To understand applied physics, and physical formulae in particular, the investigation of identifier units is beneficial. However, normally the units are not given explicitly in formulae and have to be inferred. In this paper, we investigate how this process can be automated. As an example application, we use physical formulae from Wikipedia together with information from the related knowledge-base Wikidata. We envision that this method can be generalized and describe how, in the future, hard logical constraints may be used as a feedback mechanism for statistical methods in the context of natural language processing.
Units play an essential role in the physical sciences. Especially, dimensional analysis is one of the most significant tools for comprehension and understanding of physical formulae. We claim that this technique is not only beneficial for humans on their way to physical understanding, but also to machines that are programmed to semantically enrich mathematical and especially, physical content.
In this paper, we consider physical units in Wikipedia, as a first step towards a general solution of the underlying problem. We aim to: (1) identify formulae that deal with physical relationships; (2) automatically derive the units of the identifiers used in those formulae; and (3) integrate and store the learned data in the central Wikimedia triple store Wikidata.
Our paper is structured as follows. First, we analyze related works that can
be used to complete the task at hand. In that context, we recap how possible
definitions can automatically be extracted from the text surrounding formulae.
Thereafter, we describe our method to relate the identifier to dimensions using
the Wikidata knowledge base and our approach to unit constraint propagation.
This will be followed by a refinement of the definition candidates based on the
constraints. Lastly, we describe the reinsertion of learned units into the Wikidata
knowledge base, which will be used for the formulae we processed. Finally, we
provide an outlook on how this method can be used for feedback driven
selftuning of Mathematical Language Processing [
A method to find identifier definiens tuples in natural language text is proposed
in [
This method has been applied to Wikipedia articles to enrich formulae with definitions for their included identifiers. It works for Wikipedia sites with diferent languages. The result may have more than a single possible definition for an identifier, each with a probability that expresses the likeliness for the definition to be the relevant. The selection of the correct definition is a problem addressed in our paper.
A method to map the meaning to identifiers is used in [
In [
Dimensional analysis is also a widely investigated field in the area of
programming languages. In [
As preparation of this work, a new feature in Wikidata, the data-type
mathematical expression, has been developed [
In this paper, we limit ourselves to the following definition.
Definition 1 (physical formula). A physical formula is a binary mathematical relation of type equation or inequality containing one or more physical quantities.
E = mc2, sin2 θ + cos2 θ = 1, (1) (2) mmoon < mearth < msun, (3)
λp ℏ = . (4)
2π Expressions (1), (4) are physical formulae according to Definition 1. They contain the physical quantities E, m, c, λ, p, ℏ of which c, ℏ are physical constants. Expressions (2), (3) are not physical formulae since (2) does not contain physical quantities, and (3) is not binary. Note that the above definition can be extended to non-binary relation chains without loss of generality.
This definition implies the following algorithm to identify physical formulae:
While we will apply the heuristics from [
Our main focus is on the development of an algorithm which decides if an
item is a physical quantity or entity. Our approach to address this problem is to
analyze the semantic properties of the relevant Wikidata item using the SPARQL [
The dimension of a physical quantity is an inherent property of each quantity. Dimensions are for example length L, mass M or time T . The derived quantity 3 SPARQL is used to query RDF (Ressource Description Framework) triples. They consist of subject, predicate, object. Example: Find all subjects (items) in Wikidata that have predicate "subclass of" and object "physical quantity" 4 Wikidata stores information in form of triples like ("length","subclass of","physical quantity") where the unique id of "physical quantity" in Wikidata is Q107715. If a Item like "length" has a relation like "subclass of" or "instance of" to the Item for "physical quantity" we suppose it to be of kind physical quantity. speed has the dimension LT − 1. For all physical quantities, we try to derive their dimension from Wikidata using SPARQL queries.
To obtain that, we also take unit information into account, since it is more prevalent in the Wikidata dataset compared to dimension information. Because there are multiple unit systems (e.g., imperial units using yard for length versus SI units using meter) more than one unit per dimension exists. However, physical laws are usually valid independent from the unit system that is used. In this context, it has to be noted that some adjustment needs to be done for units that disregard conceptually important physical properties. For instance the Carnot eficiency η Carnot depends on the absolute temperature scale (e.g., kelvin) of the hot TH and the cold TC reservoir via Note that the fact that those temperatures are based on an absolute temperature scale is essential. This requires that non-cardinal temperature units such as Celsius and Fahrenheit to be converted to prior to computation.
Given the extracted dimensions, the mathematical domain of physical
quantities can be specified better. Approaches to formally describe this domain are
presented in [
TC
TH
where x is the spatial part of x as defined in [
(6) 3.3
Definition 2 (valid physical formula). We call a physical formula valid, if the dimensions are compatible with the mathematical operators used in that formula. |.|1 denotes the 1-norm. class 3: map 2: restrict o 1: apply 0: unitless o o [( xa] , [ yb]) [( xa] , [ yb]) [( xa] , [ yb]) o formula contains the mathematical operators equals (=), times (· ), and power (∧). For ‘equals’, the dimensions on the right-hand side and left-hand side must be the same, for ‘times’, no restrictions apply and for ‘power’ the unit of the exponent must be 1. If any of these constraints are violated (e.g., with the incorrect E = mc) then the units or the formula can not be correct. Note also that this method is not limited to scalar physical quantities and can be applied to physical quantities of higher dimensions such as vectors like more than one possible dimension for an identifier.
The relation between energy and mass is described by the mass-energy equivalence formula E = mc2, where E is energy, m is mass, and c is the speed of light.
For Example 1 (from Wikipedia), which was also used in [
Id. Definition score Id. Definition score Id. Definition score E energy 0.42 m energy 0.35 c energy 0.30 E mass 0.42 m mass 0.35 c mass 0.35 E speed of light 0.16 m speed of light 0.30 c speed of light 0.35 All three definitions are found in the same sentence, but have diferent distances between identifier and definiens. That results in diferent scores. Note, that the actual scoring computation is more evolved and the score values have been made up for demonstration purposes. We now describe this more formally.
For a physical formula f (x1, . . . , xn, on+1, . . . , om), with the identifiers (physical quantities and other identifiers) xi and mathematical operators oj , and a set of definiens candidates N , the MLP Project returns a probability distribution for the identifiers χ (x) = ∑N∈N w0,nN , with ∑N∈N w0,N = 1. From Wikidata we get the unit and implied dimension information denoted as ρ 1(x) = ∑N∈N w0,n dim(N ) = ∑i w1,idi, where w1,j is the sum over all w0,N with dim N = dj . Next, we apply the operator compatibility rules, which afects the probability distribution according to their constraints. The dimension probability distribution of an operator o, dim (o(a, b)) implies operator dependent constraints to dim a and dim b.
class(o) = 3 =⇒ dim o(a, b) = o(dim a, dim b). class(o) = 2 =⇒ dim a = dim b. class(o) = 1 =⇒ dim b = 1.
class(o) = 0 =⇒ dim a = 1, o(a, b) = o(a).
To reflect that, we define a refined probability distribution ρ (x) = ∑i w2,idi, with ∑i w2,idi = 1. Finally, we need to solve a system of linear equations describing the w2 = Aw1. We propose the following method to realize that. Assume that the mathematical notation is good enough to extract the operator tree, with tools such as LATExml. We convert this operator to the root form and call O(f ) ∈ Nm× m the adjacent matrix. Since the tree is not directed O(f ) = O(f )T , and since identifiers are always connected by an operator, the top left entries of the adjacent matrix are zero, i.e., ∀i ≤ n ∧ j ≤ n =⇒ O(f )i,j = 0. Based on that, we will elaborate diferent strategies for the most eficient execution of the constraint propagation problem. Our first approach is the following sketch of an algorithm: 1. start with the identifiers ( x) as working set; ⎡ E ⎤ ⎢ .42 ML2T− 2 ⎥ ⎢⎣+.42 M ⎥⎦ +.16 LT− 1
= ⎡ m ⎤ ⎢ .35 ML2T− 2 ⎥ ⎢⎣+.35 M ⎥⎦ ⎡ +.30 LT− 1 ·
∧ c ⎤ 2 ⎢ .30 ML2T− 2 ⎥ ⎢⎣+.35 M ⎥⎦ +.35 LT− 1 ⎡ E ⎤ ⎢ .42 ML2T− 2 ⎥ ⎢⎣+.42 M ⎦⎥ +.16 LT− 1 ⎡ = ⎤ ⎢ ? ML2T− 2 ⎥ ⎢⎣+? M ⎥ +? LT− 1 ⎦
· ⎡ m ⎤ ⎢ .35 ML2T− 2 ⎥ ⎢⎣+.35 M ⎥ +.30 LT− 1 ⎦ ⎡ ⎤ ⎢ .30 M∧2L4T− 4 ⎥ ⎢⎣+.35 M2 ⎥
+.35 L2T− 2 ⎦ ⎡ c ⎤ 2 ⎢ .30 ML2T− 2 ⎥ ⎢⎣+.35 M ⎥
+.35 LT− 1 ⎦ ⎡
= ⎢ ? ML2T − 2 ⎥ ⎢⎣ +? M ⎥
⎦ +? LT − 1
⎤ [
] · ML2T − 2 [ c
] ∧
L2T − 2 ⎡ ⎢ .30 ML2T − 2 ⎥ ⎢⎣ +.35 M ⎥
⎦ +.35 LT − 1
⎤ 2 ⎡
E ⎢ .42 ML2T − 2 ⎥ ⎢⎣ +.42 M ⎥
⎦ +.16 LT − 1 ⎤ Fig. 2: Demonstration of our constraint propagation algorithm. 2. propagate the constraints to their parent operators of classes 0-2 and remember the parent operators of class 3 (maps); 3. update the working set to consist of the updated parents; 4. propagate the constraints downwards; 5. the changed children are now the new working set; 6. continue with upwards and downwards propagation until the working set is empty, and a steady state is reached; 7. update the working set with the class 3 operators; 8. resolve the class 3 operators, in appropriate order (note that this might lead to many possible dimensions for the class 3 operators); 9. propagate the new constraints by going back to step 2; 10. the algorithm terminates if the working set and the set of unprocessed class 3 operators is empty.
We demonstrate this algorithm in figure 2 based on Example 1. Formula (1) depends on (E, m, c, 2, =, · , ∧) and the adjacent matrix reads ⎜ ⎜E O(“E = mc2”) = O ⎜⎜ ⎜ ⎜ ⎝ ⎛ =
· m ∧ ⎟⎟
⎠ c 2 ⎞ E m c 2 = · ∧
E ⎛0 0 0 0 1 0 0⎞
⎟⎟ m 0 0 0 0 0 1 0
⎟⎟ = c ⎜0 0 0 0 0 0 1⎟
2 ⎜⎜0 0 0 0 0 0 1⎟⎟ .
= ⎜1 0 0 0 0 1 0⎟
· ⎝0 1 0 0 1 0 1⎠
∧ 0 0 1 1 0 1 0
One alternative to this approach is [
) sdt
∫ ρ 0(s)dt = ⏐⏐ ∫ ρ 0(s)dt⏐⏐ 1 = The likelihood for M T is higher in comparison to T 2, which does not seem plausible at the first place. Consequently, we will search for better suitable ways to re-scale the unit vector to length 1.
The finding of items in Wikidata can be improved by using methods like stemming to match the definition with the item caption. This is due to the fact that the definitions are extracted from free-text and may be conjugated. The identification of the units/dimensions can be done by the querying property, has quality. Due to the community-driven inserts of items, it is not guaranteed that every item has all of the possible properties. For example, the authors may omit a property such as, has quality. Instead the property, quantity symbol, may be queried.
Moreover, our algorithm can be used as a feedback mechanism for the MLP process. With the additional unit and dimension information, the algorithm will learn about mistakes from the unit checking. With this information, the algorithm will be able to tune itself. 5
The knowledge about units in physical formulae is fundamental. However, in most cases they are not marked up explicitly. We investigated how they can be determined automatically. We described a process to automatically infer the unit information for identifiers in physical formulae. This process is based on formulae and identifier-definition pairs that are extracted from text using Natural Language Processing techniques. The process may result in multiple definition candidates for identifiers. To infer a consistent identifier-definition mapping, we use an algorithm based on the principle of dimensional homogeneity in physical formulae. The unit/dimensional information for the definitions is retrieved from the structured data store Wikidata. The results are then used to extend Wikidata with the formulae and links Wikidata entries describing the identifiers. Acknowledgements. We would like to thank Volker Markl, Michael Kohlhase and Abdou Youssef for their support of this research project.