Affine-invariant querying of spatial data using a triangle-based logic

Abstract

If the same land area is photographed from two different angles, then a computer can recognize the two images to show the same land area by proving that the images are affine-invariant. In practice, images are abstracted as a type of spatial data that is a finite set of triangles. Hence we propose a simple affine-generic query language with variables over finite sets of triangles. The proposed query language is intuitive and has the same expressive power as the affine-generic fragment of first-order logic over the reals on triangle databases.

Appendix

We collect below some of the technical proofs, starting with the proof of the soundness lemma.

Proof of Lemma 4.2. Let $\hat {\sigma } = \{\hat {R}_{1}, \hat {R}_{2}, \ldots , \hat {R}_{m}\}$ be a triangle database schema. Let $\dot {R}_{i}$ be the corresponding spatial point relation names of arity $3 \cdot ar(\hat {R}_{i})$, for 1 ≤ i ≤ m, and let $\dot {\sigma }$ be the corresponding point database schema $\{\dot {R}_{1}, \dot {R}_{2}, \ldots , \dot {R}_{m}\}$. We translate each formula $\hat {\varphi }$ of FO({PartOf}, $\hat {\sigma })$ into an equivalent formula in FO{Between}, $\dot {\sigma }$. We do this by induction on the structure of FO({PartOf}, $\hat {\sigma })$-formulas.

First, we translate the variables of $\hat {\varphi }$. Each triangle variable △ is naturally translated into three spatial point variables p₁, p₂ and p₃. We allow one or more of the corner points of a triangle to coincide, so there are no further restrictions on the variables p_j, for 1 ≤ i ≤ 3.

The atomic formulas of FO({PartOf}, $\hat {\sigma })$ are equality expressions on triangle variables, expressions of the form PartOf(△₁,△₂), and expressions of the form ${\hat {R}_{i}}(\bigtriangleup _{1}, \bigtriangleup _{2}, \ldots , \bigtriangleup _{k_{i}})$, where $k_{i} = ar(\hat {R}_{i})$, for 1 ≤ i ≤ m. More complex formulas can be constructed using the Boolean operators ∧, ∨ and ¬ and existential quantification.

The translation of atomic formulas.

We first show that all atomic formulas of FO({PartOf}, $\hat {\sigma })$ can be expressed in the language FO{Between}, $\dot {\sigma }$.

• (i) The translation of (△₁ = △₂), where △₁ is translated into p_1,1, p_1,2 and p_1,3 and △₂ is translated into p_2,1, p_2,2 and p_2,3, equals

  $$ \displaylines{\qquad \bigvee_{\sigma(1,2,3) = (j_{1}, j_{2}, j_{3}), \sigma \in {\mathcal{S}}_{3}} (p_{1,1} \mathrel{=} p_{2,{j_{1}}}\land p_{1,2} \mathrel{=} p_{2,{j_{2}}}\land p_{1,3} \mathrel{=} p_{2,{j_{3}}}),\qquad} $$

  where ${\mathcal {S}}_{3}$ is the set of all permutations of {1, 2, 3}. This formula expresses, that the corner points of △₁ and △₂ coincide, when taken in the right order. The correctness of this translation follows trivially from the definition of triangle equality (see Definition 4.2).

• (ii) The translation of PartOF(△₁△₂), where △₁ is translated into p_1,1, p_1,2 and p_1,3 and △₂ is translated into p_2,1, p_2,2 and p_2,3, is

  $$ \displaylines{\qquad\bigwedge\limits_{i=1}^{3} \textbf{InTriangle}(p_{1,i}, p_{2,1}, p_{2,2}, p_{2,3}),\qquad} $$

  where the definition of InTriangle is:

  $$ \textbf{InTriangle}(p, p_{1}, p_{2}, p_{3}) := \exists p_{4}(\textbf{Between}({p_{1}}{p_{4}}{p_{2}}) \mathrel{\land} \textbf{Between}({p_{4}}{p}{p_{3}})). $$

  Figure 9 illustrates the corresponding geometric construction. The correctness of this translation follows from the definition of the predicate PartOF (see Definition 4.1).

• (iii) The translation of ${\hat {R}}_{i}(\bigtriangleup _{1}, \bigtriangleup _{2}, \ldots , \bigtriangleup _{k})$, where △_i is translated into p_i,1, p_i,2 and p_i,3 for 1 ≤ i ≤ k, is

  $$ \dot{R}_{i}(p_{1,1}, p_{1,2}, p_{1,3}, p_{2,1}, p_{2,2}, p_{2,3}, \ldots, p_{k,1}, p_{k,2}, p_{k,3}). $$

  The correctness of this translation follows from Definition 3.2 and Remark 3.1.

Fig. 9

In this illustration, the expression InTriangle(p, p₁, p₂, p₃) is true because there exists a point p₄ between p₁ and p₂ such that p lies between p₄ and p₃

The translation of composed formulas.

Assume that we already correctly translated the FO$(\{\textbf {PartOf}\}, \hat {\sigma })$-formulas $\hat {\varphi }$ and $\hat {\psi }$ into the FO({Between}, $\dot {\sigma }$)-formulas $\dot {\varphi }$ and $\dot {\psi }$. Suppose that the number of free variables in $\hat {\varphi }$ is k_φ and that of $\hat {\psi }$ is k_ψ. Therefore, we can assume that, for each triangle database $\hat {\mathcal {D}}$ over the input schema $\hat {\sigma }$, and for each k_φ-tuple of triangles (T₁, T₂,…,Tk_φ) given as $(({\mathsf {p}}_{1,1}, {\mathsf {p}}_{1,2}, {\mathsf {p}}_{1,3}), ({\mathsf {p}}_{2,1}, {\mathsf {p}}_{2,2}, {\mathsf {p}}_{2,3}), \ldots , ({\mathsf {p}}_{k_{\varphi },1}, {\mathsf {p}}_{k_{\varphi },2}, {\mathsf {p}}_{k_{\varphi },3}))$ that

$$ \begin{array}{@{}rcl@{}} \hat{\mathcal{D}}\models\hat{\varphi}[ \mathsf{T}_{1}, \mathsf{T}_{2}, \ldots, \mathsf{T}{k_{\varphi}}] \text{ if and only if }\\ \dot{\mathcal{D}} \models\dot{\varphi}[{\mathsf{p}}_{1,1}, {\mathsf{p}}_{1,2}, {\mathsf{p}}_{1,3}, {\mathsf{p}}_{2,1}, {\mathsf{p}}_{2,2}, {\mathsf{p}}_{2,3}, \ldots, {\mathsf{p}}_{k_{\varphi},1}, {\mathsf{p}}_{k_{\varphi},2}, {\mathsf{p}}_{k_{\varphi},3}] \end{array} $$

is true when $\dot {\mathcal {D}}$ is the point database over the input schema $\dot {\sigma }$, obtained from $\hat {\mathcal {D}}$ by applying the canonical bijection can_tr between $((\mathbb {R}^{2})^{3})^{k_{\varphi }}$ and $(\mathbb {R}^{2})^{3k_{\varphi }}$, on $\hat {\mathcal {D}}$. For the formula $\hat {\psi }$ the analog holds.

In the following, we omit the k_φ-tuples (resp., k_ψ-tuples) of triangles and 3k_φ-tuples (resp., 3k_ψ-tuples) of points on which the formulas are applied, to make the proofs more readable.

• (i) The translation of $\hat {\varphi } \mathrel {\land } \hat {\psi }$ is $\dot {\varphi }\mathrel {\land }\dot {\psi }$. Indeed,

  $$ \begin{array}{@{}rcl@{}} &\dot{\mathcal{D}} \models (\dot{\varphi}\mathrel{\land}\dot{\psi}) \\ iff. & \dot{\mathcal{D}} \models \dot{\varphi}\ and\ \dot{\mathcal{D}} \models\dot{\psi} \\ iff. & \hat{\mathcal{D}}\models \hat{\varphi}\ and\ \hat{\mathcal{D}}\models\hat{\psi}\\ iff. & \hat{\mathcal{D}}\models (\hat{\varphi}\mathrel{\land}\hat{\psi}). \end{array} $$

• (ii) The translation of $\hat {\varphi } \mathrel {\lor } \hat {\psi }$ is $\dot {\varphi }\mathrel {\lor }\dot {\psi }$. Indeed,

  $$ \begin{array}{@{}rcl@{}} &\dot{\mathcal{D}} \models (\dot{\varphi}\mathrel{\lor}\dot{\psi}) \\ iff. &\dot{\mathcal{D}} \models \dot{\varphi}\ or\ \dot{\mathcal{D}} \models\dot{\psi}\\ iff. & \hat{\mathcal{D}}\models \hat{\varphi}\ or\ \hat{\mathcal{D}}\models\hat{\psi} \\ iff. & \hat{\mathcal{D}}\models (\hat{\varphi}\mathrel{\lor}\hat{\psi}). \end{array} $$

• (iii) The translation of $\neg \hat {\varphi }$ is $\neg \dot {\varphi }$. Indeed,

  $$ \begin{array}{@{}rcl@{}} & \dot{\mathcal{D}} \models \neg\dot{\varphi} \\ iff. & \text{ it is not true that } \dot{\mathcal{D}} \models \dot{\varphi}\\ iff. & \text{ it is not true that } \hat{\mathcal{D}}\models \hat{\varphi}\\ iff. & \hat{\mathcal{D}}\models \neg\hat{\varphi}. \end{array} $$

• (iv) Assume that $\hat {\varphi }$ has free variables △,△₁,…,△_k and △ is translated into p₁, p₂, p₃ and △_i is translated into p_i,1, p_i,2, p_i,3, for 1 ≤ i ≤ k. The translation of

  $$\exists\bigtriangleup \hat{\varphi}(\bigtriangleup, \bigtriangleup_{1}, \bigtriangleup_{2}, \ldots, \bigtriangleup_{k})$$
  $$\text{is }\exists p_{1} \exists p_{2} \exists p_{3} \dot{\varphi}(p_{1}, p_{2}, p_{3}, p_{1,1}, p_{1,2}, p_{1,3}, \ldots, p_{k,1}, p_{k,2}, p_{k,3}).$$

  Indeed,

  $$ \begin{array}{@{}rcl@{}} & \dot{\mathcal{D}} \models \exists p_{1} \exists p_{2} \exists p_{3} \dot{\varphi}(p_{1}, p_{2}, p_{3})[{\mathsf{p}}_{1,1}, {\mathsf{p}}_{1,2}, {\mathsf{p}}_{1,3}, \ldots, {\mathsf{p}}_{k,1}, {\mathsf{p}}_{k,2}, {\mathsf{p}}_{k,3}] \\ iff. & \text{ there exist points } {\mathsf{p}}_{1}, {\mathsf{p}}_{2}, {\mathsf{p}}_{3}\ in\ \mathbb{R}^{2} \text{ such that } \\ & \dot{\mathcal{D}} \models \dot{\varphi}[{\mathsf{p}}_{1}, {\mathsf{p}}_{2}, {\mathsf{p}}_{3}, {\mathsf{p}}_{1,1}, {\mathsf{p}}_{1,2}, {\mathsf{p}}_{1,3}, \ldots, {\mathsf{p}}_{k,1}, {\mathsf{p}}_{k,2}, {\mathsf{p}}_{k,3}]\\ iff. & \text{ there exists a triangle } \mathsf{T}{} \text{ such that } \hat{\mathcal{D}}\models \hat{\varphi}[\mathsf{T}{}, \mathsf{T}_{1}, \ldots, \mathsf{T}{k}], \text{ where } \\ & \mathsf{T}{i} \text{is the triangle with corner points}\ {\mathsf{p}}_{i,1}, {\mathsf{p}}_{i,2}\ \text{and}\ {\mathsf{p}}_{i,3} for 1 \leq i \leq k\\ iff. & \hat{\mathcal{D}}\models \exists\mathsf{T}{} \hat{\varphi}(\mathsf{T}{})[ \mathsf{T}_{1}, \ldots, \mathsf{T}{k}]. \end{array} $$

To summarize the proof strategy, let $\hat {\sigma } = \{\hat {R}_{1}, \hat {R}_{2}, \ldots , \hat {R}_{m}\}$ be a triangle database schema. Let $\dot {\sigma } = \{\dot {R}_{1}, \dot {R}_{2}, \ldots , \dot {R}_{m}\}$ be the corresponding point database schema. Each formula $\hat {\varphi }$ in FO({PartOf}, $\hat {\sigma })$, with free variables △₁,△₂,…,△_k can be translated into a FO{Between}, $\dot {\sigma }$-formula $\dot {\varphi }$ with free variables p₁, p₂, p₃, p_1,1, p_1,2, p_1,3, p_2,1, p_2,2, p_2,3,…,p_k,1, p_k,2, p_k,3. This translation is such that, for all triangle databases $\hat {\mathcal {D}}$ over $\hat {\sigma }$, $\hat {\mathcal {D}}\models \hat {\varphi }$ iff. $\dot {\mathcal {D}} \models \dot {\varphi }$. Here, $\dot {\mathcal {D}}$ is the point database over $\dot {\sigma }$ which is the image of $\hat {\mathcal {D}}$ under the canonical bijection between $(\mathbb {R}^{2})^{3})^{k}$ and $(\mathbb {R}^{2})^{3k}$. This completes the soundness proof.

Next, we give the proof of the completeness lemma.

Proof of Lemma 4.3. Let $\hat {\sigma } = \{\hat {R}_{1}, \hat {R}_{2}, \ldots , \hat {R}_{m}\}$ be a triangle database schema and $\dot {\sigma }$ be the corresponding point database schema. We have to prove that we can translate every triangle database query, expressed in the language FO{Between}, $\dot {\sigma }$, into a triangle database query in the language FO({PartOf}, $\hat {\sigma })$ over triangle databases.

We first show how we can simulate point variables by a degenerated triangle, and any FO{Between}, $\dot {\sigma }$-formula $\dot {\varphi }(p_{1}, p_{2}, \ldots , p_{k})$ by a formula φ(△₁,△₂,…,△_k), where △₁,△₂,…,△_k represent triangles that are degenerated into points. We prove this by induction on the structure of FO{Between}, $\dot {\sigma }$-formulas. Initially, each FO{Between}, $\dot {\sigma }$-formula $\dot {\varphi }(p_{1}, p_{2}, \ldots , p_{k})$ will be translated into a FO({PartOf}, $\hat {\sigma })$-formula $\hat {\varphi }(\bigtriangleup _{1}, \bigtriangleup _{2}, \ldots , \bigtriangleup _{k})$ with the same number of free variables.

The translation of a point variable p is the triangle variable △, and we add the condition Point△ as a conjunct to the beginning of the translation of the formula. The definition of Point△ is

$$ \forall \bigtriangleup{^{\prime}} (\textbf{PartOf}({\bigtriangleup{^{\prime}}, \bigtriangleup}\rightarrow (\bigtriangleup \mathrel= \bigtriangleup{^{\prime}})). $$

In the following, we always assume that such formulas Point△ are already added to the translation as a conjunct.

The translation of atomic formulas.

The atomic formulas of the language FO({Between}, $\dot {\sigma }$) are equality constraints on point variables, formulas of the form Between(p₁, p₂, p₃), and formulas of the type $\dot {R}_{i}(p_{1}, p_{2}, \ldots , p_{k_{i}})$, where $k_{i} = 3 \cdot ar(\hat {R}_{i})$. We show that all of those can be simulated by an equivalent formula of FO({PartOf}, $\hat {\sigma })$.

• (i) The translation of (p₁ = p₂) is $(\bigtriangleup _{1} =_{\vartriangle } \bigtriangleup _{2})$.

• (ii) The translation of Between(p₁p₂p₃), where △₁, △₂ and △₃ (which as assumed are already declared points) are the translations of p₁, p₂ and p₃, respectively, is expressed by saying that all triangles that contain both △₁ and △₃ should also contain △₂. It then follows from the convexity of triangles (or line segments, in the degenerated case) that △₂ lies on the line segment between △₁ and △₃. Figure 10 illustrates this principle. We now give the formula translating Between(p₁p₂p₃):

  $$ \forall \bigtriangleup_{4} ((\textbf{PartOf}({\bigtriangleup_{1}}{\bigtriangleup_{4}})\land \textbf{PartOf}({\bigtriangleup_{3}}{\bigtriangleup_{4}}) \rightarrow \textbf{PartOf}({\bigtriangleup_{2}}{\bigtriangleup_{4}})). $$

  The correctness of this translation follows from the fact that triangles are convex objects.

• (iii) Let $\dot {R}_{j}$ be a relation name from $\dot {\sigma } = \{\dot {R}_{1}, \dot {R}_{2}, \ldots , \dot {R}_{m}\}$. Let $ar(\hat {R}_{j}) = k_{j}$ and thus $ar(\dot {R}_{j}) = 3k_{j}$, for 1 ≤ j ≤ m. The translation of $\dot {R}_{j}(p_{1,1}, p_{1,2}, p_{1,3}, p_{2,1}, p_{2,2}, p_{2,3}, \ldots , p_{k,1}, p_{k,2}, p_{k,3})$ is:

  $$ \exists\bigtriangleup_{1}\exists\bigtriangleup_{2}\ldots\exists\bigtriangleup_{k}({\hat{R}_{j}}(\bigtriangleup_{1}, \bigtriangleup_{2}, \ldots, \bigtriangleup_{k}) \mathrel{\land} \bigwedge\limits_{i=1}^{k} \textbf{CornerP}(\bigtriangleup_{i,1}, \bigtriangleup_{i,2}, \bigtriangleup_{i,3}, \bigtriangleup_{i})). $$

  The definition of CornerP is:

  $$ \begin{array}{@{}rcl@{}} \textbf{CornerP}(\bigtriangleup_{1}, \bigtriangleup_{2}, \bigtriangleup_{3}, \bigtriangleup):= \forall \bigtriangleup_{4} ((\textbf{Point}({\bigtriangleup_{4}}) \land \textbf{PartOf}({\bigtriangleup_{4}},{\bigtriangleup})\\ \rightarrow \text{Intriangle}_{\vartriangle}(\bigtriangleup_{4}, \bigtriangleup_{1}, \bigtriangleup_{2}, \bigtriangleup_{3})). \end{array} $$

  The predicate $\textbf {InTriangle}_{\vartriangle }$ is the translation of the predicate InTriangle of the language FO{Between} as described in the proof of Lemma 4.2, into FO({PartOF}). The FO{Between} formula expressing InTriangle only uses Between. In the previous item of this proof, we already showed how this can be translated into FO({PartOF}).

  Given a (3k)-tuple of points (p_1,1, p_1,2, p_1,3, p_2,1, p_2,2, p_2,3,…,p_a,1, p_k,2, p_k,3) in $\mathbb {R}^{2}$. There will be 6^k k-tuples of triangles (T₁, T₂,…,Tk) such that, for each of the Ti, 1 ≤ i ≤ k, the condition CornerP(Ti, 1,Ti, 2,Ti, 3,Ti) is true. However, there will only be one tuple of triangles that is the image of the 3k-tuple of points (p_1,1, p_1,2, p_1,3, p_2,1, p_2,2, p_2,3,…,p_a,1, p_k,2, p_k,3) under the inverse of the canonical bijection can_tr. This shows that the simulation is correct.

Fig. 10

Illustration of the translation of the predicate Between. The (degenerated) triangle Tq (in blue) lies between the (degenerated) triangles Tp and Tr if and only if all triangles (illustrated by the orange, red and brown triangles) that contain both Tp and Tr, also contain Tq

The translation of composed formulas.

Now suppose that we already simulated the FO{Between}, $\dot {\sigma }$ formulas $\dot {\varphi }(p_{1}, p_{2}, \ldots , p_{k_{\varphi }})$ and $\dot {\psi }(p_{1}, p_{2}, \ldots , p_{k_{\psi }})$ into formulas $\hat {\varphi }$ and $\hat {\psi }$ in FO({PartOf}, $\hat {\sigma })$ with free variables △₁, △₂, …, $\bigtriangleup _{{k_{\varphi }}}$ and $\bigtriangleup _{{1}}^{\prime }$, $\bigtriangleup _{{2}}^{\prime }$, …, $\bigtriangleup _{{k_{\psi }}}^{\prime }$, respectively. Hence, we can assume that, for each triangle database $\hat {\mathcal {D}}$ over $\hat {\sigma }$ and for each k_φ-tuple of triangles (T1, T2, …, Tk_φ) = $(({\mathsf {p}}_{1}, {\mathsf {p}}_{1}, {\mathsf {p}}_{1}), ({\mathsf {p}}_{2}, {\mathsf {p}}_{2}, {\mathsf {p}}_{2}), \ldots , ({\mathsf {p}}_{k_{\varphi }}, {\mathsf {p}}_{k_{\varphi }}, {\mathsf {p}}_{k_{\varphi }}))$, which are required to be degenerated into points, that

$$ \hat{\mathcal{D}} \models \hat{\varphi}[\mathsf{T}_{1}, \mathsf{T}_{2}, \ldots, \mathsf{T}{k_{\varphi}}] iff. \dot{\mathcal{D}}\models \dot{\varphi}[{\mathsf{p}}_{1}, {\mathsf{p}}_{2}, \ldots, {\mathsf{p}}_{k_{\psi}}]. $$

For $\hat {\psi }$ we have analogue conditions.

The composed formulas $\dot {\varphi } \land \dot {\psi }$, $\dot {\varphi } \lor \dot {\psi }$, $\neg \dot {\varphi }$ and $\exists p \dot {\varphi }$, are translated into $\hat {\varphi } \land \hat {\psi }$, $\hat {\varphi } \lor \hat {\psi }$, $\neg \hat {\varphi }$ and $\exists \bigtriangleup (\hat {\varphi })$, respectively if we assume that p is translated into △. The correctness proofs for these translations are similar to the proofs in Lemma 4.2. Therefore, we do not repeat them here. This concludes the proof of Lemma 4.3.