L. The grounding of terms and clauses is defined inductively, as follows.
Definition 1. A grounding G for a first order language L is a function from the signature of L to the real numbers that satisfies the following conditions: 1. G(c) 2 Rn for every constant symbol c 2 C; 2. G(f ) 2 Rn (f) ! Rn for every f 2 F ; 2 In logic, the term “grounding” indicates the operation of replacing the variables of a term/formula with constants. To avoid confusion, we use the term “instantiation” for this. 3. G(P ) 2 Rn (R)
! [0; 1] for every P 2 P; A grounding G is inductively extended to all the closed terms and clauses, as follows: G(f (t1; : : : ; tm)) = G(f )(G(t1); : : : ; G(tm))
G(P (t1; : : : ; tm)) = G(P )(G(t1); : : : ; G(tm)) G(:P (t1; : : : ; tm)) = 1
G(P (t1; : : : ; tm)) G( 1 _
_ k) = (G( 1); : : : ; G( k))
where is an s-norm operator, also known as a t-co-norm operator (i.e. the dual of some
t-norm operator). 3
Example 1. Suppose that O = fo1; o2; o3g is a set of documents defined on a finite
dictionary D = fw1; :::; wng of n words. Let L be the language that contains the binary
function symbol concat(x; y) denoting the document resulting from the concatenation
of documents x with y. Let L contain also the binary predicate Sim which is
supposed to be true if document x is deemed to be similar to document y. An example
of grounding is the one that associates to each document its bag-of-words vector [
For instance, if the three documents are o1 = “John studies logic and plays football”, o2 = “Mary plays football and logic games”, o3 = “John and Mary play football and study logic together”, and W =fJohn, Mary, and, football, game, logic, play, study, togetherg then the following are examples of the grounding of terms, atomic formulas and clauses.
G(o1) = h1; 0; 1; 1; 0; 1; 1; 1; 0i G(o2) = h0; 1; 1; 1; 1; 1; 1; 0; 0i
G(o3) = h1; 1; 2; 1; 0; 1; 1; 1; 1i
G(concat(o1; o2)) = G(o1) + G(o2) = h1; 1; 2; 2; 1; 2; 2; 1; 0i G(Sim(concat(o1; o2); o3) =
G(concat(o1; o2)) G(o3) jjG(concat(o1; o2))jj jjG(o3)jj max(0:86; 0:73) = 0:86 3 Examples of t-norms which can be chosen here are Lukasiewicz, product, and Go¨del.
Lukasiewicz s-norm is defined as Luk(x; y) = min(x + y; 1); Product s-norm is defined as P r(x; y) = x + y x y; Go¨del s-norm is defined as max(x; y) = max(x; y):
We start by defining ground theory and their satisfiability.
Definition 2 (Satisfiability). Let w 2 [0; 1]. We say that G satisfies if v G( ) w.
be a closed clause in L, G a grounding, and v in the confidence interval [v; w], written G j=vw ,
A partial grounding, denoted by G^, is a grounding that is defined on a subset of the signature of L. A grounded theory is a set of clauses in the language of L and partial grounding G^.
Definition 3 (Grounded Theory). A grounded theory (GT) is a pair hK; G^i where K is a set of pairs h[v; w]; (x)i, where (x) is a clause of L containing the set x of free variables, and [v; w] [0; 1] is an interval contained in [0; 1], and G^ is a partial grounding.
Definition 4 (Satisfiability of a Grounded Theory). A GT hK; G^i is satisfiabile if there exists a grounding G, which extends G^ such that for all h[v; w]; (x)i 2 K and any tuple t of closed terms, G j=vw (t).
From the previous definiiton it follows that checking if a GT hK; G^i is satisfiable
amounts to seaching for an extension of the partial grounding G^ in the space of all
possible groundings, such that all the instantiations of the clauses in K are satisfied
w.r.t. the specified interval. Clearly this is unfeasible from a practical point of view.
As is usual, we must restrict both the space of grounding and clause instantiations.
Let us consider each in turn: To check satisfiability on a subset of all the functions on
real numbers, recall that a grounding should capture a latent correlation between the
quantitative attributes of an object and its relational properties4. In particular, we are
interested in searching within a specific class of functions, in this paper based on tensor
networks, although other family of functions can be considered. To limit the number
of clause instantiations, which in general might be infinite since L admits function
symbols, the usual approach is to consider the instantiations of each clause up to a
certain depth [
When a grounded theory hK; G^i is inconsitent, that is, there is no grounding G that satisfies it, we are interested in finding a grounding which satisfies as much as possible of hK; G^i. For any h[v; w]; i 2 K we want to find a grounding G that minimizes the satisfiability error. An error occurs when a grounding G assigns a value G( ) to a clause which is outside the interval [v; w] prescribed by K. The measure of this error can be defined as the minimal distance between the points in the interval [v; w] and G( ): Loss(G; h[v; w]; i) = jx
G( )j; v
x
w
(1)
4 For example, whether a document is classified as from the field of Artificial Intelligence (AI)
depends on its bag-of-words grounding. If the language L contains the unary predicate AI(x)
standing for “x is a paper about AI” then the grounding of AI(x), which is a function from
bag-of-words vectors to [
Notice that if G( ) 2 [v; w], Loss(G; ) = 0.
The above gives rise to the following definition of approximate satisfiability w.r.t. a family G of grounding functions on the language L.
Definition 5 (Approximate satisfiability). Let hK; G^i be a grounded theory and K0 a finite subset of the instantiations of the clauses in K, i.e.
K0
fh[v; w]; (t)ig j h[v; w]; (x)i 2 K and t is any n-tuple of closed terms.g Let G be a family of grounding functions. We define the best satisfiability problem as the problem of finding an extensions G of G^ in G that minimizes the satisfiability error on the set K0, that is:
G = argmin
X G^ G2G h[v;w]; (t)i2K0
Loss(G; h[v; w]; (t)i) 4 Implementing Real Logic in Tensor Networks Specific instances of Real Logic can be obtained by selectiong the space G of groundings and the specific s-norm for the interpretation of disjunction. In this section, we describe a realization of real logic where G is the space of real tensor transformations of order k (where k is a parameter). In this space, function symbols are interpreted as linear transformations. More precisely, if f is a function symbol of arity m and v1; : : : ; vm 2 Rn are real vectors corresponding to the grounding of m terms then G(f )(v1; : : : ; vm) can be written as:
G(f )(v1; : : : ; vm) = Mf v + Nf for some n mn matrix Mf and n-vector Nf , where v = hv1; : : : ; vni.
The grounding of m-ary predicate P , G(P ), is defined as a generalization of the
neural tensor network [
G(P ) =
uTP tanh vT W P[1:k]v + VP v + BP
(2)
vwehcetorer Win P[R1:kk], iasnad 3-D tensor in Rmn mn k, VP is a matrix in Rk mn, and BP is a
is the sigmoid function. With this encoding, the grounding (i.e.
truth-value) of a clause can be determined by a neural network which first computes
the grounding of the literals contained in the clause, and then combines them using
the specific s-norm. An example of tensor network for :P (x; y) ! A(y) is shown in
Figure 1. This architecture is a generalization of the structure proposed in [
G(P (v; u) ! A(u)
max G(:P ) G(A) 1 uP th + th + th + uA th + WP1
WP2
VP1
VP2
BP1
BP2
W A1
W A2
VA1
VA2
BA1
BA2 v = hv1; : : : ; vni
u = hu1; : : : ; uni
Logic Tensor Networks have been implemented as a Python library called ltn using
Google’s TENSORFLOWTM . To test our idea, in this section we use the well-known
friends and smokers5 example [
The facts contained in the knowledge-bases should have different degrees of truth, and this is not known. Otherwise, the combined knowledge-base would be inconsistent (it would deduce e.g. S(b) and :S(b)). Our main task is to complete the knowledgebase (KB), that is: (i) find the degree of truth of the facts contained in KB, (ii) find a truth-value for all the missing facts, e.g. C(i), (iii) find the grounding of each constant symbol a; :::; n:6 To answer (i)-(iii), we use ltn to find a grounding that best 5 Normally, a probabilistic approach is taken to solve this problem, and one that requires instantiating all clauses to remove variables, essentially turning the problem into a propositional one; ltn takes a different approach. 6 Notice how no grounding is provided about the signature of the knowledge-base.
SF C Ka:::h
S(a); S(e); S(f ); S(g); :S(b); :S(c); :S(d); :S(g); :S(h); F (a; b); F (a; e); F (a; f ); F (a; g); F (b; c);
F (c; d); F (e; f ); F (g; h); :F (a; c); :F (a; d); :F (a; h); :F (b; d); :F (b; e); :F (b; f ); :F (b; g); :F (b; h); :F (c; e); :F (c; f ); :F (c; g); :F (c; h); :F (d; e); :F (d; f ); :F (d; g); :F (d; h); :F (e; g); :F (e; h); :F (f; g); :F (f; h);
C(a); C(e); :C(b); :C(c); :C(d); :C(f ); :C(g); :C(h)
SF Ki:::n
S(i); S(n); :S(j); :S(k); :S(l); :S(m); F (i; j); F (i; m); F (k; l); F (m; n); :F (i; k); :F (i; l); :F (i; n); :F (j; k); :F (j; l); :F (j; m); :F (j; n); :F (l; n); :F (k; m); :F (l; m) 8x:F (x; x); 8xy(F (x; y) ! F (y; x)); 8x9yF (x; y);
K
SF C 8xy(S(x) ^ F (x; y) ! S(y));
8x(S(x) ! C(x)) approximates the complete KB. We start by assuming that all the facts contained in the knowledge-base are true (i.e. have degree of truth 1). To show the role of background knolwedge in the learning-inference process, we run two experiments. In the first (exp1), we seek to complete a KB consisting of only factual knowledge: Kexp1 = Ka:::h [ KiS::F:n. In the second (exp1), we also include background knowledge, that is:
SF C Kexp2 = Kexp1 [ KSF C .
We confgure the network as follows: each constant (i.e. person) can have up to 30
real-valued features. We set the number of layers k in the tensor network to 10, and
the regularization parameter7 = 1 10. For the purpose of illustration, we use the
Lukasiewicz t-norm with s-norm (a; b) = min(1; a + b), and use the harmonic mean
as aggregation operator. An estimation of the optimal grounding is obtained after 5,000
runs of the RMSProp learning algorithm [
The results of the two experiments are reported in Table 1. For readability, we use boldface for truth-values greater than 0.5. The truth-values of the facts listed in a knowledge-base are highlighted with the same background color of the knowledgebase in Figure 2. The values with white background are the result of the knowledge completion produced by the LTN learning-inference procedure. To evaluate the quality of the results, one has to check whether (i) the truth-values of the facts listed in a KB are indeed close to 1.0, and (ii) the truth-values associated with knowledge completion correspond to expectation. An initial analysis shows that the LTN associated with Kexp1 produces the same facts as Kexp1 itself. In other words, the LTN fits the data. However, the LTN also learns to infer additional positive and negative facts about F and C not derivable from Kexp1 by pure logical reasoning; for example: F (c; b), F (g; b) and :F (b; a). These facts are derived by exploiting similarities between the groundings of 7 A smoothing factor jj jj22 is added to the loss function to create a preference for learned parameters with a lower absolute value.
F e F e Learning and reasoning on Kexp1 = the constants generated by the LTN. For instance, G(c) and G(g) happen to present a high cosine similarity measure. As a result, facts about the friendship relations of c affect the friendship relations of g and vice-versa, for instance F (c; b) and F (g; b). The level of satisfiability associated with cally satisfiable.
Kexp1 1, which indicates that Kexp1 is classi
The results of the second experiment show that more facts can be learned with the inclusion of background knowledge. For example, the LTN now predicts that C (i) and C (n) are true. Similarly, from the symmetry of the friendship relation, the LTN concludes that m is a friend of i, as expected. In fact, all the axioms in the generic background knowledge KSF C are satisfied with a degree of satisfiability higher than 90%, apart from the smoking causes cancer axiom - which is responsible for the classical inconsistency since in the data f and g smoke and do not have cancer -, which has a degree of satisfiability of 77%. 6
In his recent note, [
Real logic is orthogonal to the approach taken by (Hybrid) Markov Logic Networks
(MLNs) and its variations [
Bayesian logic (BLOG) [
An approach for embedding logical knowledge onto data for the purpose of
relational learning, similar to Real Logic, is presented in [
As in [
Finally, related work in the domain of neural-symbolic computing and neural
network fibring [
We have proposed Real Logic: a uniform framework for learning and reasoning. Approximate satisfiability is defined as a learning task with both knowledge and data being mapped onto real-valued vectors. With an inference-as-learning approach, relational knowledge constraints and state-of-the-art data-driven approaches can be integrated. We showed how real logic can be implemented in deep tensor networks, which we call Logic Tensor Networks (LTNs), and applied efficiently to knowledge completion and data prediction tasks. As future work, we will make the implementation of LTN available in TENSORFLOWTM and apply it to large-scale experiments and relational learning benchmarks for comparison with statistical relational learning, neural-symbolic computing, and (probabilistic) inductive logic programming approaches.