The Undecidability of Orthogonal and Origami Geometries

Abstract

In the late 1950s A. Tarski published an abstract stating that the set of first order consequences of projective (and also affine) incidence geometry is undecidable. Although his theorem is cited in many follow up papers, a detailed proof was not published. To the best of our knowledge, we have not found a detailed complete proof in the literature. In this paper we analyze what is needed to give a correct proof which is reconstructible by practitioners of AI and automated theorem proving and extend the undecidability to many other axiomatizations of geometry. These include the geometry of Hilbert and Eulidiean planes, Wu’s geometry and Origami geometry. We also discuss applications to automated theorem proving.

J. A. Makowsky—Partially supported by a grant of Technion Research Authority. Work done in part while the author was visiting the Simons Institute for the Theory of Computing in Fall 2016.

Appendices

A Axioms for Affine Geometries

In this appendix we collect some of Hilbert’s axioms of geometry which we used, and which are all true when one considers the analytic geometry of the plane with real coordinates.

1.1 A.1 Incidence Geometries

Axioms Using Only the Incidence Relation  

(I-1)::

For any two distinct points A, B there is a unique line l with $A \in l$ and $B\in l$.

(I-2)::

Every line contains at least two distinct points.

(I-3)::

There exists three distinct points A, B, C such that no line l contains all of them.

  They can be formulated in $\mathrm {FOL}$ using the incidence relation only.

Parallel Axiom. We define: $Par(l_1, l_2)$ or $l_1 \parallel l_2$ if $l_1$ and $l_2$ have no point in common.

 

(ParAx)::

For each point A and each line l there is at most one line $l'$ with $l \parallel l'$ and $A \in l'$.

  $Par(l_1, l_2)$ can be formulated in $\mathrm {FOL}$ using the incidence relation only, hence also the Parallel Axiom.

Pappus’ Axiom  

(Pappus)::

Given two lines $l, l'$ and points $A,B,C \in l$ and $A',B',C' \in l'$ such that $AC' \parallel A'C$ and $BC' \parallel B'C$. Then also $AB' \parallel A'B$.

 

Axioms of Desargues and of Infinity

• (InfLines): Given distinct A, B, C and l with $A \in l, B, C \not \in l$ we define $A_1 = Par(AB,C) \times l$, and inductively, $A_{n+1} = Par(A_nB,C) \times l$. Then all the $A_i$ are distinct.

  Note that this axiom is stronger than just saying there infinitely many points. It says that there are no lines which have only finitely many points.

• (De-1): If $AA', BB', CC'$ intersect in one point or are all parallel, and $AB \parallel A'B'$ and $AC \parallel A'C'$ then $BC \parallel B'C'$.

• (De-2): If $AB \parallel A'B'$, $AC \parallel A'C'$ and $BC \parallel B'C'$ then $AA', BB', CC'$ are all parallel.

The axiom (InfLies) is not first order definable but consists of an infinite set of first order formulas with infinitely many new constant symbols for the points $A_i$, and the incidence relation. The two Desargues axioms are first order definable using the incidence relation only.

• Affine plane: Let $\tau _{\in } \subseteq \tau$ be a vocabulary of geometry. A $\tau$-structure $\varPi$ is an (infinite) affine plane if it satisfies (I-1, I-2, I-3 and the parallel axiom (ParAx) and (InfLines). We denote the set of these axioms by $T_{affine}$.

• Pappian plane: $\varPi$ is a Pappian plane if additionally it satisfies the Axiom of Pappus (Pappus). We denote the set of these axioms by $T_{pappus}$.

In the literature the definition of affine planes vary. Sometimes the parallel axiom is included, and sometimes not. We always include the parallel axiom, unless indicated explicitly otherwise.

1.2 A.2 Hilbert Style Geometries

Axioms of Betweenness

• (B-1): If Be(A, B, C) then there is l with $A, B, C \in l$.

• (B-2): For every A, B there is C with Be(A, B, C).

• (B-3): For each distinct $A, B, C \in l$ exactly one point of the points A, B, C is between the two others.

• (B-4): (Pasch) Assume the points A, B, C and l in general position, i.e. the three points are not on one line, none of the points is on l. Let D be the point at which l and the line AB intersect. If Be(A, D, B) there is $D' \in l$ with $Be(A, D', C)$ or $Be(B, D', C)$.

The axioms of betweenness are all first order expressible in the language with incidence relation and the betweenness relation.

Congruence Axioms: Equidistance. We write for Eq(A, B, C, D) the usual $AB \cong CD$.

• (C-0): $AB \cong AB \cong BA$.

• (C-1): Given $A, B, C, C'$, l with $C, C' \in l$ there is a unique $D \in l$ with $AB \cong CD$ and $B(C, C', D)$ or $B(C, D, C')$.

• (C-2): If $AB \cong CD$ and $AB \cong EF$ then $CD \cong EF$.

• (C-3): (Addition) Given A, B, C, D, E, F with Be(A, B, C) and Be(D, E, F), if $AB \cong DE$ and $BC \cong EF$, then $AC \cong DF$.

Note that (C-1) and (C-3) use the betweenness relation Be. Hence they are first order definable using the incidence, betweenness and equidistance relation.

Congruence Axioms: Equiangularity. We denote by $\mathbf {AB}$ the directed ray from A to B, and by $\angle (ABC)$ the angle between $\mathbf {AB}$ and $\mathbf {BC}$. For the congruence of angles $An(A,B,C,A',B',C')$ we write $\angle (ABC) \cong \angle (A'B'C')$.

• (C-4): Given rays $\mathbf {AB}$, $\mathbf {AC}$ and $\mathbf {DE}$ there is a unique ray $\mathbf {DF}$ with $\angle (BAC) \cong \angle (EDF)$.

• (C-5): Congruence of angles is an equivalence relation.

• (C-6):(Side-Angle-Side) Given two triangles ABC and $A'B'C'$ with $AB \cong A'B'$, $AC \cong A'C'$ and $\angle {BAC} \cong \angle {B'A'C'}$ then $BC \cong B'C'$, $\angle {ABC} \cong \angle {A'B'C'}$ and $\angle {ACB} \cong \angle {A'C'B'}$.

Axiom E. Let A be a point and BC be a line segment. A circle $\varGamma (A,BC)$ is the set of all points U such that E(A, U, B, C). A point D is inside the circle $\varGamma (A,BC)$ if there is U with E(A, U, B, C) and Be(A, D, U). A point D is outside the circle $\varGamma (A,BC)$ if there is U with E(A, U, B, C) and Be(A, U, D).

• (AxE): Given two circles $\varGamma , \varDelta$ such that $\varGamma$ contains at least one point inside, and one point outside $\varDelta$, then $\varGamma \cap \varDelta \ne \emptyset$.

• Hilbert plane: Let $\tau$ with $\tau _{hilbert} \subseteq \tau$ be a vocabulary of geometry. A $\tau$-structure $\varPi$ is an (infinite) Hilbert plane if it satisfies (I-1, I-2, I-3), (B-1, B-2, B-3, B-4) and (C-1, C-2, C-3, C-4, C-5, C-6).

  We denote the set of these axioms by $T_{hilbert}$

• P-Hilbert plane: $\varPi$ is a P-Hilbert plane if it additionally satisfies (ParAx).

  We denote the set of these axioms by $T_{p-hilbert}$

• Euclidean plane: $\varPi$ is a Euclidean plane if it is a P-Hilbert plane which also satisfies Axiom E.

  We denote the set of these axioms by $T_{euclid}$

1.3 A.3 Axioms of Orthogonal Geometry

Congruence Axioms: Orthogonality. We denote by $l_1 \perp l_2$ the orthogonality of two lines $Or(l_1, l_2)$. We call a line l isotropic if $l \perp l$. Note that our definitions do not exclude this.

• (O-1): $l_1 \perp l_2$ iff $l_2 \perp l_1$.

• (O-2): Given O and $l_1$, there exists exactly one line $l_2$ with $l_1 \perp l_2$ and $O \in l_2$.

• (O-3): $l_1 \perp l_2$ and $l_1 \perp l_3$ then $l_2 \parallel l_3$.

• (O-4): For every O there is an l with $O \in l$ and $l \not \perp l$.

• (O-5): The three heights of a triangle intersect in one point.

Axiom of Symmetric Axis and Transposition

• (AxSymAx): Any two intersecting non-isotropic lines have a symmetric axis.

• (AxTrans): Let $l, l'$ be two non-isotropic lines with $A,O, B \in l$, $AO \cong OB$ and $O' \in l'$ there are exactly two points $A', B' \in l'$ such that $AB \cong A'B' \cong B'A'$ and $A'O' \cong O'B'$.

The two axioms are equivalent in geometries satisfying the Incidence, Parallel, Desargues and Orthogonality axioms together with the axiom of infinity.

• Orthogonal Wu plane: Let $\tau$ with $\tau _{Wu} \subseteq \tau$ be a vocabulary of geometry. A $\tau$-structure $\varPi$ is an orthogonal Wu plane if it satisfies (I-1, I-2, I-3), (O-1, O-2, O-3, O-4, O-5), the axiom of infinity (InfLines), (ParAx), and the two axioms of Desargues (D-1) and (D-2).

  We denote the set of these axioms by $T_{o-wu}$

• Metric Wu plane: $\varPi$ is a metric Wu plane if it satisfies additionally the axiom of symmetric axis (AxSymAx) or, equivalently, the axiom of transposition (AxTrans).

  We denote the set of these axioms by $T_{m-wu}$

The axiomatization of orthogonal is due to Wu [Wu86, Wu94, WG07], see also [Pam07].

1.4 A.4 The Origami Axioms

A line which is obtained by folding the paper is called a fold. The first six axioms are known as Huzita’s axioms. Axiom (H-7) was discovered by K. Hatori. Jacques Justin and Robert J. Lang also found axiom (H-7), [Wik]. The axioms (H-1)-(H-7) only express closure under folding operations, and do not define a geometry. To make it into an axiomatization of geometry we have to that these operations are performed on an affine plane.

We follow here [GIT+07]. The original axioms and their expression as first order formulas in the vocabulary $\tau _{origami}$ are as follows:

• (H-1): Given two points $P_1$ and $P_2$, there is a unique fold (line) that passes through both of them.

  $$ \forall P_1, P_2 \exists ^{=1} l (P_1 \in l \wedge P_2 \in l) $$

• (H-2): Given two points $P_1$ and $P_2$, there is a unique fold (line) that places $P_1$ onto $P_2$.

  $$ \forall P_1, P_2 \exists ^{=1} l SymLine(P_1, l, P_2) $$

• (H-3): Given two lines $l_1$ and $l_2$, there is a fold (line) that places $l_1$ onto $l_2$.

  $$ \forall l_1, l_2 \exists k \forall P \left( P \in k \rightarrow Peq(l_1,P, l_2) \right) $$

• (H-4): Given a point P and a line $l_1$, there is a unique fold (line) perpendicular to $l_1$ that passes through point P.

  $$ \forall P, l \exists ^{=1} k \forall P (P \in k \wedge Or(l, k)) $$

• (H-5): Given two points $P_1$ and $P_2$ and a line $l_1$, there is a fold (line) that places $P_1$ onto $l_1$ and passes through $P_2$.

  $$ \forall P_1, P_2 l_1 \exists l_2 \forall P (P_2 \in l_2 \wedge \exists P_2 (SymLine(P_1, l_2, P_2) \wedge P_2 \in l_1)) $$

• (H-6): Given two points $P_1$ and $P_2$ and two lines $l_1$ and $l_2$, there is a fold (line) that places $P_1$ onto $l_1$ and $P_2$ onto $l_2$.

  $$\begin{aligned}&\forall P_1, P_2 l_1, l_2 \exists l_3 \left( (\exists Q_1 SymLine(P_1, l_3, Q_1) \wedge Q_1 \in l_1) \wedge \right. \\&\quad \left. (\exists Q_2 SymLine(P_2, l_3, Q_2) \wedge Q_2 \in l_2) \right) \end{aligned}$$

• (H-7): Given one point P and two lines $l_1$ and $l_2$, there is a fold (line) that places P onto $l_1$ and is perpendicular to $l_2$.

  $$ \forall P, l_2, l_2 \exists l_3 \left( Or(l_2, l_3) \wedge (\exists Q SymLine(P, l_3, Q) \wedge Q \in l_1) \right) $$

• Affine Origami plane: Let $\tau$ with $\tau _{origami} \subseteq \tau$ be a vocabulary of geometry. A $\tau$-structure $\varPi$ is an affine Origami plane if it satisfies (I-1, I-2, I-3), the axiom of infinity (InfLines), (ParAx) and the Huzita-Hatori axioms (H-1) - (H-7).

  We denote the set of these axioms by $T_{a-origami}$

Proposition 4

The relations SymLine and Peq are first order definable using Eq and Or with existential formulas over $\tau _{f-field}$: Hence the axioms (H-1)–(H-7) are first order definable in $\mathrm {FOL}(\tau _{wu})$.

Proof

• (i) $SymLine(P_1, \ell , P_2)$ iff there is a point $Q \in \ell$ such that $Or((P_1,Q), \ell )$, $Or((P_2,Q), \ell )$ and $Eq(P_1,Q, P_2, Q)$.

• (ii) $Peq(\ell _1, P, \ell _2)$ iff there exist points $Q_1, Q_2$ such that $Or((P, Q_1), \ell _1)$, $Or((P, Q_2), \ell _2)$, $Eq(P, Q_1)$ and $Eq(P, Q_2)$. $\Box$

B Translation Schemes

We first introduce the formalism of translation schemes, transductions and translation. In [TMR53] this was first used, but not spelled out in detail. The details appear first in [Rab65]r. Our approach follows [Mak04, Sect. 2]. To keep it notationally simple we explain on an example. Let $\tau$ be a vocabulary consisting of one binary relation symbol R, $\sigma$ be a vocabulary consisting of one ternary relation symbol S. We want to interpret a $\sigma$ structure on k-tuples of elements of a $\tau$-structure.

A $\tau -\sigma$-translation scheme $\varPhi =(\phi , \phi _S)$ consists of a formula $\phi (\bar{x})$ with k free variables and a formula $\phi _S$ with 3k free variables. $\varPhi$ is quantifier-free if all its translation formulas are quantifier-free.

Let $\mathcal {A}=\langle A, R^A \rangle$ be a $\tau$-structure. We define a $\sigma$-structure $\varPhi ^*(\mathcal {A})= \langle B, S^B \rangle$ as follows: The universe is given by

$$ B= \{ \bar{a} \in A^k : \mathcal {A}\models \phi (\bar{a}) \} $$

and the relation is given by

$$ S^B = \{ \bar{b} \in A^{k\times 3}: \mathcal {A}\models \phi _S(\bar{b}) \}. $$

$\varPhi ^*$ is called a transduction.

Let $\theta$ be a $\sigma$-formula. We define a $\tau$-formula $\varPhi ^{\sharp }(\theta )$ inductively by substituting occurrences of $S(\bar{b}$ by their definition via $\phi _S$ where the free variables are suitable named. $\varPhi ^{\sharp }$ is called a translation.

The fundamental property of translation schemes, transductions and translation is the following:

Proposition 5

(Fundamental Property of Translation Schemes). Let $\varPhi$ be a $\tau -\sigma$-translation scheme, and $\theta$ be a $\sigma$-formulas.

$$ \mathcal {A}\models \varPhi ^{\sharp }(\theta ) \text{ iff } \varPhi ^*(\mathcal {A}) \models \theta $$

If $\theta$ has free variables, the assignment have to be chosen accordingly. Furthermore, if $\varPhi$ is quantifier-free, and $\theta$ is a universal formula, $\varPhi ^{\sharp }(\theta )$ is also universal.

In order to use translation schemes to prove decidability and undecidability of theories we need two lemmas.

Lemma 2

Let $\varPhi$ be a $\tau -\sigma$-translation scheme.

• (i) Let $\mathcal {A}$ be a $\tau$-structure. If the complete first order theory $T_0$ of $\mathcal {A}$ is decidable, so is the complete first order theory $T_1$ of $\varPhi ^*(\mathcal {A})$.

• (ii) There is a $\tau$-structure $\mathcal {A}$ such that the complete first order theory $T_1$ of $\varPhi ^*(\mathcal {A})$ is decidable, but the complete first order theory $T_0$ of $\mathcal {A}$ is undecidable.

• (iii) If however, $\varPhi ^{\sharp }$ is onto, i.e., for every $\phi \in \mathrm {FOL}(\tau )$ there is a formula $\theta \in \mathrm {FOL}(\sigma )$ with $\varPhi ^{\sharp }(\theta )$ logically equivalent to $\phi$, then the converse of (i) also holds.

• (iv) Let $T \subseteq \mathrm {FOL}(\tau )$ be a decidable theory and $T' \subseteq \mathrm {FOL}(\sigma )$ and $\varPhi ^*$ be such that $\varPhi ^*|_{Mod(T)}: Mod(T) \rightarrow Mod(T')$ be onto. Then $T'$ is decidable.

Remark 1

The condition that $\varPhi ^{\sharp }$, resp. $\varPhi ^*$ have to be onto is often overlooked in the literature⁴.

We shall need one more observation:

Lemma 3

Let $T \subseteq \mathrm {FOL}(\tau )$ and $\phi \in \mathrm {FOL}(\tau )$. Assume T is decidable. Then $T \cup \{ \phi \}$ is also decidable.