Modal Logics of Topological Relations

Transcription

1 Ö Ò ÍÒÚÖ ØÝ Ó ÌÒÓÐÓÝ ÁÒ ØØÙØ ÓÖ ÌÓÖØÐ ÓÑÔÙØÖ ËÒ Ö ÓÖ ÙØÓÑØ ÌÓÖÝ ÄÌËßÊÔÓÖØ Modal Logics of Topological Relations Carsten Lutz and Frank Wolter LTCS-Report ÄÖ ØÙÐ ĐÙÖ ÙØÓÑØÒØÓÖ ÁÒ ØØÙØ ĐÙÖ ÌÓÖØ ÁÒÓÖÑØ ÌÍ Ö Ò ØØÔ»»ÐغҺØÙ¹Ö Òº ÀÒ ¹ÖÙÒ¹ËØÖº ¾ ¼½¼¾ Ö Ò ÖÑÒÝ

2 Modal Logics of Topological Relations Carsten Lutz Institute of Theoretical Computer Science TU Dresden, Germany Frank Wolter Department of Computer Science University of Liverpool, UK Abstract The eight topological Ê (or Egenhofer-Franzosa)- relations between spatial regions play a fundamental role in spatial reasoning, spatial and constraint databases, and geographical information systems. In analogy with Halpern and Shoham s modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Ê-relations. The semantics is based on region spaces induced by standard topological spaces, in particular the real plane. We investigate the expressive power and computational complexity of the logics obtained in this way. It turns our that, similar to Halpern and Shoham s logic, the expressive power is rather natural, but the computational behavior is problematic: topological modal logics are usually undecidable and often not even recursively enumerable. This even holds if we restrict ourselves to classes of finite region spaces or to substructures of region spaces induced by topological spaces. We also analyze modal logics based on the set of Ê-relations, with similar results. 1 Introduction Reasoning about topological relations between regions in space is recognized as one of the most important and challenging research areas within Spatial Reasoning in AI and Philosophy, Spatial and Constraint Databases, and Geographical Information Systems (GISs). Research in this area can be classified according to the logical apparatus employed: General first-order theories of topological relations between regions are studied in AI and Philosophy [7; 29; 28], Spatial Databases [27; 33] and, from, an algebraic viewpoint, in [9; 34]; Purely existential theories formulated as constraint satisfaction systems over jointly exhaustive and mutually disjoint sets of topological relations between regions [10; 31; 18; 33; 29] Modal logics of space with operators interpreted by the closure and interior operator of the underlying topological space and propositions interpreted as subsets of the topological space, see e.g., [21; 5; 1]. A similar classification can be made for Temporal Reasoning: general first-order theories [3], temporal constraint systems [2; 36; 26] and modal temporal logics like Prior s tense logics, LTL, and CTL [16; 12]. However, one of the most important and influential approaches in temporal reasoning has not yet found a fully developed analogue on the spatial reasoning research agenda: Halpern and Shoham s Modal Logic of intervals [19], in which propositions are interpreted as sets of intervals (those in which they are true) and reference to other intervals is enabled by modal operators interpreted by Allen s 13 relations between intervals. Despite its bad computational behavior (undecidable, usually not even r.e.), this framework proved extremely fruitful and influential in temporal reasoning, see e.g. [35; 4; 22]. To develop an equally powerful and useful modal language for reasoning about topological relations between regions, we first have to select a set of basic relations. In the initially mentioned research areas, there seems to be consensus that the eight Ê-relations, which are also known as Egenhofer-Franzosa -relations and have been independently introduced in [29] and [11], are very natural and important both from a theoretical and a practical viewpoint, see e.g. [27; 10]. Thus, in this paper we will consider modal logics with eight modal operators interpreted by the eight Ê-relations, and whose formulas are interpreted as sets of regions (those in which they are true). This modal framework for reasoning about regions has been suggested in an early paper by Cohn [8] and further considered in [37]. However, it proved difficult to analyze the computational behavior of such logics and, despite several efforts, to the best of our knowledge no results have been obtained so far. To relate this approach to previous and ongoing work on first-order theories of regions [29; 28; 27; 33], it is important to observe that the modal logic we propose is a fragment of

3 Ö Ö Ö Ö ÔÓ Ö Ö Ö Ö Õ Ö Ö ØÔÔ Ö Ö ÒØÔÔ Ö Ö ØÔÔ Ö Ö ÒØÔÔ Figure 1. Examples for the Ê relations. first-order logic with the eight binary Ê-relations and infinitely many unary predicates. More precisely, we will show that our logic has exactly the same expressive power as the two-variable fragment of this FO logic although the latter is exponentially more succinct. Since usual first-order theories of regions admit arbitrarily many variables but no unary predicates, their expressive power is incomparable to the one of our modal logics. We argue that the availability of unary predicates is essential for a wide range of application areas: in contrast to describing only purely topological properties of regions, it allows to also capture other properties such as being a country (in a GIS), a ball (for a soccerplaying robot), or a protected area (in a spatial database). In our modal logic, we can then formulate constraints such as there are no two overlapping regions that are both countries and every river is connected to an ocean or a lake. The purpose of this paper is to introduce modal logics of topological relations in a systematic way, perform an initial investigation of their expressiveness and relationships, and analyze their computational behavior. More precisely, this paper is organized as follows: in Section 2, we introduce region spaces, which form the semantical basis for our logics. The modal language is introduced in Section 3, and a brief analysis of its expressiveness is performed. In Section 4, we identify a number of natural logics induced by different classes of region structures, and analyze their relationship. In Section 5, we then prove the undecidability of our logics. This is strengthened to a ½ ½-hardness proof in Section 6, where it is shown that only very few of our logics are recursively enumerable. We prove undecidability of classes of finite region spaces in Section 7. Finally, in Section 8 we consider modal logics based on the non-topological, but spatial Ê-relations and show that they, too, are usually undecidable. 2 Structures We want to reason about models whose domains consist of regions that are related by the eight Ê-relations Æ ØÔÔ ØÔÔ ÔÓ ÒØÔÔ ÒØÔÔ,,,,,,,, ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ,,,,,ÔÓ,,, ÔÓ, ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ØÔÔ,, ÔÓ,ØÔÔ, ØÔÔ, ÒØÔÔ ØÔÔ,Õ ÒØÔÔ ÒØÔÔ ÒØÔÔ,,,,,, ØÔÔ, ØÔÔ,ÒØÔÔ ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ÒØÔÔ ÔÓ,ØÔÔ, ØÔÔ,Õ ÒØÔÔ ÒØÔÔ,,,ÔÓ, ÔÓ,Õ, ÔÓ, ÔÓ, ØÔÔ ÔÓ,ØÔÔ, ØÔÔ, ØÔÔ, ØÔÔ,ÒØÔÔ ØÔÔ, ØÔÔ, ÒØÔÔ ÒØÔÔ ÒØÔÔ ØÔÔ ÒØÔÔ ÒØÔÔ,,,, ÔÓ,,, ÔÓ,,, ÔÓ ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ØÔÔ, ÔÓ,ØÔÔ, ØÔÔ, ÔÓ,ØÔÔ, ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ,,,, ÒØÔÔ ÒØÔÔ ÔÓ,ØÔÔ, ÔÓ,ØÔÔ, ÒØÔÔ ÒØÔÔ ÒØÔÔ,, ÔÓ, ÔÓ, ÔÓ, ÔÓ, ØÔÔ, ÒØÔÔ ÔÓ,ØÔÔ, ØÔÔ, ØÔÔ, ÒØÔÔ ØÔÔ, ØÔÔ,ÒØÔÔ, ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ ÒØÔÔ,Õ Figure 2. The Ê composition table. ( disconnected ), ( externally connected ), ØÔÔ ( tangential proper part ), ØÔÔ ( inverse of tangential proper part ), ÔÓ ( partial overlap ), Õ ( equal ), ÒØÔÔ ( non-tangential proper part ), and ÒØØÔ ( inverse of non-tangential proper part ). Figure 1 gives examples of the Ê relations in the real plane Ê ¾, where regions are rectangles. Different spatial ontologies give rise to different notions of regions and, therefore, different classes of models. Almost all definitions of regions provided in the literature, however, have in common that the resulting models are region structures Ê Ï Ê Ê where Ï is a non-empty set (of regions) and the Ö Ê are binary relations on Ï that are mutually disjoint (i.e., Ö Ê Õ Ê, for Ö Õ), jointly exhaustive (i.e., the union of all Ö Ê is Ï Ï ), and satisfy the following: Õ is interpreted as the identity on Ï,,, and ÔÓ are symmetric, and ØÔÔ and ÒØÔÔ are the inverse relations of ØØÔ and ÒØÔÔ, respectively; the rules of the Ê composition table (Figure 2) are satisfied in the sense that, for any entry Õ ½ Õ in row Ö ½ and column Ö ¾, the first-order sentence ÜÝÞ Ö ½ Ü ÝµÖ ¾ Ý Þµµ Õ ½ Ü Þµ Õ Ü Þµµ is valid ( is the disjunction over all Ê-relations). Denote the class of all region structures by ÊË. Although of definite interest as a basic class of models representing the relation between regions in space, often more restricted definitions of region structures are considered. On the one

4 hand, one can consider further first-order conditions on region structures, say, (fragments) of the Ê-theory [29]. Another possibility is to consider only region structures that are induced by (classes of) topological spaces. Recall that a topological space is a pair Ì Í Áµ, where Í is a set and Áis an interior operator on Í, i.e., for all Ø Í, we have Á Í µ Í Á µ Á µ Á ص Á ص ÁÁ µ Á µ The closure µ of is then µ Í Á Í µµ Of particular interest are Ò-dimensional Euclidean spaces Ê Ò based on Cartesian products of the real line with the standard topology induced by the Euclidean metric. Depending on the application domain, different definitions of regions in topological spaces have been introduced. A rather general notion identifies regions with non-empty, regular closed sets, i.e. non-empty subsets Í such that Á µ. We write Ì Ö to denote the set of non-empty, regular closed subsets of the topological space Ì. Various more restrictive definitions of regions are important in the Euclidean spaces Ê Ò, e.g., the set Ê Ò of non-empty convex regular closed subsets of Ê Ò ÓÒÚ ; the set Ê Ò of closed hyper-rectangular subsets ÖØ of Ê Ò, i.e., regions of the form É Ò ½, where ½ Ò are non-singleton closed intervals in Ê. In both cases we allow unbounded regions, in particular Ê Ò. However, we should note that the technical results proved in this paper also hold if we consider bounded regions, only. Given a topological space Ì and a set of regions Í Ì in Ì as introduced above, we obtain a region structure Ê Ì Í Ì µ Í Ì Ì by putting: ص ¾ Ì iff Ø Øµ ¾ Ì iff Á µ Á ص Ø Øµ ¾ ÔÓ Ì iff Á µ Á ص Ò Ø Ø Ò Øµ ¾ Õ Ì iff Ø Øµ ¾ ØÔÔ Ì iff Ø Á ص ص ¾ ÒØÔÔ Ì iff Á ص ص ¾ ØÔÔ Ì iff Ø µ ¾ ØÔÔ Øµ ¾ ÒØÔÔ Ì iff Ø µ ¾ ÒØÔÔ Ê Ì Í Ì µ is called the region structure induced by Ì Í Ì µ. It is easy (but tedious) to verify that the conditions of region structures are satisfied. We set Ì ÇÈ Ê Ì Ì Ö µ Ì topological space 3 Languages The modal language Ä Ê extends propositional logic with countably many variables Ô ½ Ô ¾ and the Boolean connectives and by means of the modal operators,, etc. (one for each Ê relation). A region model Å Ê Ô Å ½ ÔÅ for Ä ¾ Ê consists of a region structure Ê Ï Ê and the interpretation Ô Å of the variables of Ä Ê as subsets of Ï. A formula ³ is either true at ¾ Ï (written Å ³) or false at (written Å ³), the inductive definition being as follows: 1. If ³ is a prop. variable, then Å ³ iff ¾ ³ Å. 2. Å ³ iff Å ³. 3. Å ³ ½ ³ ¾ iff ³ ½ and ³ ¾. 4. Å Ö ³ iff, for all Ø ¾ Ï, ص ¾ Ö Ê implies Å Ø ³. We use the usual abbreviations: ³ for ³ and Ö³ for Ö ³. The discussion of the expressive power of our logic starts with three simple examples. First, the useful universal box ¾ Ù ³ has the following semantics: Å ¾ Ù ³ iff Å Ø ³ for all Ø ¾ Ï In our logic, it can obviously be expressed as Î Ö ³ Ö¾Ê Second, we can express that a formula ³ holds in precisely one region (is a nominal) by Ö ³µ ÒÓÑ ³µ Ù ³ Ö¾ÊÒÕ where Ù ³ ¾ Ù ³. The definability of nominals means, in particular, that we can express Ê-constraints [31] in our language: just observe that constraints Ü Ö Ýµ, where Ö is an Ê-relation, correspond to the assertion Ô Ü ÖÔ Ý µ ÒÓÑ Ô Ü µ ÒÓÑ Ô Ý µ Another main advantage of having nominals is that we can introduce names for regions; e.g., the formulas ÒÓÑ Elbeµ ÒÓÑ Dresdenµ state that Elbe (the name of a river) and Dresden each apply to exactly one region. Third, it is useful to define operators ÔÔ and ÔÔ as abbreviations: ÔÔ ³ ØÔÔ ³ ÒØØÔ ³ ÔÔ ³ ØÔÔ ³ ÒØØÔ ³ As in the temporal case [19] and following Cohn [8], we can classify propositions ³ according to whether they are homogeneous, i.e. they hold continuously throughout regions: ¾ Ù ³ ÔÔ ³µ. they are anti-homogeneous, i.e. they hold only in regions whose interiors are mutually disjoint: ¾ Ù ³ ÔÔ ³ ÔÓ ³µ

5 Instances of anti-homogeneous propositions are river and city, while occupied-by-water is homogeneous. The following are some example statements in our logic (neglecting for simplicity that existence of sea harbors): ¾ Ù harbor-city city ÒØÔÔ harbor riverµµµ ¾ Ù Dresden harbor-cityµ ¾ Ù Elbe riverµ ¾ Ù Dresden ÔÓElbe ÔÓ river Elbeµµµ ¾ Ù Dresden ÔÔ riverµ From these formulas, it follows that Dresden has a harbor that is related via to the river Elbe. We now relate the expressive power of the modal language Ä Ê to the expressive power of first-order languages over region structures. Since spatial first-order theories are usually formulated in first-order languages equivalent to Ç Ê with eight binary relations for the Ê relations and no unary predicates [27; 28; 33; 29], we cannot reduce Ä Ê to such languages. A formal proof is provided by the observation that Ç Ê is decidable over the region space consisting of rectangles in Ê ¾ (in fact it is reducible to the decidable first order theory of Ê ), while in Section 6 we show that Ä Ê is not even r.e. over that space. Thus, the proper first-order language to compare Ä Ê with is the monadic extension Ç Ñ Ê of Ç Ê that is obtained by adding unary predicates Ô ½ Ô ¾. By well-known results from modal correspondence theory [15], any modal formula ³ can be polynomially translated into an equivalent formula ³ of Ç Ñ with only two variables Ê such that, for any region model Å and any region, Å ³ iff Å ³ More surprisingly, the converse holds as well: this follows from recent results of [23] since the Ê relations are mutually exclusive and jointly exhaustive. Theorem 1. For every Ç Ñ -formula ³ ܵ with free Ê variable Ü that uses only two variables, one can effectively construct a Ä Ê -formula ³ of length at most exponential in the length of ³ ܵ such that, for every region model Å and any region, Å ³ iff Å ³ A proof sketch can be found in Appendix A. There, we also argue that, due to a result of Etessami, Vardi, and Wilke [13], there exist properties that can be formulated exponentially more succinct in the two-variable fragment of Ç Ñ Ê than in Ä Ê. 4 Logics In this section, we analyze the impact of choosing different underlying classes of region structures. As discussed in Section 2, the most important such classes are induced by topological spaces. A formula ³ is valid in a class of regions structures Ë if it is true in all points of all models based on region structures from Ë. We use Ä Ê Ëµ to denote the logic of the class Ë, i.e. the set of all Ä Ê -formulas valid in Ë. If Ë Ê Ì Í Ì µ for some topological space Ì with regions Í Ì, then we write Ä Ê Ì Í Ì µ instead of Ä Ê Ëµ The basic logic we consider is Ä Ê Ê˵, the logic of all region structures. On arbitrary topological spaces, we investigate Ä Ê Ì Çȵ, the logic of all region structures induced by topological spaces in which regions are nonempty regular closed sets. On Ê Ò, Ò ½, we investigate the family of logics Ä Ê Ê Ò Í Ò µ where Ê Ò Ö Í Ò Ê Ò ÖØ In particular, we may have Í Ò Ê Ò ÓÒÚ. In many applications, it does not seem natural to enforce the presence of all regions with some characteristics (say, non-empty and regular closed) in every model. Instead, one could include only those regions that are relevant for the application. Thus, given a class Ë of region structures, we are interested in the classes Ë Ëµ of all sub- Ë structures of structures in Ë. Then we write ÄÊ Ëµ as abbreviation of Ä Ê Ë Ëµµ. Going one step further, one could even postulate that the set of relevant regions is finite (but unbounded). Thus we use Ë Ò Ëµ to denote all finite substructures of structures in Ë and write Ä Ò Ê Ëµ for Ä Ê Ë Ò Ëµµ. It is natural to ask for the relationship between the logics just introduced. We start with two examples: first, Ä Ò Ê Ê˵ (and any other logic of spaces with finitely many regions) differs from Ä Ê Ê˵, Ä Ê Ì Çȵ and the Ä Ê Ê Ò Í Ò µ since ÔÔ ÔÔ Ô Ôµ ÔÔ Ô is valid in Ë Ò Ê˵ (it states that there does not exist an infinite ascending ÔÔ-chain). Second, the logic Ä Ê Ê˵ differs from Ä Ê Ì Çȵ and the Ä Ê Ê Ò Í Ò µ since Ù Ô Õµ Ù ÔÔÔ ÔÔÕµ is not valid in ÊË (it states that any two disconnected regions are proper parts of a region). These and some more relationships are summarized in Figure 3. Perhaps most interesting is the fact that Ä Ò Ê Ê˵ and Ä Ê Ê˵ can be regarded as logics of topological spaces, and even of Ê Ò : Theorem 2. For Ò ¼: (i) Ä Ò Ê Ê˵ Ä Ò Ê Ì Çȵ Ä Ò Ê ÊÒ Ê Ò µ Ö (ii) Ä Ê Ê˵ Ä Ë Ê Ì Çȵ ÄË Ê ÊÒ Ê Ò µ Ö

6 Ä Ê ÊÊ Ò Öص Ä Ò Ä Ò Ê ÊÊ ÓÒÚµ Ê Ê ¾ Ê ¾ Öص Ä Ê Ê Ò ¾ Ê ¾ ÓÒÚµ Ä Ò Ê Ê Ê Öص Ä Ê Ê Ò Ê ÓÒÚµ Ê Ê˵ Ä Ò Ê Ì Çȵ Ä Ê Ê Ò Ò Ê Ò Öµ Ä Ò Ä Ê Ê˵ Ä Ë Ê Ì Çȵ Ä Ë Ê Ê Ò Ê Ò Öµ Ä Ê Ì Çȵ Ä Ê Ê Ò Ê Ò Öµ Figure 3. Inclusions between logics. A proof of the theorem and a justification of the inclusions in Figure 3 (in particular the fact that they are proper) can be found in Appendix B. It is out of the scope of this paper to fully complete the picture given by Figure 3. Yet, there are a few more interesting things to be noted. For example, using the same formulas as in the corresponding finite cases, it can be shown that Ä Ê Ê Ò Ê Ò µ Ä ÓÒÚ Ê Ê Ò Ê Ò µ and Ö Ä Ê Ê Ò Ê Ò µ Ä ÖØ Ê Ê Ò Ê Ò Ö µ. But in contrast to the finite case, the converse inclusions do not hold either because the following formula is valid in Ê Ê Ò Ê Ò µ: Ö ÒÓÑ Ô µ ½ 5 Undecidability ½ Ù Ô Ô µµ Ù ÔÔÔ ½ ÔÔÔ ¾ ÔÔÔ µ We now establish the undecidability of all logics whose region classes contain certain region spaces based on Ê Ò. Indeed, an even weaker (but less natural) condition is established in Appendix C. Theorem 3. Let Ê Ê Ò Í µ ¾ Ë ÊË with Ê Ò ÖØ Í, for some Ò ¼. Then Ä Ê Ëµ is undecidable. Ë Thus the logics Ä Ê Ëµ and ÄÊ Ëµ are undecidable, for Ë one of ÊË, Ì ÇÈ, Ê Ê Ò Ê Ò µ, Ö Ê ÊÒ Ê Ò µ, ÓÒÚ and Ê Ê Ò Ê Ò ÖØ µ, with Ò ¼. The proof is by reduction of the domino problem that requires tiling of the first quadrant of the plane to the satisfiability problem. Definition 4. Let Ì À Î µ be a domino system, where Ì is a finite set of tile types and À Î Ì Ì represent the horizontal and vertical matching conditions. We say that tiles the first quadrant of the plane iff there exists a mapping Æ ¾ Ì such that, for all Ü Ýµ ¾ Æ ¾ : Figure 4. Enumerating tile positions. if Ü Ýµ Ø and Ü ½ ݵ Ø ¼, then Ø Ø ¼ µ ¾ À if Ü Ýµ Ø and Ü Ý ½µ Ø ¼, then Ø Ø ¼ µ ¾ Î Such a mapping is called a solution for. For reducing this domino problem to satisfiability in Ä Ê logics, we fix an enumeration of all the tile positions in the first quadrant of the plane as indicated in Figure 4. The function takes positive integers to Æ Æ-positions, i.e. ½µ ¼ ¼µ, ¾µ ½ ¼µ, µ ½ ½µ, etc. Our proof strategy is inspired by [25; 32]. Let Ì À Î µ be a domino system. In the reduction, we use the following propositional letters: for each tile type Ø ¾ Ì, a letter Ô Ø ; propositional letters,, and that are used to mark certain, important regions; propositional letters ÛÐÐ and ÓÓÖ that are used to identify regions corresponding to tiles with positions from the sets ¼ Æ and Æ ¼, respectively. The reduction formula ³ is defined as ÛÐÐ ÓÓÖ ÒØÔÔ ¾ Ù where is the conjunction of a number of formulas. We list these formulas together with some intuitive explanations: 1. ensure that the regions ¾ Ï Å are ordered by the relation ÔÔ (i.e. the union of ØÔÔ and ÒØÔÔ): ÔÓ (1) 2. enforce that the regions Å are discretely ordered by ÒØÔÔ. These regions will correspond to positions of the grid. In order to ensure discreteness, we use sequence of alternating and regions as shown in Figure 5. ØÔÔ µ (2) ØÔÔ µ (3) ØÔÔ µ (4) ØÔÔ µ (5)

7 pos. 1 aaa ½ ¾ pos. 2 pos. 3 Figure 6. Two going right regions. Figure 5. A discrete ordering in the plane. If we are at an region, we can access the region corresponding to the next grid position (w.r.t. the fixed ordering) and to the previous grid position using ³µ ØÔÔ ØÔÔ ³µµ ³µ ØÔÔ ØÔÔ ³µµ 3. we need a way to go right in the grid. To this end, we introduce additional regions satisfying as displayed in Figure 6. For example, Grid cell 2 in the figure is right of Grid cell 1, and Grid cell 4 is right of Grid cell 2. ØÔÔ (6) ØÔÔ µ (7) ÔÓ ØÔÔ ØÔÔ (8) We can go to the right and upper element with Ê ³µ ØÔÔ ØÔÔ ³µµ Í ³µ Ê ³µµ Similarly, we can go to the left and down: Ä ³µ ØÔÔ ØÔÔ ³µµ ³µ Ä ³µµ Considering Formulas (6) to (8), it can be checked that going to the right is a monotone and injective total function (see Appendix C). 4. axiomatize the behavior of tiles on the floor and on the wall to enforce that our going to the right relation brings us to the expected position: ÒØÔÔ ÓÓÖ ÛÐеµ (9) ÛÐÐ ÓÓÖ (10) ÛÐÐ Í ÛÐе (11) ÒØÔÔ ÛÐÐ ÛÐеµ (12) Ê ÛÐе (13) ÛÐÐ Ä (14) 5. finally, we enforce the tiling: Ô Ø Ô Ø ¼µ (15) ØØ¾Ì ØØ ¼ µ¾à ØØ ¼ µ¾î Ô Ø Ê Ô Ø ¼ (16) Ô Ø Í Ô Ø ¼ (17) The main strength of our reduction is that it requires only very limited prerequisites. Indeed, we will show that satisfiability of ³ in any region model implies that has a solution. Thus, to prove undecidability of some logic Ä Ê Ëµ, it suffices to show that ³ is satisfiable in Ë if has a solution. This can be done for each region space Ê Ê Ò Í µ with Ê Ò ÖØ Í and Ò ¼: Lemma 5. Let be a domino system. Then: (i) if the formula ³ is satisfiable in a region model, then the domino system has a solution; (ii) if the domino system has a solution, then the formula ³ is satisfiable in a region model based on Ê Ê Ò Í µ, for each Ò ¼ and each Í with Ê Ò ÖØ Í.

8 Obviously, Theorem 3 is an immediate consequence of this lemma. A proof can be found in Appendix C, where indeed a more general variant of Lemma 5 is proved since the restriction in Point 2 is weakened. 6 Axiomatizability In this section, we show that many of the introduced logics are ½ ½-hard, thus highly undecidable and not even recursively enumerable. We start with some easy positive results and then prove a general negative result. First, we remind the reader of the following consequence of the translation of Ä Ê into Ç Ñ Ê : Proposition 6. If a class Ë of region structures is characterized by a finite set of axioms from Ç Ê, then Ä Ê Ëµ is recursively axiomatizable. Recall that ÊË was defined by first-order axioms. Hence, Ä Ê Ê˵ and any Ä Ê Ëµ with Ë a first-order definable subclass of Ë are recursively enumerable. Actually, using general results on modal logics with names [17] and the fact that ÊË is axiomatized by universal first-order sentences, it is not difficult to provide a finitary axiomatization of Ä Ê Ê˵ using non-standard rules. By Theorem 2, we obtain axiomatizations for ÄÊ Ì Çȵ and ev- Ë ery Ä Ë Ê ÊÒ Ê Ò Ö µ, Ò ¼. We now establish a non-axiomatizability result that applies to many logics Ä Ê Ëµ whose class of region spaces Ë is induced by a class of topological spaces: Theorem 7. The following logics are ½ ½-hard: Ä Ê Ì Çȵ and Ä Ê Ê Ò Í Ò µ with Í Ò ¾ Ê Ò Ö Ê Ò ÓÒÚ Ê Ò ÖØ and Ò ½. To prove this result, the domino problem of Definition 4 is modified by requiring that, in solutions, a distinguished tile Ø ¼ ¾ Ì occurs infinitely often in the first column of the grid. It has been shown in [20] that this variant of the domino problem is ½ ½-hard. Since we reduce it to satisfiability, this yields a ½ ½-hardness bound for validity. As a first step toward reducing this stronger variant of the domino problem, we extend ³ with the following conjunct: ¾ Ù ÒØÔÔ ÛÐÐ Ôؼ µ ÒØÔÔ ÛÐÐ Ô Ø¼ µ ÒØÔÔ ÛÐÐ Ô Ø¼ µ (18) However, this is not yet sufficient: in models of ³, we can have not only one discrete ordering of regions, but rather many stacked such orderings (c.f. Point 5 of Claim 1 in the proof of Lemma 17). Due to this effect, the above formula does not enforce that the main ordering (there is only one for which we can ensure a proper going to the right relation ) has infinitely many occurrences of Ø ¼. It is thus obvious that we have to prevent stacked orderings. This is done by enforcing that there is only one limit region, i.e. only one region approached by an infinite sequence of -regions in the limit. We add the following formula to ³ : ¾ Ù ØÔÔ ÔÓ ØÔÔ ÒØÔÔ µ (19) Let ³ ¼ be the resulting extension of ³. The classes of region spaces to which the extended reduction applies is more restricted than for the original one. We adopt the following property: Definition 8 (Closed under infinite unions). Suppose that Ê Ï Ê Ê is a region space. Then Ê is called closed under infinite unions if Ê Ê Ì Í Ì µ is a region space induced by a topological space Ì, and, additionally, Ê satisfies the following property: for any sequence Ö ½ Ö ¾ ¾ Ï such that Ö ½ ÒØÔÔ Ö ¾ ÒØÔÔ Ö, we have Á ˾ Ö µµ ¾ Ï. We can now formulate the first part of correctness for the extended reduction. The proofs of this and the following lemma can be found in Appendix D. Lemma 9. Let Ê Ì Í Ì µ Ï Ê Ê be a region space that is closed under infinite unions such that all regions in Í Ì are regular closed. Then the formula ³ ¼ is satisfiable in a region model based on Ê only if the domino system has a solution with Ø ¼ occurring infinitely often on the wall. For the second part of correctness, we again consider region spaces Ê Ê Ò Í µ with Ê Ò ÖØ Í. Note that we can not generalize this to a larger class of topological spaces in the same way as in the proof of Point 2 of Lemma 5 (Appendix C). Lemma 10. If the domino system has a solution with Ø ¼ occurring infinitely often on the wall, then the formula ³ ¼ is satisfiable in a region model based on Ê ÊÒ Í µ, for each Ò ½ and each Í with Ê Ò ÖØ Í Ê Ò Ö. Note that the region spaces Ê Ê Ò Ê Ò µ, ÖØ Ê ÊÒ Ê Ò µ ÓÒÚ and Ê Ê Ò Ê Ò µ are closed under infinite unions. Since ÖØ Ê Ò ÖØ Ê Ò ÓÒÚ Ê Ò Ö, Lemmas 9 and 10 immediately yield Theorem 7. It is worth noting that there are a number of interesting region spaces to which this proof method does not apply. Interesting examples are the region space based on simply connected regions in Ê ¾ [33] and the space of polygons in Ê ¾ [28]. Since these spaces are not closed under infinite unions, the above proof does not show the nonaxiomatizability of the induced logics. We conjecture, however, that slight modifications of the proof introduced here can be used to prove their ½ ½-hardness as well.

9 7 Finite Region Spaces We now consider logics of classes of finite region spaces. In this case, we can establish a quite general undecidability result. Moreover, undecidability of such a logic implies that it is not recursively enumerable if it is based on a class of region structures Ë Ò Ëµ with Ë first-order definable. Theorem 11. If Ë Ò Ê Ê Ò Ê Ò ÖØ µµ Ë Ë Ò Ê˵ for some Ò ½, then Ä Ê Ëµ is undecidable. Thus, the following logics are undecidable for each Ò ½: Ä Ò Ä Ò Ê Ê˵, Ê Ì Çȵ, Ä Ò Ê ÊÒ Ê Ò µ, Ö Ä Ò Ê ÊÒ Ê Ò ÓÒÚ µ, and Ä Ò Ê ÊÒ Ê Ò µ. ÖØ To prove this result, we reduce yet another variant of the domino problem. For ¾ Æ, the -triangle is the set µ Æ ¾. The task of the new domino problem is, given a domino system Ì À Î µ, to determine whether tiles an arbitrary -triangle, ¾ Æ, such that the position ¼ ¼µ is occupied with a distinguished tile ¼ ¾ Ì, and some position is occupied with a distinguished tile ¼ ¾ Ì. It is shown in Appendix E that the existence of such a tiling is undecidable. The reduction formula ³ is defined as ÛÐÐ ÓÓÖ ¼ ÒØÔÔ ¾ Ù ¼ ÒØÔÔ ¼ µµ where is the conjunction of the Formulas (1), (3) to (5), and (7) to (17) of Section 5, and the following formulas: The first tile that has no tile to the right is on the floor: Ê ÒØÔÔ µ Ê µ ÓÓÖ (20) If a tile has no tile to the right, then the next tile (if existent) also has no tile to the right: Ê µ Ê µ (21) The last tile is on the wall and we have no stacked orderings: µ ÛÐÐ ÒØÔÔ µµ (22) The proof of the following lemma is now a variation of the proof of Lemma 5. Details are left to the reader. Lemma 12. Let be a domino system. Then: (i) if the formula ³ is satisfiable in a finite region model, then tiles a -triangle as required; (ii) if tiles a -triangle, then ³ is satisfiable in a region model based on a structure from Ë Ò Ê Ê Ò Ê Ò ÖØ µµ, for each Ò ½. Æ Ö ÔÓ ÔÔ ÔÔ Ö Ö,ÔÓ,ÔÔ Ö,ÔÓ,ÔÔ Ö ÔÓ Ö,ÔÓ,ÔÔ, ÔÓ,ÔÔ Ö,ÔÓ,ÔÔ ÔÔ Ö Ö,ÔÓ,ÔÔ ÔÔ ÔÔ Ö,ÔÓ,ÔÔ, ÔÓ,ÔÔ Õ,ÔÓ,ÔÔ,ÔÔ ÔÔ Figure 7. The Ê composition table. Obviously, Theorem 11 is an immediate consequence of Lemma 12. Since ÊË is first-order definable, we can enumerate all finite region models. Thus, Theorems 11 and Theorem 2 give us the following: Corollary 13. The following logics are not r.e., for each Ò ½: Ä Ò Ä Ò Ê Ê˵, Ê Ì Çȵ, Ä Ò Ê ÊÒ Ê Ò µ. Ö 8 The Ê set of Relations For several applications, the Ê relations are weakened into a set of only 5 relations called Ê (or medium resolution topological relations) [18; 9]. This is done by keeping the relation Õ and ÔÓ but coarsening (1) the ØÔÔ and ÒØÔÔ relations into a new proper-part of relation ÔÔ; (2) the ØÔÔ and ÒØÔÔ relations into a new has properpart relation ÔÔ; and (3) the and relations into a new disjointness relation Ö. The modal language Ä Ê for reasoning about Ê-style region structures Ê Ï Ê thus extends propositional logic with the operators Ö, where Ö ranges over the five Ê-relations. They are interpreted by the relations Õ Ê, ÔÓ Ê, and Ö Ê Ê Ê ; ÔÔ Ê ØÔÔ Ê ÒØØÔ Ê ; ÔÔ Ê ØÔÔ Ê ÒØØÔ Ê. Given a class Ë of region structures, we denote by Ä Ê Ëµ the set of Ä Ê -formulas which are valid in all Ë members of Ë. The sets ÄÊ Ëµ and Ä Ò Ê Ëµ are defined analogously to the Ê case. A number of results from our investigation of Ä Ê have obvious analogues for Ä Ê : First, we can characterize the logics ÄÊ Ì Çȵ and Ä Ò Ë Ê Ì Çȵ by means of a composition table: denote by ÊË the class of all structures Ê Ï Ö Ê Õ Ê ÔÔ Ê ÔÔ Ê ÔÓ Ê where Ï is non-empty and the Ö Ê are mutually exclusive and jointly exhaustive binary relations on Ï such that (1) Õ is interpreted as the identity relation on Ï, (2) ÔÓ and Ö are symmetric, (3) ÔÔ is the inverse of ÔÔ and (4) the rules of the Ê-composition table (Figure 7) are valid. Second, it is

10 possible to prove an analogue of Theorem 2, i.e. that, for Ò ½, we have iµ Ä Ò Ê ÊË µ Ä Ò Ê Ì Çȵ Ä Ò Ê ÊÒ Ê Ò Ö µ iiµ Ä Ê ÊË µ Ä Ë Ê Ì Çȵ ÄË Ê ÊÒ Ê Ò Ö µ Third, on region models, Ä Ê has the same expressive power as the two-variable fragment of Ä Ñ, i.e. the firstorder language with the five binary Ê-relation symbols Ê and infinitely many unary predicates. We now investigate the computational properties of logics based on Ä Ê. Analogously to the Ê case, the most natural logics are undecidable. Still, our Ê undecidability result is less powerful than the one for Ê. More precisely, we have to restrict ourselves to region structures with certain properties: denote by ÊË the class of all region structures Ê Ï Ê such that, for any set Ë Ï of cardinality two or three, there exists a unique region ËÙÔ Ëµ such that Õ ËÙÔ Ëµ or ÔÔ ËÙÔ Ëµ for each ¾ Ë; for every region Ø ¾ Ï with ÔÔ Ø for each ¾ Ë, we have ËÙÔ Ëµ Õ Ø or ËÙÔ Ëµ ÔÔ Ø, for every region Ø ¾ Ï with Ø Ö for each ¾ Ë, we have Ø Ö ËÙÔ Ëµ. It is easy to verify that Ì ÇÈ ÊË and Ê Ê Ò Ê Ò Ö µ ¾ ÊË for each Ò ¼. Theorem 14. Suppose Ê Ê Ò Ê Ò Ö µ ¾ Ë ÊË, for some Ò ½. Then Ä Ê Ëµ is undecidable. Thus, the following logics are undecidable, for each Ò ½: Ä Ê Ì Çȵ and Ä Ê Ê Ò Ê Ò Ö µ. The proof is by reduction of the satisfiability problem for the undecidable modal logic Ë (see [24] for the original proof in an algebraic setting. We use the modal notation of [14]). Due to space limitations, we refer the reader to [14] or to Appendix F for a formal definition of Ë, and just recall here that the domain of Ë is a product Ï ½ Ï ¾ Ï, and that there are three modal operators for referring to triples that are identical to the current one, but for one component. With every Ë -formula ³, we associate a Ä Ê - formula ¾ Ù ³ ( ) such that ³ is Ë -satisfiable iff ¾ Ù ³ is satisfiable in a model from Ë. In ( ), is the conjunction of the following formulas: 1. Each sets Ï of Ë -models is simulated by the set Ö ¾ Ï Å Ö. Thus, we introduce fresh variables, ½ ¾, and state ½¾ ÔÔ ÔÔ ÔÓ µ (23) ½ ¾ ¾ ¾ (24) ½¾ Ù (25) 2. the set Ï ½ Ï ¾ Ï is simulated by a fresh variable, so we add ÔÔ µ ÔÔ ½¾ ½¾ ÔÔ µ (26) 3. the sets Ï Ï, ½ are simulated by fresh variables, so we add ÔÔ µ ÔÔ Now, we define ³ inductively by ÔÔ µ (27) Ô Ô ³µ ³ ³ µ ³ ½ ³µ ÔÔ ¾ ÔÔ ³ µµ ¾ ³µ ÔÔ ½ ÔÔ ³ µµ ³µ ÔÔ ½¾ ÔÔ ³ µµ The following Lemma immediately yields Theorem 14 and is proved in Appendix F. Lemma 15. Suppose Ê Ê Ò Ê Ò Ö µ ¾ Ë ÊË, for some Ò ½. Then an Ë -formula ³ is satisfiable in an Ë - model iff ¾ Ù ³ is satisfiable in Ë. 9 Conclusion We first compare our results with Halpern and Shoham s on interval temporal logic [19]: Theorems 3, 7, and 11 apply to logics induced by the region space Ê Ê Ê ÓÒÚ µ, which is clearly an interval structure. Interestingly, on this interval structure our results are stronger than those of Halpern and Shoham in two respects: first, we only need the Ê relations, which can be viewed as a coarsening of the Allen interval relations used by Halpern and Shoham. Second and more interestingly, by Theorem 3 we have also proved undecidability of the substructure logic Ä Ë Ê Ê Ê ÓÒÚµ, which is a natural but much weaker variant of the full (interval temporal) logic Ä Ê Ê Ê ÓÒÚ µ, and not captured by Halpern and Shoham s undecidability proof. Several open questions for future research remain. Similar to the temporal case, the main challenge is to exhibit a decidable and still useful variant of the logics proposed in this paper. Perhaps the most interesting candidate is Ä Ê Ê˵, which coincides with the substructure logics

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12 A Expressive Completeness The proof of the following theorem is an adaption of the proof in [13], and a minor variant of the proof in [23] that is provided here for convenience. Throughout this section, we use ¾Ç Ñ Ê to denote the two-variable fragment of Ç Ñ and assume that its two variables are called Ü Ê and Ý. Theorem 16. For every Ç Ñ -formula ³ ܵ with free Ê variable Ü that uses only two variables, one can effectively construct a Ä Ê -formula ³ of length at most exponential in the length of ³ ܵ such that, for every region model Å and any region, Å ³ iff Å ³ Proof. A formula is called a unary atom if it is of the form Ê Ü Üµ, Ê Ý Ýµ, ܵ, or ݵ. A ¾Ç Ñ -formula Ê Ü Ýµ is called a binary atom if it is an atom of the form Ö Ü Ýµ, Ö Ý Üµ, or Ü Ý.. We assume ³ ܵ is built using Let ³ ܵ ¾ ¾Ç Ñ Ê,, and only. We inductively define two mappings Ü and Ý where the former one takes each ¾Ç Ñ -formula Ê ³ ܵ to the corresponding Ä Ê -formula ³ Ü, and the latter does the same for ¾Ç Ñ Ê-formulas ³ ݵ. We only give the details of Ü since Ý is defined analogously by switching the roles of Ü and Ý.: If ³ ܵ Ô Üµ, then put ³ ܵµ Ü Ô. If ³ ܵ Ö Ü Üµ, then put ³ ܵµ Ü if Ö Õ, and ³ ܵµ Ü otherwise. If ³ ܵ ½ ¾, then put ³ ܵµ Ü Ü ½ If ³ ܵ, then put ³ ܵµ Ü µ Ü. Ü ¾. If ³ ܵ Ý Ü Ýµ, then Ü Ýµ can be written as Ü Ýµ ½ Ö ½ ܵ Рܵ ½ ݵ ݵ i.e. as a Boolean combination of, ܵ, and ݵ; the are binary atoms; the ܵ are unary atoms or of the form Ý ¼; and the ݵ are unary atoms or of the form Ü ¼. We may assume that Ü occurs free in ³ ܵ. Our first step is to move all formulas without a free variable Ý out of the scope of : obviously, ³ ܵ is equivalent to Û ½Û ¾ ½ Û µ Ý ½ Ö Û ½ Û Ð ½ µµ Now we guess a relation Ö that holds between Ü and Ý, and then replace all binary atoms by either true or false. For Ö a topological relation and ½ Ö, let Ö if Ö Ü Ýµ; Ö if Ö Ý Üµ for Ö ¾ ÔÓ; Ö if ØÔÔ Ý Üµ and Ö ØÔÔ or ÒØÔÔ Ý Üµ and Ö ÒØÔÔ; Ö if is Ü Ý and Ö Õ; Ö otherwise. Using this notiation, our last formula is equivalent to Ï Û ½Û ¾ ν Û µ Ï Ö¾Ê Ý Ö Ü Ýµ Now compute, recursively, Ü Ø ½ Ø Ö Û ½ Û Ð ½ µµµ as Ï Û½Û¾ ν Ü Û µ and Ý, and define ³ ܵ ÏÖ¾ÊÖ Ø ½ Ø Ö Û ½ Û Ð Ý ½ Ý µµ Now for the succinctness of ¾Ç Ñ. In [13], Etessami, Ê Vardi, and Wilke show that, on infinite words, the twovariable fragment of first-order logic with binary predicates successor and as well as an infinite number of unary predicates (called ¾Ç Ò in the following) have the same expressive power as temporal logic. Here, temporal logic refers to the variant with operators next, previously, sometime in the future, and sometime in the past, but without until and since. Etessami et al. also show that the sequence of ³ Ò µ Ò½ of first-order sentences with two variables defined by ³ Ò ÜÝ Ô Üµ Ô Ýµµ Ô Ò Üµ Ô Ò Ýµµ Ò is such that the shortest temporal logic formulas equivalent to ³ Ò have size ¾ Å Òµ. Intuitively, this formula states that any two points agreeing on Ô ¼ Ô Ò ½ also agree on Ô Ò. Since the above formula does not involve the successor and relations, it is not hard to prove that this result carries over to our case: the ¾Ç Ñ Ê -formulas ³ Òµ Ò½ above are such that the shortest Ä Ê formulas equivalent to ³ Ò have size ¾ Å Òµ. B Logics Theorem 2. For Ò ¼: (i) Ä Ò Ê Ê˵ Ä Ò Ê Ì Çȵ Ä Ò Ê ÊÒ Ê Ò µ Ö (ii) Ä Ê Ê˵ Ä Ë Ê Ì Çȵ ÄË Ê ÊÒ Ê Ò µ Ö Proof. (i) follows from [6; 30], where it is proved that Ë Ò Ê˵ Ë Ò Ì Çȵ Ë Ò Ê Ê Ò Ê Ò Ö µµ We now prove (ii) with the help of (i) and the proof of [14] Theorem Obviously, ÊË Ë Ì Çȵ Ë Ê Ê Ò Ê Ò Ö µµ

13 Hence, by Löwenheim-Skolem it is sufficient to show that every countable region space Ê Ï Ê is isomorphic to some substructure of Ê Ê Ò Ê Ò µ. However, it is Ö proved in [14] Theorem that every at most countable set of Ê-constraints of the form Ü Ö Ýµ, Ö an Ê relation, is satisfiable in Ê Ê Ò Ê Ò µ provided that every Ö finite subset of is satisfiable in Ë Ì Çȵ. Now our claim follows immediately with the help of (i). We now provide formulas which prove the inequalities of Figure 3 which were not yet considered. Since Ê constraints can be expressed in Ä Ê, we can use constraints Ü Ö Ýµ, where Ö is an Ê-relation and Ü Ý are individual variables for regions. C Ä Ò Ê ÊÒ ½ Ê Ò ½ ÖØ µ Ä Ò Ê ÊÒ Ê Ò ÖØ µ, for all Ò ¼: let, for a set of Ò distinct individual variables Ü ½ Ü Ò, Ò Ü Ü µ ½ Ò Then ¾ Ò is satisfiable in Ê Ê Ò Ê Ò ÖØ µ, but ¾ Ò ½ is not satisfiable in Ê Ê Ò Ê Ò µ. ÖØ Ä Ò Ê ÊÒ Ê Ò µ Ä Ò ÖØ Ê ÊÑ Ê Ñ ÓÒÚ µ, for all Ñ Ò ¾: this follows from the observation that all Ò, Ò, are satifisable in Ê Ê Ò Ê Ò ÓÒÚ µ, Ò ¾. Ê Ê˵, for all Ò ¼. Iden- Ä Ò Ê ÊÒ Ê Ò ÖØ tical to previous case. Ä Ò Ê Ê Ê µ ÓÒÚ µ Ä Ò Ä Ò Ê Ê¾ Ê ¾ ÓÒÚ µ: take variables Ü, ½. Then the union of, and Ü ÔÔ Ü µ Ü ÔÔ Ü µ ½ Ü Ü µ ½ ¾ ½ ¾ is satisfiable in Ê Ê Ê ÓÒÚ µ but not in Ê Ê¾ Ê ¾ ÓÒÚ µ. Ä Ê Ê Ò Ê Ò µ Ä Ö Ê Ì Çȵ, for all Ò ¼: ÔÔ is valid in Ê Ê Ò Ê Ò µ, but not in Ì ÇÈ. Ö Undecidability of Ê logics To ease notation, throughout the appendix we denote accessibility relations in models simply with,, etc., instead of with Ê, Ê, etc. The purpose of this section is to prove Lemma 5. Indeed, we prove the two stated Points as independent lemmas and, as announced in Section 5, even establish a stronger variant of the second Point. Lemma 17. If the formula ³ is satisfiable in a region model, then the domino system has a solution. Proof. Let Å Ê Ô Å ½ ÔÅ ¾ be a region model of ³ with Ê Ï. Claim 1. There exists a sequence Ö ½ Ö ¾ ¾ Ï such that 1. Å Ö ½ ³, 2. Ö ½ ÒØÔÔ Ö ¾ ÒØÔÔ Ö ÒØÔÔ, 3. Å Ö for ½. 4. for each ½, there exists a region ¾ Ï such that (a) Ö ØÔÔ, (b) Å, (c) ØÔÔ Ö ½, (d) for each region with Ö ØÔÔ and Å, we have, and (e) for each region Ö with ØÔÔ Ö and Å Ö, we have Ö Ö ½, 5. for all Ö ¾ Ï with Å Ö, we have that Ö Ö for some ½ or Ö ÒØÔÔ Ö for all ½. Proof: Points 1 to 4 of this claim can be proved using a simple induction. We only do the induction start since the induction step is identical. Since Å is a model of ³, there is a region Ö ½ such that Å Ö ½ ³. By definition of ³, Point 3 is satisfied. Due to Formulas (2) and (3), there are regions ½ and Ö ¾ such that Ö ½ ØÔÔ ½, Å ½, ½ ØÔÔ Ö ¾, and Å Ö ¾. We show that all necessary Properties are satisfied: Point 2. Since Ö ½ ØÔÔ ½ and ½ ØÔÔ Ö ¾, we have Ö ½ ØÔÔ Ö ¾ or Ö ½ ÒØÔÔ Ö ¾ according to the composition table. But then, the first possibility is ruled out by Formula (5). Point 4d. Suppose there is an ½ with Ö ½ ØÔÔ and Å. Since Ö ½ ØÔÔ ½, ½ and are related via one of ÔÓ, ØÔÔ, and ØÔÔ by the composition table. But then, the first option is ruled out by Formula (1) and the last two by Formula (4). Point 4e. Analogous to the previous case. This finishes the induction, and it thus remains to prove Point 5. Assume that there is a region Ö such that Å Ö, Ö Ö for all ½, and Ö ÒØÔÔ Ö does not hold for some ½. Since Ö ÒØÔÔ Ö does not hold and Ö Ö, Ö and Ö are related by one of,, ÔÓ, ØÔÔ, ØÔÔ, and ÒØÔÔ. The first three possibilities are ruled out by Formula (1), and ØÔÔ and ØÔÔ are ruled out by Formula (5). It thus remains to treat the case Ö ÒØÔÔ Ö. Consider the relationship between Ö ½ and Ö. Since Ö ½ Ö and due to Formulas (1) and (5), there are only two possibilities for this relation;

14 Ö ½ ÒØÔÔ Ö. Impossible by ³ s subformula ÒØÔÔ. Ö ÒØÔÔ Ö ½. Then we have Ö ½ ÒØÔÔ Ö ÒØÔÔ Ö. Take the maximal such that Ö ÒØÔÔ Ö and the minimal such that Ö ÒØÔÔ Ö. Since Ö Ö Ò for all Ò ½, we have ½. By Point 4, there thus is a region with Ö ÒØÔÔ, Å, and ÒØÔÔ Ö. Clearly, we have ÔÓ Ö which is a contradiction to Formula (1). Claim 2. For each ½, there exist regions Ø and Ù such that 1. Ö ØÔÔ Ø, 2. Å Ø, 3. for each region Ø with Ö ØÔÔ Ø and Å Ø, we have Ø Ø, 4. Ø ØÔÔ Ù, 5. Å Ù, 6. for each region Ù with Ø ØÔÔ Ù and Å Ö, we have Ù Ù. Proof: Let ½. By Formula (6), there is a Ø with Ö ØÔÔ Ø and Å Ø. Let us show that Ø satisfies Property 3. To this end, let Ø Ø such that Ö ØÔÔ Ø and Å Ø. Then Ø and Ø are related via one of ÔÓ, ØÔÔ, and ØÔÔ. But then, all these options are ruled out by Formula (8). Now for Points 4 to 6. By Formula (7), there is an Ö such that Ø ØÔÔ Ö and Å Ö. Point 6 can now be be proved analogously to Point 3, using Formulas (1) and (4) instead of Formula (8). This finishes the proof of Claim 2. Before proceeding, let us introduce some notation. for ¼, we write µ if the tile position µ can be reached from µ by going one step to the right. Similarly, we define a relation for going one step up; for ¼ we write Ö Ö if Ù Ö. Similarly, we write Ö Ö if Ö Ö ½. Clearly, the and relations are partial functions by Claims 1 and 2. The following claim establishes some other important properties of : first, it may only move ahead in the sequence Ö ½ Ö ¾, but never back. And second, it is monotone and injective. Claim 3. Let ½. Then the following holds: 1. if Ö Ö, then ; 2. if, Ö Ö, and Ö Ö, then. Proof: First for Point 1. Suppose Ö Ö and. Then Ù Ö and, by Claim 2, Ö ØÔÔ Ø ØÔÔ Ö, which is clearly impossible: the composition table then yields that Ö is related to itself via ØÔÔ or ÒØÔÔ, in contrast to the fact that Ö Õ Ö, and the relations are mutually disjoint. Now suppose Ö Ö and. Then Ù Ö and Claim 2 yields Ö ØÔÔ Ø ØÔÔ Ö. Since, Claim 1 gives us Ö ÒØÔÔ Ö : a contradiction. Now for Point 2. Assume Ö Ö, Ö Ö, and. This means that Ù Ù Ö. By Claim 2, we have ØÔÔ Ù and ØÔÔ Ù. Thus, or and are related by one of ÔÓ, ØÔÔ, and ØÔÔ. The last three possibilities are ruled out by Formula (8). Thus we get. This, however, is a contradiction to the facts that, and, by Claims 1 and 2, Ö ÒØÔÔ Ö, Ö ØÔÔ, and Ö ØÔÔ. Now assume Ö Ö, Ö Ö, and. By Claim 1, we have Ö ÒØÔÔ Ö. By Claim 2, we have Ö ØÔÔ Ø and Ö ØÔÔ Ø. It is easily verified that Ø and Ø are thus related by one of, ÔÓ, ØÔÔ, and ÒØÔÔ. All possibilities but ÒØÔÔ are ruled out by Formula (8), and hence Ø ÒØÔÔ Ø. We now make another derivation for the relationship between Ø and Ø, and, in this way, obtain a contradiction. Since Ö Ö and Ö Ö, we have Ù Ö and Ù Ö. By Claim 2, we thus get Ø ØÔÔ Ö and Ø ØÔÔ Ö. By Claim 1 and since, we have Ö ÒØÔÔ Ö. Thus, we obtain that Ø and Ø are related by one of ÒØÔÔ, ØÔÔ, and ÔÓ. This is a contradiction to the previously derived Ø ÒØÔÔ Ø, thus finishing the proof of Claim 3. The following lemma establishes the core part of the proof: the fact that the relation coincides with the µ relation. More precisely, this follows from Point 3 of the following claim. For technical reasons, we simultaneously prove some other, technical properties. The remainder closely follows the lines of Marx and Reynolds [25]. Claim 4. Let ½ and µ. Then the following holds: 1. if µ is on the floor, then Å Ö ÓÓÖ; 2. Å Ö ÛÐÐ; 3. Ö Ö and Ö Ö ½. 4. if ½µ is on the wall, then Å Ö ½ ÛÐÐ Proof: All subclaims are proved simultaneously by induction on. First for the induction start. Then we have ½ and ¾. 1. Clearly, ¾µ is on the floor. Since Å Ö ½ ³, we have Å Ö ½ ÛÐÐ. Thus Formula (10) yields Å Ö ¾ ÓÓÖ. 2. We have ½ µ ¾. Point 1 gives us Å Ö ¾ ÓÓÖ. Since Ö ½ ÒØÔÔ Ö ¾, we also have Å Ö ¾ ÒØÔÔ. Thus, Formula (9) yields Å Ö ¾ ÛÐÐ.