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# How to define the operation of division apart from the inverse of multiplication?

Sorry if this question is too far out there, but I'm looking for a rigorous definition of the division operation. As I have seen it before, $a/b$ is the solution to the equation $a=xb$. While I am okay with this sort of defintion, it is tantamount to saying that division is the inverse operation of multiplication. (at least that's how I see it, comments on this assertion are welcome.)

Instead, what I seek is a different sort of formalization of the definition of division of real numbers. Whereas the definition above is nearly equivalent to defining division as the inverse operation of multiplication (again, maybe i'm wrong), I seek a definition that does not rely on the definition of multiplication already being asserted.

Motivation: Consider the fundamental theorem of calculus that essentially states that the indefinite integral is the inverse of the derivative. This is a highly non-trivial theorem, and while one may be able to define the indefinite integral as the inverse of the derivative (could be wrong here as well), it is much more meaningful to define the two operations separately and then show that they're actually just inverses of one another. Does anything like this exist for multiplication and division? Or are these operations just too fundamental?

Question: Does there exist a definition for $a/b$ where $a,b \in \mathbb{R}$ that does not require multiplication to already be defined?

I know, I'm grasping at straws, but maybe there is something here that really enhances my understanding of what division is in general. Or maybe there are some interesting definitions involving limits or something that I have never seen before.

• The first question is, how did you define multiplication? (Good question BTW) – Daniel Aug 5 '15 at 15:09
• I would define it as if it were just notation for repeated addition, but i would only know how to define it for integers, and wouldn't know how to extend it to real numbers in general. – Paddling Ghost Aug 5 '15 at 15:11
• @PaddlingGhost: Your point about being able to use repeated addition to justify multiplication for integers but not real numbers is indeed a crucial point. One either goes through conceptual division of repetition (meaning that if I have a procedure $P$, is there a procedure $Q$ such that repeating $Q$ $2$ times is the same as performing $P$?) to get to rationals and then treat real numbers as everything that can be approximated by rationals, or one goes through geometric intuition as 'experienced' in the real world (as I chose in my answer). – user21820 Aug 5 '15 at 16:04
• Saying "I'm looking for a rigorous definition of the division operation" is misleading, because that is not what you are looking for - you are interested in a definition that does not make reference to multiplication. No, I do not believe it is natural at all to detach division from multiplication. – anon Aug 6 '15 at 23:58

One approach is to define a new notion: a Number is a pair of real numbers, $(a, b)$, with $b \ne 0$, and $(a, b)$ is "the same as" $(c, d)$ if $ad = bc$. (The idea here is that the Number $(a, b)$ represents the thing you want to call $a/b$, by the way).

We can add Numbers: $$(a, b) + (p, q) = (aq + pb, bq)$$

and multiply: $$(a, b) * (p, q) = (ap, bq)$$

and it turns out that these operations are independent of representations (i.e., if two Numbers are "the same", then when added to another Number, the results will be "the same"). You have to prove that of course.

Once you've done all this -- shown that the set of equivalence classes of Numbers is the same as the set of reals with addition and multiplication under the correspondence $r \mapsto (r, 1)$ -- you're ready to define division:

$$(a, b) / (c, d) = (ad, bc).$$

And then you can prove that division has all the properties that you want.

This is really the idea that @Travis is suggesting in his comment, but skipping the visit to $\mathbb Q$.

BTW, Spivak's Caculus, in its last chapter, has a lovely problem about the "high school student's real numbers" in which one defines a real to be a sequence of digits, defines addition and multiplication carefully, and then proves that these form a complete ordered field. In that problem, "division" is essentially the long-division algorithm, and proving that it does the opposite of long-multiplication is a major pain in the neck. Perhaps this is, in some way, what you're seeking.

Post-comment remark

One more thing -- it's worth looking at Edwin Moise's "Elementary Geometry from an Advanced Standpoint", in which one of the later chapters describes the work of Eudoxus (I think!) on defining real numbers via (geometric) ratios. It's quite remarkable work, and hints at the idea of Dedekind cuts that were developed only after a couple of millenia.

Further post-comment remark

Here's the text of problem 29-2 from Michael Spivak's Calculus.

"This problem outlines a construction of "the high-school student's real numbers." We define a real number to be a pair $(a, \{b_n\})$ where $a$ is an integer and $\{b_n\}$ is a sequence of natural numbers from 0 to 9,. with the proviso that the sequence is not eventually $9$; intuitively, this pair represents $$a + \sum_{n = 1}^\infty b_n 10^{-n}.$$ With this definition, a real number is a very concrete object, but the difficulties involved in defining additional and multiplication are formidable (how do you add infinite decimals without worrying about carrying digits infinitely far out?). A reasonable approach is outlined below; the trick is to use least upper bounds right from the start.

(a) Defined $(a, \{b_n\}) << (c, \{d_n\})$ if $a < c$ or if $a = c$ and for some $n$ we have $b_n < d_n$, but $b_j = d_j$ for $1 \le j < n$. Using this definition, prove the least upper bound property.

(b) Given $\alpha = (a, \{b_n\})$, define $\alpha_k = a + \sum_{n = 1}^k b_n 10^{-n}$; intuitively, $\alpha_k$ is the rational number obtained by changing all decimal places after the $k$th to $0$. Conversely, given a rational number $r$ of the form $a + \sum_{n = 1}^k b_n 10^{-n}$, let $r'$ denote the real number $(a, \{b'_n\})$, where $b'_n = b_n$ for $1 \le n \le k$, and $b'_n = 0$ for $n > k.$

Now for $\alpha = (a, \{b_n\})$ and $\beta = (c, \{d_n\})$ define \begin{align} a ++ b &= \sup\{ (\alpha_k + \beta_k)': k \text{ a natural number} \} \end{align}

(the least upper bound exists by part (a)). If multiplication is defined similarly, then the verification of all conditions for a fields is a straightforward task, not highly recommended. Once more, however, existence of multiplicative inverses will be the hardest."

Note: Spivak uses a boldface "<" where I've used $<<$, and a bold plus sign where I've used $++$, but I cannot make those appear here.

So that doesn't really do what you've asked, but I think it gets to the main point: you can do all this stuff more or less algorithmically, at least if you're willing to invoke upper-bound ideas (which is somewhat the trick Eudoxus uses as well, amazingly enough). But mostly it's a pain in the neck. :)

• Yes, that last paragraph, i think, summarizes exactly what i'm looking for. – Paddling Ghost Aug 5 '15 at 15:15
• @PaddlingGhost: I don't agree that division should ever be thought of as long-division. Otherwise it makes no sense to ask what $\frac{1}{2+3i}$ is, or the so-called rational function $\frac{x^2+1}{x^3-4}$ where $x$ is an indeterminate. – user21820 Aug 5 '15 at 16:09
• Since PaddingGhost was explicitly asking about division for the real number system, I think that long division, properly described, is one pretty reasonable definition. Once you've gotten past that, and seen that the algorithmic and "algebraic" definitions match, you can use the algebraic definition for a while, and then choose it as the thing to generalize. – John Hughes Aug 5 '15 at 17:49
• @John Hughes if someone were to clearly define a division algorithm that will always terminate then I would certainly consider this a satisfactory answer. Even further, if one could show me how this division algorithm ends up being multiplication's inverse, I would have no choice but to mark it as the accepted answer. – Paddling Ghost Aug 5 '15 at 18:33
• @PaddlingGhost: As I said in my comments, the division algorithm is based on inverting multiplication. At each step you choose some amount that when multiplied is less than the remainder. – user21820 Aug 6 '15 at 0:22

I guess your problem is actually that you don't understand what is so good about having inverses. That is the key. We always want to know whether we can undo something. Multiplication can be thought of as a scaling (repeated addition only works for integers, but the idea of scaling works for even complex numbers), and the natural question is when it can be undone. For amounts in the real world, it seems we can sort of scale by any real value, for example by moving closer to an object it becomes larger in our visual field. We can even scale by negative real values using say a lens. We can even scale by zero, such as focusing the sun onto a single point (a point is an ideal concept; of course there will be imperfections in the lens and our setup). We intuitively see that scaling by any nonzero amount can be undone, in the sense that we can figure out the original location before scaling. However, after a scaling by zero, all points are now at the centre of scaling and we cannot tell where they came from. Division is just the name we give to the inverse of multiplication, which is the name we might give to scaling. Namely when we scale a real position $p$ by a real amount $r$, we define the resulting position to be the result of multiplying the two reals as $p \times r$. Similarly to undo a scaling so that from the resulting position of $p \times r$ we get $p$ we define the original position as the result of dividing that by $r$, namely the original position is $p = ( p \times r ) \div r$. As explained, this holds for any real values $p,r$ such that $r \ne 0$.

Same goes for subtraction as the inverse of addition. We can consider addition of a real value to correspond to translation (shifting without changing shape) by that amount in a specific direction, and subtraction of that same real value to correspond to undoing the translation, which is equivalent to translating by the same amount in the opposite direction. This time every translation can be undone, and similarly every addition can be undone. In mathematical symbols $p = ( p + a ) - a$ for any real values $p,a$. This translation concept extends automatically and naturally to complex numbers.

To be precise for complex numbers, addition of a complex number corresponds to translation of the 2d plane in some direction in the same plane. The order of adding does not matter because the order of performing translations does not. Multiplication of a complex number corresponds to a scaling and rotation about the centre $0$. (It is beyond the scope of this answer to explain the rotation part.) Again, order of multiplication does not matter because scalings and rotations can be swapped without changing the result. However, the order of additions and multiplications cannot be swapped. This is also the case for real numbers where you are basically translating or scaling a 1d line along itself.

Now the next part of the question is concerning integrals. There are different types of integrals. The one you are referring to is probably the Riemann integral, which is the natural outcome of trying to define area under a curve. Why do we want that? Partly historical curiosity and Newtonian mechanics, at least. The latter is because we intuitively know that distance is speed times time taken for constant speed. Extending it to non-constant speed results in approximation concepts that lead to the Riemann sum. Of course in the real world everything converges so we do not need to care whether velocity is integrable; of course it is! But actually in hindsight from Einstein's theory of relativity it is better to start with position and define velocity as simply the rate of change of position, similarly force as rate of change of momentum, because the conventional formulae in the reverse direction fail spectacularly. It turns out that the Riemann integral is an anti-derivative under certain conditions, but there are functions whose derivative is not integrable! (see Volterra's function)

• @PaddlingGhost: You will also notice that the other answer is slightly circular because there is a priori no reason to consider that notion he defined! Why on earth should we treat $(a,b)$ the same as $(c,d)$ if $ad=bc$ without already having an understanding of ratio? What is a ratio? – user21820 Aug 5 '15 at 16:13
• @PaddlingGhost: The real-world geometric intuition tells us that when we scale something, shape is preserved. What does preserving shape mean? If we think carefully about it we can condense it down to ratios of lengths being preserved. What is length? Length is what is invariant under translation and rotation. We make rulers because we know that no matter where we bring our ruler, its markings remain at the same distance from each other, so that we can measure distances of other objects. Scaling preserves ratios of distances. That in a sense defines ratios. – user21820 Aug 5 '15 at 16:18
• So then you answer my question in they negative by saying no, division is simply the inverse of multiplication and here's why. So, this does answer the question, but it is unlikely to earn the green check mark as someone has already answered in the affirmative and given a suitable definition that does not involve inverses. – Paddling Ghost Aug 5 '15 at 16:31
• Just read your comments and they were helpful. I too find problems with the other answer . – Paddling Ghost Aug 5 '15 at 16:33
• @PaddlingGhost: You are simply wrong. The other answer used multiplication. What else does $ad = bc$ mean if not $a \times d = b \times c$?? – user21820 Aug 5 '15 at 16:46


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# Geodesic of a curved surface

I'm trying to read Lambourne's Relativity, Gravitation and Cosmology, but as this seems more of a maths question I've posted it here rather than in the physics forum.

The author talks about affinely parameterized geodesics and then, in Exercise 3.9, shows that the equator on the surface of a sphere is a geodesic.

My question is this:

How do you find the geodesic of a simple curved surface that isn't a sphere (for example, $z = x^2 + y^2$)?

I'm guessing that because the geodesic equation contains Christoffel symbols I would first need to find the metric for that particular surface, but I don't know how to do that.

I'm therefore guessing that my second question would need to be how do you find the metric for a simple curved surface (again for example $z = x^2 + y^2$).

Apologies if I've asked anything ridiculous.

Thank you.

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Do you understand the spherical case? –  M.B. Jul 23 '11 at 21:29
Also, understand that you can very rarely get an explicit formula for geodesics on a surface. I suppose you could solve the differential equations numerically to get a good approximation though. –  Vhailor Jul 23 '11 at 21:33
you solve differential equations. the simplest case would be finding the geodesics of the plane (and hope you get straight lines!) then the sphere and hyperbolic plane if you want to understand how it is done (in some cases where you can get nice solutions). you can try solving them using variational techniques (solving the euler-lagrange equations for the appropriate functional) or solving the geodesic equation $\nabla_{\dot{\gamma}}\dot{\gamma}=0$ –  yoyo Jul 24 '11 at 14:09
Thanks. I think I can follow the explanation of the plane and spherical cases (exercises 3.8 and 3.9 in Lambourne - very helpful as these are worked examples). However, I confess I can't quite see why a geodesic curve is defined as one that parallel transports its own tangent vector. –  Peter4075 Jul 24 '11 at 19:06

As others have mentioned, it is usually not easy to find the geodesics of a given surface, so let me address some of your other questions.

Let's start with a simple case. If a surface $S$ in $\mathbb{R}^3$ is the graph of a function $z = f(x,y)$, then the metric $g$ can be represented in coordinates by the matrix $$g = \begin{pmatrix} 1 + (f_x)^2 & f_xf_y \\ f_xf_y & 1 + (f_y)^2 \end{pmatrix},$$ where subscripts denote partial derivatives.

In the case of $z = x^2 + y^2$, for example, we have $$g = \begin{pmatrix} 1 + 4x^2 & 4xy \\ 4xy & 1 + 4y^2 \end{pmatrix}.$$

More generally, if a surface $S$ in $\mathbb{R}^3$ is given by a local parametrization $\phi(x,y) = (\phi^1(x,y), \phi^2(x,y), \phi^3(x,y))$, then the metric $g$ can be represented by the matrix $$g = \begin{pmatrix} \langle \phi_x, \phi_x\rangle & \langle\phi_x, \phi_y\rangle \\ \langle\phi_x, \phi_x\rangle & \langle\phi_y, \phi_y\rangle \end{pmatrix},$$ where I am using the notation $\langle v, w\rangle = v \cdot w$ to denote the inner product (or "dot product") in $\mathbb{R}^3$. For example, $$\langle \phi_x, \phi_y\rangle = \langle (\phi^1_x, \phi^2_x, \phi^3_x), (\phi^1_y, \phi^2_y, \phi^3_y)\rangle = \frac{\partial\phi^1}{\partial x}\frac{\partial\phi^1}{\partial y} + \frac{\partial\phi^2}{\partial x}\frac{\partial\phi^2}{\partial y} + \frac{\partial\phi^3}{\partial x}\frac{\partial\phi^3}{\partial y}.$$ The case above where the surface is given as the graph of the function $z = f(x,y)$ is the special case where $\phi(x,y) = (x,y, f(x,y)).$

Note that the metric (tensor) was classically called the first fundamental form.

Now, I personally like to think of geodesics as unit speed curves that have zero geodesic curvature (rather than as curves which satisfy the geodesic equation).

To say that a curve $\alpha(t) = (x(t), y(t), z(t))$ is unit speed means that $|\alpha'(t)| \equiv 1$. (It is a fact that all geodesics have constant speed.) To say that a curve $\alpha$ has zero geodesic curvature means, well, that the geodesic curvature $\kappa_g \equiv 0$, where $$\kappa_g = \langle N \times \alpha', \alpha'' \rangle,$$ where $N$ is the unit normal vector to the surface. A nice apparatus for thinking about this definition is the Darboux frame.

Anyway, here are three more geometric facts that might help you in your search for geodesics, or at least your intuition for them:

Fact 1 (Corollary to Meusnier's Theorem): Let $\Pi$ be a plane that intersects a surface $S$. If at each point of intersection the plane $\Pi$ is perpendicular to the tangent plane to $S$, then the curve of intersection is a geodesic. (Such a curve is called a normal section.)

Fact 2: On a surface of revolution, every meridian is a geodesic.

Fact 3 (Clairaut's Theorem): Let $S$ be a surface of revolution, let $\alpha$ be a curve on $S$ with unit speed, let $\rho\colon S \to \mathbb{R}$ be the distance of a point of $S$ to the axis of rotation, and let $\psi$ be the angle between $\alpha'$ and the meridians of $S$.

Conclusion: If $\alpha$ is a geodesic, then $\rho \sin \psi$ is constant along $\alpha$. Conversely, if $\rho \sin\psi$ is constant along $\alpha$, and if no part of $\alpha$ is part of some parallel of $S$, then $\alpha$ is a geodesic. (cf. Clairaut's Relation.)

Remarks: I usually visualize Fact 1 by considering a right circular cylinder (try this!). Fact 2 should give you some of the geodesics on the elliptic paraboloid $z = x^2 + y^2$. Fact 3 can be used to determine the geodesics on the pseudosphere.

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Jesse - thank you. There's a huge amount to take in there, but I now have the metric of my function, plus a general formula for the metric. I clicked on your link to "first fundemental form" and that looked interesting as well. –  Peter4075 Jul 29 '11 at 10:57

First of all, as Vhailor said, it is difficult to find an explicit formula for geodesics on a surface in general. This is related to the fact that by definition geodesics are solutions to a differential equation which can be quite ugly depending on the shape of the surface. In this case you can follow Vhailor's advice and solve the equation numerically using a computer.

I will not give you a complete answer for which I would need several pages, but I will try to give you some ideas. Consider also the notes by Misha Rudnev which seem to solve your question.

Concerning your second question, you are right. You need the Christoffel symbols of your surface which depends on the metric. In general, if you have an embedding of a surface into $\mathbb{R}^n$, your metric is the restriction of the Euclidean metric of $\mathbb{R}^n$. In the case of the cone $z=x^2+y^2$, such an embedding is for example given by $$(\theta,z)\in[0,2\pi[\times\mathbb{R}^+\stackrel{\phi}{\longrightarrow}(\sqrt{z}\cdot \sin(\theta),\sqrt{z}\cdot \cos(\theta),z)\in\mathbb{R}^n$$ Now you can deduce the metric of the cone by the general formula $$g_{c}(X,Y)=g_0\big(\phi_{*}(X),\phi_{*}(Y)\big)\;\;\;\;\;\;\;(1)$$ where $\phi_*$ denotes the differential of $\phi$, $g_c$ the metric of the cone, $g_0$ the Euclidean metric, (i.e. $g_0(e_i,e_j)=\delta_{i,j}$) and $X$ and $Y$ are tangent vectors to $[0,2\pi[\times\mathbb{R}^+$. Try to do the explicit computation as an exercise. ($\it{Hint}$: If you write a vectors $X$ in coordinates, $\phi_*(X)$ is just the Jacobian matrix of $\phi$ applied to $X=(X_1,X_2,X_3)^t$).

Now the Christoffel symbols are given in general by $$\Gamma ^k_{ij}=\frac{1}{2}g^{kl}\big(-\partial_lg_{ij}+\partial_ig_{jl}+\partial_jg_{il}\big)$$

You see that this becomes a lot of computation if you want to solve the differential equation for geodesics afterwards, even for an "easy" shape. However, sometimes you can use tricks to find geodesics explicitely without solving any equation. For example a geodesic $\gamma:[a,b]\rightarrow M$ is a curve whose length is less then the lenght of every other curve that joins $\gamma(a)$ and $\gamma(b)$. If you haven't done it already, a good exercise is to try to apply this to the sphere in order to find geodesics. (Hint: The image of a minimal geodesic joining two points $x$ and $y$ on the sphere by the orthogonal reflection relatively to a plane cutting the sphère into two equal hemispheres and passing by $x$ and $y$ is a geodesic. More general, the image of a geodesic by a (local) isometry is a geodesic).

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Julien - thank you, but I'm afraid this is way above my head. Eg, what does "restriction of the Euclidean metric" mean, and why is z = x^2 + y^2 a cone? I thought it was some sort of 3d parabola (paraboloid?). I'm happy to accept that the answer to the geodesic bit of my question is beyond me but is there a simpler explanation of how to find the metric of a curved surface? –  Peter4075 Jul 24 '11 at 19:40
Of course you are right! In my enthusiasm I mixed up the equations of the cone and the paraboloid:/ Nevertheless, the approach remains the same. –  Julien Meyer Jul 24 '11 at 21:53
"restriction of the Euclidean metric" is exactly what states formula (1). $\phi_*(X)$ and $\phi_*(Y)$ are tangent vectors to your surface and you use the Euclidean metric $g_0$ to multiply them. One can rephrase all this using differential forms. Do you know a bit about them? –  Julien Meyer Jul 24 '11 at 22:03
Julien - you're talking to someone who thinks 'Jacobian' relates to 17th century Scottish politics! Can we do this in baby steps? If I take partial derivatives of my precious surface z = x^2 + y^2 (which I can do), is there then some way to find the metric of the surface by using the Euclidean metric? I could then (theoretically at least) find the Christoffel symbols for the surface. Do I need to parametize the surface first before taking partial derivatives? Am I on the right track? Thanks for your patience. –  Peter4075 Jul 25 '11 at 14:09
How about this: for $z=F(x,y)$, a curve is given by
$$X(t)= ( x(t), y(t), F(x(t),y(t)) )$$ The surface normal is $$N = \left( \frac{dF}{dx}, \frac{dF}{dy}, -1 \right)$$
$$|X'|^2=1$$ Want curve acceleration parallel to the normal $( \frac{dF}{dx}, \frac{dF}{dy}, -1 )$, so
$$\operatorname{cross}( X'', N )=0$$ That get's messy quickly - four non-linear PDE's. But it is fun to play with eg $z=x^2+y^2$

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### Conservative Extensions in Horn Description Logics with Inverse Roles

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF . The same results hold for inseparability and entailment.

### Knowledge Graphs @ ICLR 2020

It's great to see more research and more datasets on complex QA and reasoning tasks. Whereas last year we saw a surge of multi-hop reading comprehension datasets (e.g., HotpotQA), this year at ICLR there is a strong line-up of papers dedicated to studying compositionality and logical complexity: and here KGs are of big help! Keysers et al study how to measure compositional generalization of QA models, i.e., when train and test splits operate on the same set of entities (broadly, logical atoms), but the composition of such atoms is different. The authors design a new large KGQA dataset CFQ (Compositional Freebase Questions) comprised of about 240K questions of 35K SPARQL query patterns. Several fascinating points 1) the questions are annotated with EL Description Logic (yes, those were the times around 2005 when DL meant mostly Description Logic, not Deep Learning); 2) as the dataset is positioned towards semantic parsing, all questions already have linked Freebase IDs (URIs), so you don't need to plug in your favourite Entity Linking system (like ElasticSearch).

### Detecting fake news for the new coronavirus by reasoning on the Covid-19 ontology

In the context of the Covid-19 pandemic, many were quick to spread deceptive information. I investigate here how reasoning in Description Logics (DLs) can detect inconsistencies between trusted medical sources and not trusted ones. The not-trusted information comes in natural language (e.g. "Covid-19 affects only the elderly"). To automatically convert into DLs, I used the FRED converter. Reasoning in Description Logics is then performed with the Racer tool.

### CQE in Description Logics Through Instance Indistinguishability (extended version)

We study privacy-preserving query answering in Description Logics (DLs). Specifically, we consider the approach of controlled query evaluation (CQE) based on the notion of instance indistinguishability. We derive data complexity results for query answering over DL-Lite$_{\mathcal{R}}$ ontologies, through a comparison with an alternative, existing confidentiality-preserving approach to CQE. Finally, we identify a semantically well-founded notion of approximated query answering for CQE, and prove that, for DL-Lite$_{\mathcal{R}}$ ontologies, this form of CQE is tractable with respect to data complexity and is first-order rewritable, i.e., it is always reducible to the evaluation of a first-order query over the data instance.

### Reasoning about Typicality and Probabilities in Preferential Description Logics

In this work we describe preferential Description Logics of typicality, a nonmonotonic extension of standard Description Logics by means of a typicality operator T allowing to extend a knowledge base with inclusions of the form T(C) D, whose intuitive meaning is that "normally/typically Cs are also Ds". This extension is based on a minimal model semantics corresponding to a notion of rational closure, built upon preferential models. We recall the basic concepts underlying preferential Description Logics. We also present two extensions of the preferential semantics: on the one hand, we consider probabilistic extensions, based on a distributed semantics that is suitable for tackling the problem of commonsense concept combination, on the other hand, we consider other strengthening of the rational closure semantics and construction to avoid the so called "blocking of property inheritance problem".

### Large-scale Ontological Reasoning via Datalog

Reasoning over OWL 2 is a very expensive task in general, and therefore the W3C identified tractable profiles exhibiting good computational properties. Ontological reasoning for many fragments of OWL 2 can be reduced to the evaluation of Datalog queries.

### Axiom Pinpointing

Axiom pinpointing refers to the task of finding the specific axioms in an ontology which are responsible for a consequence to follow. This task has been studied, under different names, in many research areas, leading to a reformulation and reinvention of techniques. In this work, we present a general overview to axiom pinpointing, providing the basic notions, different approaches for solving it, and some variations and applications which have been considered in the literature. This should serve as a starting point for researchers interested in related problems, with an ample bibliography for delving deeper into the details.

### When is Ontology-Mediated Querying Efficient?

In ontology-mediated querying, description logic (DL) ontologies are used to enrich incomplete data with domain knowledge which results in more complete answers to queries. However, the evaluation of ontology-mediated queries (OMQs) over relational databases is computationally hard. This raises the question when OMQ evaluation is efficient, in the sense of being tractable in combined complexity or fixed-parameter tractable. We study this question for a range of ontology-mediated query languages based on several important and widely-used DLs, using unions of conjunctive queries as the actual queries. For the DL ELHI extended with the bottom concept, we provide a characterization of the classes of OMQs that are fixed-parameter tractable. For its fragment EL extended with domain and range restrictions and the bottom concept (which restricts the use of inverse roles), we provide a characterization of the classes of OMQs that are tractable in combined complexity. Both results are in terms of equivalence to OMQs of bounded tree width and rest on a reasonable assumption from parameterized complexity theory. They are similar in spirit to Grohe's seminal characterization of the tractable classes of conjunctive queries over relational databases. We further study the complexity of the meta problem of deciding whether a given OMQ is equivalent to an OMQ of bounded tree width, providing several completeness results that range from NP to 2ExpTime, depending on the DL used. We also consider the DL-Lite family of DLs, including members that admit functional roles.

### Querying and Repairing Inconsistent Prioritized Knowledge Bases: Complexity Analysis and Links with Abstract Argumentation

In this paper, we explore the issue of inconsistency handling over prioritized knowledge bases (KBs), which consist of an ontology, a set of facts, and a priority relation between conflicting facts. In the database setting, a closely related scenario has been studied and led to the definition of three different notions of optimal repairs (global, Pareto, and completion) of a prioritized inconsistent database. After transferring the notions of globally-, Pareto- and completion-optimal repairs to our setting, we study the data complexity of the core reasoning tasks: query entailment under inconsistency-tolerant semantics based upon optimal repairs, existence of a unique optimal repair, and enumeration of all optimal repairs. Our results provide a nearly complete picture of the data complexity of these tasks for ontologies formulated in common DL-Lite dialects. The second contribution of our work is to clarify the relationship between optimal repairs and different notions of extensions for (set-based) argumentation frameworks. Among our results, we show that Pareto-optimal repairs correspond precisely to stable extensions (and often also to preferred extensions), and we propose a novel semantics for prioritized KBs which is inspired by grounded extensions and enjoys favourable computational properties. Our study also yields some results of independent interest concerning preference-based argumentation frameworks.

### An Overview of Distance and Similarity Functions for Structured Data

The notions of distance and similarity play a key role in many machine learning approaches, and artificial intelligence (AI) in general, since they can serve as an organizing principle by which individuals classify objects, form concepts and make generalizations. While distance functions for propositional representations have been thoroughly studied, work on distance functions for structured representations, such as graphs, frames or logical clauses, has been carried out in different communities and is much less understood. Specifically, a significant amount of work that requires the use of a distance or similarity function for structured representations of data usually employs ad-hoc functions for specific applications. Therefore, the goal of this paper is to provide an overview of this work to identify connections between the work carried out in different areas and point out directions for future work.


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# Recessional velocity

Recessional velocity is the rate at which an astronomical object is moving away, typically from Earth. It can be measured by shifts in spectral lines or estimated by general reddening of a galactic spectra.

Galaxies outside our galaxy often have their recessional velocity calculated relative to the cosmic microwave background.[1]

## Application to cosmology

Recessional velocity is most pertinent to distant galaxies, which (due to Hubble's Law) redshift proportionally to their distance from the Earth. The redshift is usually interpreted as due to recessional velocity, which can be calculated according to the formula:

${\displaystyle v=H_{0}D\ }$

where ${\displaystyle H_{0}}$ is the Hubble constant, ${\displaystyle D}$ is the intervening distance in Mpc, and ${\displaystyle v}$ is the recessional velocity, generally measured in km/s.

The recessional velocity of a galaxy is usually calculated from the redshift observed in its emitted electromagnetic radiation. The distance to the galaxy is then estimated using Hubble's Law.