Three-Dimensional Affine Spatial Logics

T H R E E - D I M E N S I O NA L A FFI N E S P A T I A L L O G I C S A P R EP R I N T Adam T rybus Institute of Philosophy Jagiellonian Univ ersity 52 Grodzka St., Kraków 31-044, Poland adam.trybus@uj.edu.pl March 18, 2026 A B S T R AC T W e focus on a branch of region-based spatial logics dealing with af fine geometry . The research on this topic is scarce: only a handful of papers in vestigate such systems, mostly in the case of the real plane. Our long-term goal is to analyse certain family of af fine logics with inclusion and conv exity as primiti ves interpreted ov er real spaces of increasing dimensionality . In this article we sho w that logics of dif ferent dimensionalities must hav e dif ferent theories, thus justifying further work on diff erent dimensions. W e then focus on the three-dimensional case, exploring the e xpressiv eness of this logic and consequently showing that it is possible to construct formulas describing a three-dimensional coordinate frame. The final result, making use of the high e xpressiv e power of this logic, is that e very region satisfies an af fine complete formula, meaning that all regions satisfying it are af fine equiv alent. Keyw ords spatial logic · affine geometry · qualitativ e spatial reasoning 1 Introduction History and philosoph y Spatial logic, as the term has been used, can be vie wed as built using any first-order language with geometrical interpretation, where v ariables range over geometrical entities and relation and function symbols are interpreted as geometrical relations and functions. Ho wev er , in most instances, the name accompanies region-based, rather that point-based systems, meaning that the geometrical entities v ariables range over are not points but some collections of point. Although the name itself might be an in vention of the early twenty-first century (see Aiello et al. [2007]), spatial logics have rich and di verse background: after all, the notion of space is a staple in philosophy . 1 One of the most famous treatments of space was proposed by Kant, who in Critique of Pure Reason (Kant [1998]) ar gues that geometry is both synthetic and a priori (using his, now well-kno wn, labels). Since the argument seem to be hinging on the e xistence of only one type of geometry , the de velopment of ne w , non-Euclidean geometries in the nineteenth century was considered as a threat to Kantian views. Bertrand Russell, in one of his earliest publications, tries to defend Kant’ s approach (see Russell [1897]. Moreov er , the approach of man y nineteenth-century geometers was very philosophically informed. For example, Moritz P asch — most famous perhaps for figuring out the gaps in Euclid’ s reasoning — vie wed geometry as having a decidedly emprical basis (see e.g. Pasch [2010]). As a consquence, his analysis starts not with the Euclidean but rather with, what no w is known as, af fine geometry: one, where the notion of metric is not important. Thus, af fine geometry can be viewed as emphasising qualitative rather than quantitative aspects of geometrical thinking. This theme is also important for contemporary researchers working within the so-called qualitative spatial r easoning field. Logical formalisms that came from that field are sometimes called spatial logics, and those spatial logics that apply af fine notions are the focus of our article. Although there has been some interest in such type of logic (see e.g. Davis et al. [1999], Bennett and Cohn [1999] as well as Pratt [1999], T rybus [2016]), it should be said that it pales in comparison with the research on topological spatial systems (see e.g. Aiello et al. [2007] for a wide selection of topic related to topological formalisms). W e believ e that while there are good reasons for topological analysis, philosophical 1 See Casati and V arzi [1999] for an excellent introduction into the intersection of philosophical and formal approaches to spatial reasoning. Three-Dimensional Affine Spatial Logics A P R E P R I N T in vestigations pro vide additional justification for extending the work on affine systems. For e xample, Bertrand Russell, no doubt influenced by Pasch and others, 2 took up the idea of the importance of non-numerical, qualitati ve, geometry and argued extensi vely for the primacy of projecti ve and af fine notions ov er the Euclidean ones (see Russell [1897], Russell [1903] and T rybus [2021] for a discussion). Moreo ver , while Alfred N. Whitehead’ s complex philosophical ideas influenced the de velopment of region-based theories of space (see Whitehead [1929]), which are closely related to topology and mereology (a theory of part-whole relations, see Simons [2003]), he also de voted his attention to af fine and projectiv e geometry in Whitehead [1907] and Whitehead [1913] respeciv ely . Finally , affine geometry can be said to be an intermediate geometry between the projecti ve and Euclidean ones. Hence, it retains the status of non-numerical geometry and at the same time is less general than projecti ve geometry and topology , thus remaining closer to our ev ery-day experiences. The region-based af fine spatial logics — the focus of our article — are not the first attempts at logical analysis of this type of geometry . Alfred T arski mentions affine geometry in his comparison between the de velopments in logic and geometry (see T arski [1986], which is a written account of a talk he gav e, which in turn relfects his ideas from before the war). Moreover , together with his student Lesław Szczerba, T arski worked on point-based af fine spatial logics (see Szczerba and T arski [1979]), which built on T arski’ s earlier work on formalising Euclidean geometry (see T arski [1959] and McFarland et al. [2014] for a detailed look at T arski’ s in volvement in geometry). 3 Constructing a spatial logic If we were to custom-build a spatial logic, the first problem we are going to f ace is the choice of underlying geometric space. Many approaches have been studied, in most of them howe ver either R n for some n or some more general topological space is considered. Ha ving set on the underlying geometric space, say X , we are faced with another decision. Should the variables range ov er elements of X or some subset S ⊆ 2 X ? In the first case we would be talking about point-based spatial logics, in the second about re gion-based spatial logics, which is our focus here. As mentioned abov e, we place our work in the qualitative spatial r easoning (QSR) subarea of symbolic AI. The adjecti ve qualitative in this conte xt means that all the primiti ve relations and functions are of non-numerical nature. For example, consider a language with a single relation symbol C understood as the contact relation. Intuitiv ely two sets are in contact if their boundaries share at least one point. This spatial logic was inv estigated under many guises, most notably within the qualitativ e spatial reasoning paradigm. W e are now f aced with the following question: precisely what sort of re gions should we consider? W e could obviously decide to consider all S ⊆ 2 X for a giv en space X . Are there any reasons to consider a special class of regions rather than giv e them all an equal footing? One such reason is the admittedly vague notion of well-behavedness . First of all, to smooth out the reasoning with regions, we would lik e to weed out as man y “special cases” as possible. Assuming we are working with some topological space, this can be done by considering only r e gular subsets of that space as plausible re gion-candidates. This gets rid of many a “strange” set e.g. of fractal nature. In the next step we need to decide whether we consider our regions to contain their boundaries or not. In the first case we end up with re gular closed sets and in the second case with regular open sets. From a formal point of vie w , this is not an essential choice. In the remainder we will consider mainly regular open v ariants (and ev erything we say can be applied mutatis mutandis to the regular closed case). The class of all regular open subsets of some topological space is already a good choice for the well-behav ed regions. 4 Apart from what has been mentioned already , by a well-kno wn result the elements of the class of regular open subsets of some topological space form a Boolean Algebra, that is, operations of sum, product and complement of re gular open sets conform to the laws of Boolean Algebra. W e can do better still. W e can look inside this class for some more refined region candidates. As is customary , we single out tw o classes: (regular open) polygons and (regular open) rational polygons (limiting ourselv es to rational numbers). The fact that it is countable, mak es the second subclass especially interesting from the point of view of computer science applications. The choice of geometric space and either point- or region-based approach dictates the choice of relations and functions that we are presented with. W ithin the qualitati ve spatial reasoning paradigm, non-numerical predicates on re gions are considered, most notably contact and connectedness. T raditionally , spatial logics ov er languages containing relation and function symbols interpreted as relations and functions in v ariant under certain geometric transformations (topological, Eucidean etc.) are called accordingly as e.g. Euclidean, topological (spatial) logic. W e follow this con vention here. For example, consider an affine spatial logic constructed in the follo wing manner . Start with a language with two primitiv e symbols conv and ≤ . Let them denote the follo wing predicates defined on re gular open rational polygonal subsets of R 2 . The symbol conv ( a ) is to be understood as “region a is con ve x” and the symbol a ≤ b as “region a is a subset of region b ”. It is an af fine spatial logic, since con vexity is an af fine-in v ariant property . This spatial logic is in fact one that we are concerned the most within this article. The last choice made in constructing a spatial logic concerns the syntactical comple xity of the language we want to use. In our case, we work with standard first-order logic. 2 Notably by M. Pieri and F . Klein. F or a more detailed description of their work see: Marchisotto and Smith [2007] and Klein [2004], respectiv ely . 3 The article Nagel [1939] is a fascinating summary of the influence of geometrical results on the de velopment of logic. 4 This is by no means the final word in the quest for well-beha vedness, see Lando and Scott [2019]. 2 Three-Dimensional Affine Spatial Logics A P R E P R I N T The focus of this article The order of the article is as follows. After some more technical remarks regarding region-based theories of space and affine geometry , finally definining the structures that are important for us. Then, we describe in short the most important results obtained in T rybus [2016]. This is done partially to introduce certain (visual) intuitions that are easier to grasp in the tw o dimensional case b ut that carry o ver , to some extent, to the three-dimensional case. Next, we describe some more general results regarding the family of structures that we have defined: namely that they all ha ve dif ferent theories. Finally , we fix our attention on the three-dimensional extension of the two-dimensional logic analysed in T rybus [2016]. W e prov e a number of expressi veness results that are helpful in establishing a result similar to one of the main theorems of Pratt [1999], regarding the existence of formulas that are satisfied only by affine-equi valent regions. 2 General setup Let L conv , ≤ be a first-order language with two predicates: binary ≤ and unary conv . W e work with an L conv , ≤ -structure with v ariables ranging over the set of re gular open rational polygons of the real plane; ≤ interpreted as the inclusion relation and conv as a property of being con ve x. W e start with defining a notion of a r e gular open set . Definition 2.1. Let S be a subset of some topological space. W e denote the interior of S by S 0 and the closur e of S by S − . S is called re gular open if S = ( S ) − 0 . The following result is standard. Proposition 2.2. The set of r e gular open sets in X forms a Boolean algebr a RO ( X ) with top and bottom defined by 1 = X and 0 = ∅ , and Boolean operations defined by a · b = a ∩ b , a + b = ( a ∪ b ) − 0 and − a = ( X \ a ) 0 . W e restrict our attention to certain well-beha ved regular open sets (see our remarks in the introduction). Let us start with a staple topological space used in QSR , namely R 2 . Note that every line in R 2 divide s R 2 into two domains, called half-planes . Open half-planes are regular open sets, hence we can speak about the sums, products and complements of such half-planes in RO ( R 2 ) . By a r e gular open rational polygon we mean a Boolean combination in RO ( R 2 ) of finitely man y half-planes bounded by lines with rational coefficients in R 2 . W e denote the set of all re gular open rational polygons in R 2 by RO Q ( R 2 ) . Note that RO Q ( R 2 ) is a Boolean subalgebra of RO ( R 2 ) . The notion of regular open rational polygon can be easily extended to that of a polytope , when considering dimensions greater than 2 . In general, we write RO Q ( R n ) , n ∈ N , to denote the set of all regular open rational polytopes of dimension n (all the mentioned results carry ov er from the two-dimensional case). Definition 2.3. A set S ∈ R n is called con vex if for all λ 1 , λ 2 ∈ R , such that λ 1 , λ 2 ≥ 0 and λ 1 + λ 2 = 1 and for all x ∈ S , λ 1 x + λ 2 y ∈ S. Finally , let us introduce the family of n -dimensional structures we will be interested in. Definition 2.4. Let M n = ⟨ RO Q ( R n ) , conv M , ≤ M ⟩ , where ≤ M = {⟨ a, b ⟩ ∈ RO Q ( R n ) × RO Q ( R n ) | a ⊆ b } ; conv M = { a ∈ RO Q ( R n ) | a is conv ex } . W e sometimes refer to M n as a rational model (of dimension n ) and often drop the associated superscripts and subscripts if it does not lead to confusion. In our exposition we follo w the standard notational con ventions. In particular , if ϕ is a formula, ϕ ( x 1 , . . . , x n ) means that ϕ has at most n variables: x 1 , . . . , x n . Also, if an n -tuple of regions a 1 , . . . , a n satisfy ϕ in M , we write M | = ϕ [ a 1 , . . . , a n ] . Howe ver , we eschew formal clutter and whenev er possible av oid dissecting the text with lemmas in fa vour of v erbal description of results (especially the simpler ones) preserving the flo w of thought. It should be noted, howe ver , that all results can be easily con verted into a more formalised description. W e also need a few simple facts regarding af fine geometry . The follo wing generalises the notion of an affine transformation in R 2 to any dimension n . Recall that an n × n matrix A is in vertible if there e xists a n × n matrix B with AB = I , where I is the identity matrix; A is orthogonal if AA T = I , where A T is the transpose of A . 3 Three-Dimensional Affine Spatial Logics A P R E P R I N T Definition 2.5. An (n-dimensional) affine tr ansformation of R n is a function τ : R n → R n of the form τ ( x ) = A x + b, where A is an in vertible n × n matrix and b ∈ R n . Note that affine transformations map straight lines to straight lines, preserve parallelism and ratios of lengths along parallel straight lines. 5 Moreov er , it is a standard result that the set of af fine transformations forms a group under the operation of composition of functions. W e say that tw o regions are affine-equivalent if there is an affine transformation from one region to another (this notion naturally e xtends to sequences of regions). 3 T wo dimensions The papers Davis et al. [1999], Pratt [1999] together with T rybus [2016] deal with v arious systems related to M 2 . The two-dimensional rational model turns out to be very expressi ve. Firstly note that the Boolean operations are clearly L conv , ≤ -definable (as are 0 and 1 ; for details see below , Theorem 4.2). The paper Pratt [1999] showed that a number of interesting properties are definable in M 2 . It is easy to see that we can define a formula satisfied in the two-dimensional rational model if and only if the respectiv e region is a half-plane (half-plane is the only region such that both it and its complement are con vex). For the remainder of this paragraph, we use letters l, m, n etc. (possibly with subscripts) to denote such half-planes but sometimes we abuse the con vention and use the same symbols to denote the lines bounding the half-planes in question. W ith that in mind, Pratt [1999] showed that there is a formula in volving two v ariables satisfiable in the two-dimensional rational model if and only if the two re gions are half-planes with coincident bounding lines. Similarly , there is a formula, such that the two regions in volv ed are half-planes with parallel bounding lines. Note that in affine geometry a coor dinate frame is defined as follo ws. Definition 3.1. Let l, m, n be any non-parallel, non-coincident lines with l ∩ m = O , l ∩ n = I and m ∩ n = J . W e say that l, m, n form a coordinate fr ame . Fig. 1 provides some examples of coordinate frames. Since the construction in volves all the notions e xpressible in the two-dimensional model, we can “talk” about coordinate frames within that spatial logic. O J I (a) O J I (b) Figure 1: Example coordinate frames ( m is the horizontal and l the vertical line in (a), the image in (b) can be thought of as an affine transformation of that of (a)). No w , the papers Pratt [1999] and T rybus [2016] sho w , in a sequence of results, that there e xist formulas that allo w fixing any rational half-plane with respect to a gi ven coordinate frame. This is done by further e xploring the expressi vity of the model. Note that Davis et al. [1999] sho ws that if two regions are af fine-equiv alent, then for certain affine spatial logics these satisfy the same formulas. An analogous theorem, relating the language L conv , ≤ is prov ed in Pratt [1999]. Using the fixing formulas, the con verse theorem is sho wn to hold in the case of M 2 . Theorem 3.2 (Pratt [1999]) . Every n -tuple in M 2 satisfies an L conv , ≤ -formula ϕ with the following pr operty: any two n -tuples satisfying ϕ ar e affine-equivalent. 5 Hence, the properties of being a straight line, of lines being parallel and of being a ratio of a certain type are all affine-in variant. 4 Three-Dimensional Affine Spatial Logics A P R E P R I N T As indicated abov e, the proof relies on constructing certain formulas that allow us to “talk” about rational polygons and fixing their bounding lines in a certain manner , heavily relying on the e xpressivity results outlined above. (The paper T rybus [2016] provides details for this construction.) Moreov er , the paper T rybus [2016] uses these “fixing” formulas to provide an axiom system for the tw o-dimensional model, which is proved to be sound and complete. The axioms express a number of properties e.g. that there are at least three regions such that lines bounding them form a coordinate frame or that if a re gion is a Boolean combination of half-planes, then it is con vex if and only if it is a product of some of these half-planes. Ho wev er , for the most part, the axioms secure certain properties of these fixing formulas. The axiom system is also equipped with tw o infinitary rules of inference stating that e very half-plane can be fixed in reference to a gi ven coordinate frame and that e very region is a Boolean combination of some half-planes. Let us finally note that our main result in the present article closely mimicks that described in Theorem 3.2. 4 Bey ond two dimensions What can be known about the rational models of dimensions greater than two? Even at this stage, one can indeed make some statements about the relations among such models. Recall the well-known Helly’ s theorem. Theorem 4.1 (Helly) . Let A be a finite class of N con vex sets in R n such that N ≥ n + 1 and each n + 1 -element subclass of A has a non-empty intersection. Then all N elements of A have a non-empty intersection. Figure 2: A very simple example of Helly’ s Theorem in R 2 . First off, note that for all n , we ha ve the following easy result. Theorem 4.2. Let M n be a rational polygonal model. Then, the Boolean operator s: pr oduct ( · ), sum ( + ) and complement ( − ) ar e definable in M n . Mor eover the top (1) and bottom (0) elements ar e also definable. Pr oof. For simplicity , we shall represent such formulas in their (infix) form as x · y , x + y and − x and top and bottom as 1 and 0 , instead of more correct but cumbersome standard notation as formulas in our language (which is what they really are), abusing the symbol of equality to also render situations like x · y = 0 . Now for the definitions of formulas. The following formula is satisfiable in M n if and only if the region represented by the variable m is the product of the regions represented by x and y respectiv ely: ( m ≤ x ) ∧ ( m ≤ y ) ∧ ∀ w ( w ≤ x ∧ w ≤ y → w ≤ m ) . An analogous formula can be constructed for the sum (all easily expressible in our language). The top can be defined as ∀ y ( y ≤ x ) and the bottom as ∀ y ( y = x ∨ ¬ ( y ≤ x )) . Moreover , the follo wing formula is satisfiable in M n if and only if the region represented by y is the complement of the re gion represented by x : y · x = 0 ∧ ∀ w ( w · x = 0 → w ≤ y ) . Now , consider the following formula ϕ ( x 1 , . . . , x N ) := ^ I ⊆ 2 S Y i ∈ I x i  = 0 ∧ ^ 1 ≤ j ≤ N conv ( x j ) → Y 1 ≤ j ≤ N x j  = 0 , where S = { 1 , . . . , N } . 5 Three-Dimensional Affine Spatial Logics A P R E P R I N T This formula 6 “says” in any n -dimensional model that regions r 1 , . . . , r N hav e non-empty intersection if each r j is con ve x and for ev ery subset of { r 1 , . . . , r N } , its elements hav e a non-empty intersection. Theorem 4.3. F or a given n ther e exists a set of formulas Φ n expr essing the Helly’ s theorem in M n . T o see that this is the case, consider ϕ N := ∀ x 1 . . . ∀ x N ϕ ( x 1 , . . . , x N ) . W e define Φ n = { ϕ N | N ≥ n + 1 and | I | = n + 1 } . Recall that the theory of a structure is the set of all sentences valid in that structure. Theorem 4.4. The theory of M n  = M n +1 for all n . T o see that this is the case, observe that for some ϕ N ∈ Φ n we hav e M n | = ϕ N but M n +1 | = ϕ N . Therefore, we can say that these models are indeed dif fer ent . Howe ver , if one were to extend the axiomatisation results from T rybus [2016] to dimensions greater than two — and this is indeed our long-term goal — the models should be also sho wn to be similar in some other respect. Namely , the first task would be to see whether the notion of a coordinate frame can be expressed in such models in general. 5 Three dimensions Basic expressi vity First of all, notice that the formula conv ( x ) ∧ conv ( − x ) is satisfiable in M 3 only by regions that are half-spaces. Since the plane bounding such half-spaces is unique, this also allo w us to talk indirectly about such planes. It is con venient to be able to talk about a number of different half-spaces (planes); hence we introduce the following abbre viation. hs n ( x 1 , . . . , x n ) := ^ 1 ≤ i ≤ n conv ( x i ) ∧ conv ( − x i ) ∧ ^ 1 ≤ i ≤ n, 1 ≤ j ≤ n, i  = j x i  = x j ∧ x i  = − x j Next, we see that we can talk about parallel planes by means of the follo wing formula. hs 2 ( x, y ) ∧ (( x · y = 0 ∨ x · − y = 0) ∨ ( − x · y = 0 ∨ − x · − y = 0)) This formula “says” in M 3 that the two re gions are distinct half-spaces (with distinct bounding planes) and that it is either that the first region has a non-empty intersection with the other or that its complement has this property . (Note that the main disjunction in the brack ets operates really as an e xclusiv e “or” when the two half-spaces are dif ferent.) Similarly , the following formula e xpresses the fact that two planes meet in the single line. Since such lines are unique, we can also — albeit indirectly — talk about lines in M 3 . line ( x, y ) := hs 2 ( x, y ) ∧ ¬ (( x · y = 0 ∨ x · − y = 0) ∨ ( − x · y = 0 ∨ − x · − y = 0)) Finally , consider the case when the following is satisfied: line ( y 1 , y 2 ) ∧ line ( y 1 , y 3 ) ∧ line ( y 2 , y 3 ) (we thus assume this piece of formalism to be a part of all formulas described in the remainder of this paragraph). This formula simply says that all the planes bounding the three half-space hav e a non-empty intersection with each other . Consider the following three configurations: (i) a fan: where all the planes meet in a single line; (ii) a prism: where two of the planes meet in a line not on the third plane and meet the third plane in two separate, parallel lines; (iii) a corner: where two of the planes meet the third plane in two separate, non-parallel lines and meet each other in a line that passes through the third plane. 6 W e use Q and P as abbreviations for finite products and sums, respectively . W e also use ± a to denote region a or its complement. 6 Three-Dimensional Affine Spatial Logics A P R E P R I N T Since in all the above cases, the number of domains into which the entire space is being partitioned changes (6 domains for a fan, 7 for a prism and 8 for a corner) and it can be expressed in terms of products of respectiv e half-spaces or their complements, one can build formulas describing all three cases in M 3 . Noting that there are 8 non-empty intersections possible in total, in the case of a corner , one enforces a non-empty intersection of all the half-spaces by adding the formula ¬∃ x ¬∃ y ¬∃ z ((( x = y 1 ∨ x = − y 1 ) ∧ ( y = y 2 ∨ y = − y 2 ) ∧ ( z = y 3 ∨ z = − y 3 )) ∧ ( x · y · z = 0)) Directly , and assuming that line ( y 1 , y 2 ) ∧ line ( y 1 , y 3 ) ∧ line ( y 2 , y 3 ) is satisfied, this formula “says” that any three (out of six in total — remember we al way hav e a half-space and its complement) half-spaces bounded by some planes hav e a non-empty intersection. Next, in the case of a prism, one simply adds the formula ∃ x ∃ y ∃ z ((( x = y 1 ∨ x = − y 1 ) ∧ ( y = y 2 ∨ y = − y 2 ) ∧ ( z = y 3 ∨ z = − y 3 )) ∧ ( x · y · z = 0)) forcing the existence of a non-empty intersection. Ho wev er , when paired with ¬∃ x ′ ¬∃ y ′ ¬∃ z ′ ((( x ′ = y 1 ∨ x ′ = − y 1 ) ∧ ( y ′ = y 2 ∨ y ′ = − y 2 ) ∧ ( z ′ = y 3 ∨ z ′ = − y 3 )) ∧ ( x  = x ′ ∨ y  = y ′ ∨ z  = z ′ ) ∧ ( x ′ · y ′ · z ′ = 0)) . Thus, the end ef fect is only one empty intersection. Finally , in the case of a f an, one has to force precisely tw o non-empty intersections. This is done by stringing together the following: ∃ x ∃ y ∃ z ((( x = y 1 ∨ x = − y 1 ) ∧ ( y = y 2 ∨ y = − y 2 ) ∧ ( z = y 3 ∨ z = − y 3 )) ∧ ( x · y · z = 0)) and ∃ x ′ ∃ y ′ ∃ z ′ ((( x ′ = y 1 ∨ x ′ = − y 1 ) ∧ ( y ′ = y 2 ∨ y ′ = − y 2 ) ∧ ( z ′ = y 3 ∨ z ′ = − y 3 )) ∧ ( x ′ · y ′ · z ′ = 0)) with the condition that ( x  = x ′ ∨ y  = y ′ ∨ z  = z ′ ) together with ¬∃ x ′′ ¬∃ y ′′ ¬∃ z ′′ ((( x ′′ = y 1 ∨ x ′′ = − y 1 ) ∧ ( y ′′ = y 2 ∨ y ′′ = − y 2 ) ∧ ( z ′′ = y 3 ∨ z ′′ = − y 3 )) ∧ (( x  = x ′′ ∨ y  = y ′′ ∨ z  = z ′′ ) ∨ ( x ′  = x ′′ ∨ y ′  = y ′′ ∨ z ′  = z ′′ )) ∧ ( x ′′ · y ′′ · z ′′ = 0)) The abo ve are admittedly long-winded but relatively simple and repetitive formulas. W e hide the details under the self-explanatory abbre viations fan ( x, y , z ) , prism ( x, y , z ) and corner ( x, y, z ) . Having established this, let us note that the case (iii) provides a basis for a coordinate frame. For the remainder of this section, we focus on fleshing out one of the ways of defining a coordinate frame in M 3 . Consider the formula frame ( y 1 , y 2 , y 3 , y ′ ) := corner ( y 1 , y 2 , y 3 ) ∧ line ( y 1 , y ′ ) ∧ line ( y 2 , y ′ ) ∧ line ( y 3 , y ′ ) It is satisfiable by a tuple of elements a 1 , a 2 , a 3 , a ′ only when these are half-spaces such that the planes bounding the first three of them form a corner and the plane bounding the fourth half-space form a prism with each pair of these planes. The three lines that lie at the pairwise intersections of the planes a 1 , a 2 , a 3 will determine the axes of the coordinate frame. Next, a ′ meets the remaining planes at three distinct lines that intersect pairwise on each of the axes: the points of intersection of each pair of such lines and an axis will be marked with a point, called the unit of measur ement (akin to the points I and J in Figure 1 but for all the three planes in volv ed). Thus, there are three axes and three units of measurement. 7 Three-Dimensional Affine Spatial Logics A P R E P R I N T Addition and multiplication Consider two planes intersecting a third one in tw o lines. These lines are coincident, if the planes themselves are. Let us assume that line ( y 1 , y ) ∧ line ( y 2 , y ) is satisfied. By adding y 1 = y 2 ∨ y 1 = − y 2 we define a relev ant formula, denoted coincident 2 ( y 1 , y 2 , y ) . 7 Similarly , such lines are parallel, if the planes are. Therefore if we add ¬ ( y 1 = y 2 ∨ y 1 = − y 2 ) ∧ ( y 1 · y 2 = 0 ∨ y 1 · − y 2 = 0 ∨ − y 1 · y 2 = 0 ∨ − y 1 · − y 2 = 0) we obtain a formula (denoted parallel 2 ( y 1 , y 2 , y ) ) satisfied in M 3 if and only if the relation of parallelism holds between the respectiv e lines. Also, when lines in a plane are not coincident or parallel, they hav e to meet in a single point. Thus, we can add the following constraints ¬ coincident 2 ( y 1 , y 2 , y ) ∧ ¬ parallel 2 ( y 1 , y 2 , y ) , defining a formula point 2 ( y 1 , y 2 , y ) with the obvious interpretation. W e need these expressivity results to define important operations on line segments found on the planes forming the coordinate frame. W e start with defining addition in a plane (following Bennett [1995]): Definition 5.1. W e say that OC is the result of the addition of O A and OB and write O A + OB = OC if and only if the following lines can be found (see Fig. 3): (a) l 1 , l 3 meeting at a point O ; (b) m parallel to l 3 ; (c) l A , meeting l 3 at a point A and parallel or coincident with l 1 ; (d) l B , meeting l 3 at a point B and such that l B , l 1 , m meet at a single point J ; (e) l C , meeting l 3 at a point C and parallel or coincident with l B and such that l A , l C and m meet at a single point M . O I A B C J M m l 3 l 1 l 2 l B l C l A Figure 3: OA + OB = OC . Say , for simplicity , that we adopted the same notational conv entions for objects in the structure M 3 (remembering that these can refer to the rele vant lines only indirectly and that directly these denote objects in the three-dimensional space!). Furthermore, let M 3 | = frame [ l 1 , l 3 , n, n ′ ] , for some n, n ′ (with the plane bounding n ′ meeting the plane bounding n in the line l 2 , thus defining the unit of measurement): this takes care of (a). 8 W e can enforce (b) by M 3 | = parallel [ m, l 3 , n ]; (c) with M 3 | = point 2 [ l A , l 3 , n ] ∧ ( parallel 2 [ l A , l 1 , n ] ∨ coincident 2 [ l A , l 1 , n ]); (d) with M 3 | = point 2 [ l B , l 3 , n ] ∧ corner 2 [ l B , l 1 , m ]; 7 coincident 2 should be understood as defining coincidence in two dimensions. Similarly for other notions used in this paragraph. 8 Note that our entire two-dimensional construction is ‘happening’ in the plane bounding n . 8 Three-Dimensional Affine Spatial Logics A P R E P R I N T and (e) with M 3 | = point 2 [ l C , l 3 , n ] ∧ ( parallel 2 [ l C , l B , n ] ∨ coincident 2 [ l C , l B , n ]) . Thus, we can construct a formula add 2 ( y 1 , y 3 , y A , y B , y C , y , y ′ , z ) such that M 3 | = add 2 [ l 1 , l 3 , l A , l B , l C , n, n ′ , m ] if and only if O A + OB = OC (assuming the naming con ventions abov e). Similarly , again after Bennett [1995], let us define multiplication in a plane. Definition 5.2. W e say that OC is the result of multiplication of O A and OB and write O A · OB = OC if and only if the following lines can be found (see Fig. 4): (a) l 1 , l 3 meeting at a point O and l 2 meeting l 1 at a point J and l 3 at a point I ; (b) l A meeting l 3 at a point A and parallel or coincident with l 2 ; (c) l B meeting l 3 at a point B and such that l B , l 1 , l 2 meet at a single point ( J ); (d) l C meeting l 3 at a point C , parallel or coincident with l B and such that l C , l A , l 1 meet at a single point M . O I A B C J l 3 l 1 l 2 l B l C l A M Figure 4: OA · OB = OC . Gi ven the similarity of the constraints for multiplication to those for addition, it should be clear now that there is a formula multiply 2 ( y 1 , y 3 , y A , y B , y C , y , y ′ ) such that M 3 | = multiply 2 [ l 1 , l 3 , l A , l B , l C , n, n ′ ] if and only if O A · OB = OC (assuming the naming con ventions abo ve). Corollary 5.3. Addition and multiplication are definable in e very plane bounding the half-spaces used in defining a coor dinate frame in M 3 . T o obtain this simple consequence one needs to change what counts as the plane of reference (where these operations are defined by means of the abov e-described formulas). Affine completeness In this part, we show ho w to obtain results analogous to those presented in Pratt [1999] with regard to M 2 . Assume for no w that we work in a specified plane of reference with the coordinate frame defined as abov e. How one would go about actually defining the numbers on the x -axis (the y -axis being analogous)? W ell, we can start by defining these in terms of distance. So 0 would be OO and 1 would be OI with the rest of the natural numbers obtained by “repeating” the construction of OI . This is in f act how things are done in Pratt [1999] and Tryb us [2016], so the interested reader is encouraged to consult these sources. In this article, ho wev er , we propose a slightly different solution, using the fact that addition is expressible to define a successor formula instead. Say , we defined 0 as the point of intersection of the two ax es, and 1 as the point of intersection of l 3 (assuming previous con ventions) with the x -axis represented by l 1 . The successor formula can be defined so that, starting with OO as the base case (this is done by simply constructing a formula satisfiable only by those regions whose associated lines determine the same point as the lines determining the axes), one keeps on adding OI . More formally (yet still, with a lot of ancillary details left out for readability), assume that a line m crosses the line l 1 at a point M , such that OM = n OI , n ∈ N . W e then define the successor formula that in volves two important elements: the regions representing m and another line m ′ , such that m ′ is the result of adding OM and OI (clearly doable, by the above). 9 This takes care of the natural numbers on the line. The following result sho ws that we can extend this to an y rational number . 9 It should be obvious at this point that the theory of M 3 is undecidable. 9 Three-Dimensional Affine Spatial Logics A P R E P R I N T Theorem 5.4. Assuming the coor dinate frame setup above and all the intr oduced shorthands, let m be a line cr ossing the axis at a point M . Then ther e exists a formula satisfiable in M 3 if and only if OM = n OI , n ∈ Q . Pr oof. For the proof of the above, the case when n ∈ N has been outlined. Consider n = p q , with p, q ∈ N (these do not hav e to be relativ ely prime). W e get q OM = p OI , that is, in our parlance, OQ · OM = OP · OI . OP and OQ are clearly expressible using the successor formula. The formula capturing the abo ve equality must simply enforce that the same point ( M ) is the result of both multiplications (and multiplications are e xpressible). What remains is the case when n is negati ve. It is enough to make a cop y of the triangle forming the coordinate frame’ s units of measurement (clearly doable) and perform the operations “in rev erse”. Giv en an y line in a plane, and any coordinate frame, this line can cross both axes at some points; cross one of the axis and be parallel to the other; cross through the origin and be either parallel or cross the line forming the units of measurements at some point; or equal one of the formulas in volved in the construction of the coordinate frame. Any intersection points can be captured numerically using the formulas described in the outline of the proof of Theorem 5.4 (if needs be, changing what counts as the axis, see T rybus [2016]) and the remaining parallel cases can also be dealt with as parallelism is expressible in our language. Theorem 5.5. Let h, h ′ ∈ ROQ ( R 3 ) be half-spaces. Then ther e is a formula satisfiable in M 3 , such that (1) h satisfies this formula and (2) if h ′ satisfies the formula, then h ′ = h . Pr oof. Repeating the construction from Theorem 5.5 on the other planes of reference means that the resulting compound formula fixes the bounding plane of h , yielding h ′ = h or h ′ = − h . The final disambiguation can be done by insisting that there is no half-space that is contained in h and not contained in h ′ . Such formulas are sometimes called fixing formulas. Note that this notion can be e xtended from half-planes to arbitrary regions from the domain. In addition we obtain an analogue of the result from Pratt [1999]. Let us say that a formula is affine-complete (in M 3 ), if for any two re gions r , r ′ ∈ R O Q ( R 3 ) satisfying it, there is an affine transformation mapping r to r ′ (and, of course, vice versa). Our final result is that — just as in the case of M 2 — ev ery region satisfies an affine-complete formula. Theorem 5.6. Every r ∈ RO Q ( R 3 ) satisfies an affine-complete formula in M 3 . Pr oof. Recall that ev ery element of RO Q ( R 3 ) can be represented as a Boolean combination of half-spaces. Consider a formula stating that certain regions form a coordinate frame (as done above), fixing all the half-spaces that are inv olved in the construction of r with respect to the resulting coordinate frame (again, as outlined above), and describing the exact Boolean combination resulting in r (clearly expressible). Such a formula has m + n free v ariables, where m is the number of variables in volv ed in the construction of the coordinate frame and the remaining n v ariables representing the lines fixed with respect to the coordinate frame (for simplicity , allowing for repetitions of v ariables in both groups). Consider no w an tw o m + n -tuples satisfying this formula. W e sho w that the elements from both tuples are af fine equi valent. Recall that all tetrahedra (essentially: corners in our terminology) are affine-equi v alent. Therefore, there is a (unique) affine transformation, say τ , taking the m -elements of the first tuple to the m -elements of the other . Moreover , since the remaining n half-spaces from the second tuple are fix ed with respect to the coordinate frame, by Theorem 5.5, the τ -transformed n half-spaces from the first tuple must be the same as the n half-spaces from the second tuple. That is, these are also affine equi valent. Therefore, τ takes all the elements from the first m + n -tuple to the second one. The final formula existentially binds all the v ariables apart from the one representing r . This result can be easily extended from a single region to formulas of arbitrary arity , as in Pratt [1999], thus pro viding an exact match to Theorem 3.2 mentioned abo ve. 6 Open Problems Thus, the stage is set for the task of axiomatising the theory of M 3 . This might be no easy feat, considering how much simpler it is to talk about coordinate frames in M 2 compared to M 3 . If in the due process, some re gularities regarding the constructions are observed, this could be the basis for extending the results to other dimensions. 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