$(\mathbb {Z},\text {succ},U), (\mathbb {Z},E,U)$, and Their CSP’s

Abstract

When we expand the simple structure ($\mathbb {Z}$,succ) with one unary predicate U, its CSP (Constraint Satisfaction Problem) may vary in complexity. We find some sufficient conditions for its tractability, prove bounds on its complexity, and then generalize our results to more complicated structures. We also give a Karp-equivalent characterization of CSP$(\mathbb {Z},\text {succ},U)$’s.

Appendices

A Proof Details of Interesting Claims

1.1 A.1 Details of Section 3, Proposition 1 and Corollary 1

First, we show the details of the proof that if $U=\{u(i)\}_{i<\omega }$ is ELG-SR, then CSP$(\mathbb {Z},\text {succ},U) \le ^p_TU$. On $\mathfrak {D}=(D,\text {succ}^\mathfrak {D},U^\mathfrak {D})$, for each component $\mathfrak {D}_e$:

• 1. If $D_e\cap U^\mathfrak {D}=\emptyset$, run CSP$(\mathbb {Z},\text {succ})$ decider on $(D_e,\text {succ}^\mathfrak {D}\cap D_e^2)$; if $|D_e\cap U^\mathfrak {D}|=1$, run the CSP$(\mathbb {Z},\text {succ})$ algorithm beginning with assigning the unique $d_e\in D_e\cap U^\mathfrak {D}$ to an arbitrary $b\in U$.

  Correctness follows from the observation that, in these cases we are still free to translate a $\text {succ}$-homomorphism as long as one exists.

• 2. Now $|D_e\cap U^\mathfrak {D}|\ge 2$. Since U is ELG, by definition there exists $c\in \omega$ witnessing it, i.e. gaps in U are large after index c. Pick arbitrary $d_e\in D_e\cap U^\mathfrak {D}$. Run the following:

  

  And $\mathfrak {D}$ is accepted if and only if each $\mathfrak {D}_e$ is accepted. To see efficiency: since U has SR, $|\langle u(i)\rangle |$ is polynomial in i, and i is linear in $|D_e|\le |D|$. If a $\{\text {succ}\}$-homomorphism h is found, we have polynomially many calls to the U-oracle; note that each $h(d'_e)\in [u(i)-n,u(i)+n]$, so it remains polynomial size. The whole algorithm is therefore poly-calls to U with poly-overhead.

  To see correctness, first observe the following fact: in a connected component of size n, for any vertices $x\ne y$ the longest “signed path” (vertices $x_0:=x,x_1,\dots , x_k:=y$ s.t. for each i, either $(x_i,x_{i+1})\in E$ or $(x_{i+1},x_i)\in E$) has at most $n-1$ edges, and the signed length (computed by adding 1 if $(x_i,x_{i+1})\in E$ and subtracting 1 if $(x_{i+1},x_i)\in E$).¹⁰ is at most $n-1$. Whenever the component homomorphically maps to $(\mathbb {Z},\text {succ})$ via some h, the signed length of any signed path must be $h(y)-h(x)$.

  Now for any $i>\max \{c,|D_e|\}$, $|u(i)-*|>i>|D_e|$ for any $*\in U\smallsetminus \{u(i)\}$, and with $|D_e\cap U^\mathfrak {D}|\ge 2$, this gap is too big for there to exist a homomorphism sending $d_e$ to u(i). In other words, $\mathfrak {D}_e\in$ CSP$(\mathbb {Z},\text {succ},U)\iff \exists h\in \text {Hom}_{\text {succ}}(\mathfrak {D}_e,(\mathbb {Z},\text {succ},U))$ s.t. $h(d_e)\in \{u(i)\}_{i\le \max \{|D_e|,c\}}$, which is what the algorithm checks.

  As for the corollary claiming efficiency, note that the above has reduced CSP$(\mathbb {Z},\text {succ},U)$ to O(|D|)-many tests of whether each $h(d'_e)\in U$. But this can be done by asking “is $h(d'_e)=u(j)?$” for each $j\in [0,\max \{c,|D_e|\}+1]$”. Both $h(d'_e)$ and each u(j) are of size polynomial in |D|, thanks to SR and the connectedness of $\mathfrak {D}_e$.

1.2 A.2 Details of Section 3, Theorem 1 and Corollary 2

Proof

(Section 3, Theorem 1). Again, thanks to EMLG one needs to shift a $\text {succ}$-homomorphism, if found, at most linearly many times, and SR bars exponential blow-ups.

Upon input $\mathfrak {D}=(D,\text {succ}^\mathfrak {D},U_1^\mathfrak {D},\dots , U_k^\mathfrak {D})$, for each component $\mathfrak {D}_e$:

• 1. If $|D_e\cap (\bigcup _jU_j^\mathfrak {D})|\le 1$, like in Proposition 1 the gaps within and across $U_j$’s make no impact on existence of homomorphisms. One just needs to run the CSP$(\mathbb {Z},\text {succ})$ decider / searcher starting with assinging $d_e$ to an arbitrary $b\in U_j,$ where $U_j^ \mathfrak {D}$ is the unique unary intersecting $D_e$ with $d_e$ the unique intersection.

• 2. Otherwise, let of the $\le k+{k\atopwithdelims ()2}$ constants witnessing EMLG. Then for any $(j,l)\in k^2$, any , any we have . It follows that one needs at most $O(|D_e|)\sim O(|D|)$ iterations, for .

  Fix arbitrary $d_e\in D_e\cap (\bigcup _jU_j^\mathfrak {D})$ before iterating. Within each iteration, run CSP$(\mathbb {Z},\text {succ})$ assigning $d_e$ to some u(n, y) where $d_e\in D_e\cap U_y^\mathfrak {D}$, n is the iteration number ($n\le \max (c,|D_e|))$. Since $U_y$ has SR, u(n, y) has size polynomial in n, and n is linear in |D|. If a $\text {succ}$-homomophism h is found, verify whether $h(d'_e)\in U_{y'}$ for each $d'_e\in D_e\cap U_{y'}^\mathfrak {D}$. Via $U_{y'}\le ^p_m$ (hence $\le ^p_T$) $\varinjlim \nolimits _jU_j$, the decision of $U_{y'}$-membership of $h(d'_e)$ reduces to calls to the $\varinjlim \nolimits _jU_j$-oracle. Note each $h(d'_e)\in [u(n,y)-|D_e|,u(n,y)+|D_e|]$, so the reduction is of time polynomial w.r.t. |D|.   $\square$

Proof

(Section 3,Corollary 2). As in Corollary 1, for each $h(d'_e)$, ask if $h(d'_e)=u(s,t)$ for each $t\in [1,k]$ involved, $s\in [0,\max \{c,|D_e|\}+1]$. By SR and connectedness, both $h(d'_e)$ and each u(s, t) have size polynomial in |D|.   $\square$

1.3 A.3 Details of Section 4, Proposition 2 and Corollary 3

Proof

(Section 4, Proposition 2). Upon each input $\mathfrak {D}=(D,\text {succ}^\mathfrak {D},U^\mathfrak {D},\textbf{0}^\mathfrak {D})$, let $\mathfrak {D}_0:=$ the unique component containing $\textbf{0}^\mathfrak {D}$. Call CSP$(\mathbb {Z},\text {succ},U)$-oracle on $\mathfrak {D}\smallsetminus \mathfrak {D}_0$ and echo if rejects.

Call CSP$(\mathbb {Z},\text {succ})$-solver on $(D_0,\text {succ}^\mathfrak {D}\cap D_0^2)$ initiated by sending $\textbf{0}^\mathfrak {D}$ to 0, and echo if rejects; otherwise, the unique $\text {succ}$-homomorphism $h:(D_0,\text {succ}^\mathfrak {D}\cap D_0^2)\rightarrow (\mathbb {Z},\text {succ})$ that satisfies $h(\textbf{0}^\mathfrak {D})=0$ has been found, and one just needs to verify if for each d we have $h(d)\in U$ by calling the U-oracle. Note each $h(d)\in [0-|D_0|, 0+|D_0|]$ so input size remains polynomial in $|D_0|\le |D|$.

The only oracles we called above are the U-oracles and CSP$(\mathbb {Z},\text {succ},U)$-oracles, both of which Karp (hence Cook) reduces to the $\varinjlim$-oracle on RHS.

Proof

(Section 4,Corollary 3). If $U\in \texttt{P}$, calls to the U-oracle counts as polynomial time overhead; if $U\le ^p_T$ CSP$(\mathbb {Z},\text {succ},U)$, note the direct limit can be computed as $\left\{ \begin{array}{cc} \chi _U(\frac{n-1}{2}) &{}\text { if }n=2k+1 \\ \chi _{\text {CSP}(\mathbb {Z},\text {succ},U)}(\frac{n}{2}) &{}\text {o.w.} \end{array}\right.$, where $\chi _S(\bullet )$ is the membership function of set S, and $\chi _U$ can be computed by computing $\chi _{\text {CSP}(\mathbb {Z},\text {succ},U)}$ now at polynomial cost.

1.4 A.4 Details of Section 4, Theorem 2

Proof

Before going into the full details, we illustrate the reductions with the following example (Fig. 1):

Fig. 1.

Example of Theorem 2 Reduction

• 1. $\le ^p_m$: upon $(n,S_1,S_2)$, construct $D:=\{1,\dots ,n\},\text {succ}^\mathfrak {D}=\emptyset , U^\mathfrak {D}:=D$. Then for each $(i<j)\in S_1$:

  • (a) If $S_2(i,j)>0$, add intermediate vertices and edges $i=\big ((i,j),0\big ){\mathop {\longrightarrow }\limits ^{\text {succ}^\mathfrak {D}}} \big ((i,j),1\big ){\mathop {\longrightarrow }\limits ^{\text {succ}^\mathfrak {D}}} \dots {\mathop {\longrightarrow }\limits ^{\text {succ}^\mathfrak {D}}}\big ((i,j),S_2(i,j)\big )=j$.

  • (b) If $S_2(i,j)<0$, add intermediate vertices and edges $i=\big ((i,j),0\big ){\mathop {\longleftarrow }\limits ^{\text {succ}^\mathfrak {D}}} \big ((i,j),1\big ){\mathop {\longleftarrow }\limits ^{\text {succ}^\mathfrak {D}}} \dots {\mathop {\longleftarrow }\limits ^{\text {succ}^\mathfrak {D}}}\big ((i,j),|S_2(i,j)|\big )=j$.

  • (c) if $S_2(i,j)=0$, collapse vertices $i\sim j$ (so now the “vertex classes” ${[i]} = {[j]})$.

  Now $D=\Big \{[1],\dots , [n]\Big \}\cup \Big \{\big ((i,j),m\big )\Big \}_{ S_2(i,j)\ne 0,m\in [1,|S_2(i,j)|-1]}$ and $U^\mathfrak {D}=\Big \{[1],\dots , [n]\Big \}$. Note $|D|\le n+n\times {n\atopwithdelims ()2}$, and as $S_1$ is part of the tuple, |D| is polynomial in $|\langle (n,S_1,S_2)\rangle |$, so $(n,S_1,S_2)\mapsto \mathfrak {D}=(D,\text {succ}^\mathfrak {D},U^\mathfrak {D})$ is a polynomial-time construction. Further,

  • (a) If $(n,S_1,S_2)\in$ Gap(U) witnessed by $t\in U^n$, let $h:D\rightarrow \mathbb {Z}$, $[j]\mapsto t(j)$ ($j\in [1,n]$) and extend naturally to the intermediates, i.e. $\big ((i,j),m\big )\mapsto t(i)+$ sign$(S_2(i,j))\times m$. The map puts $\mathfrak {D}\in$ CSP$(\mathbb {Z},\text {succ},U)$.

  • (b) If $\mathfrak {D}\in$ CSP$(\mathbb {Z},\text {succ},U)$ by h, then $\big (t(i)=h([i])\big )_{i\in [1,n]}$ witnesses $(n,S_1,S_2)\in$ Gap(U).

• 2. $_m^p\ge$: Observe that $s_1:=\Big (3,{3\atopwithdelims ()2}, \big (\underbrace{1}_{(1,2)},\underbrace{3}_{(1,3)}, \underbrace{1}_{(2,3)}\big )\Big )\notin$ Gap(U) due to triangle-inequality, and taking arbitrary $u_1\ne u_2\in U$ (not $U^\mathfrak {D}$), $s_2:=\Big (2, \big \{(1,2)\big \}, \big (u_2-u_1\big )\Big )\in$ Gap(U). $s_1,s_2$ are input-independent.

  Now let $\mathfrak {D}=(D,\text {succ}^\mathfrak {D},U^\mathfrak {D})$. We want to produce a short tuple $s_\mathfrak {D}=(n,S_1,S_2)$ in time polynomial in |D|. We present an algorithm for that.

  • (a) Run CSP$(\mathbb {Z},\text {succ})$ decider/searcher on $\mathfrak {D}$, and halt setting $s_\mathfrak {D}:=s_1$ if reject was returned; else, we obtain an $h\in \text {Hom}_{\text {succ}}\big ((D,\text {succ}^\mathfrak {D}),(\mathbb {Z},\text {succ})\big )$ from the run.

  • (b) Throw away the components $\mathfrak {D}_e$ with $|D_e\cap U^\mathfrak {D}|\le 1$, keeping $\mathfrak {D}':=\coprod _{e:|D_e\cap U^\mathfrak {D}|\ge 2}\mathfrak {D}_e$. If $\mathfrak {D}'=\emptyset$, halt setting $s_\mathfrak {D}:=s_2$.

  • (c) Let $n:=|U^\mathfrak {D}\cap D'|$; now $n\in [2,|D'|]$. Enumerate elements in $U^\mathfrak {D}\cap D'$ as $v_1,\dots , v_n$; let $S_1:=\{(i<j): v_i,v_j\text { are from the same component }\}$. Let $S_2(i,j):=h(v_j)-h(v_i)$ with h from Step 2a. Let $s_\mathfrak {D}:=\langle (n,S_1,S_2)\rangle$.

  $s_\mathfrak {D}$ has size polynomial in |D| and the algorithm is efficient. Correctness:

  • (a) Assume $\mathfrak {D}\in$ CSP$(\mathbb {Z},\text {succ},U)$ witnessed by $h'$, then $s_\mathfrak {D}\ne s_1$ for $h'\in \text {Hom}_{\text {succ}}\big ((D,\text {succ}^\mathfrak {D}),$ $(\mathbb {Z},\text {succ})\big )$ and the CSP$(\mathbb {Z},\text {succ})$-decider should’ve caught a homomorphism.

    • If $\mathfrak {D}'\ne \emptyset$: for each $v_i\ne v_j$ from the same component in $\mathfrak {D}'$, $h'(v_j)-h'(v_i)=h(v_j)-h(v_i)$. Therefore setting $t_i=h'(v_i)$ for each $i\in [1,n]$, we get a witness for $(n,S_1,S_2)\in$ Gap(U).

    • If $\mathfrak {D}'=\emptyset$, we exited with an $s_2\in$ Gap(U).

  • (b) Assume $s_\mathfrak {D}\in$ Gap(U), then again $s_\mathfrak {D}\ne s_1$.

    • If $s_\mathfrak {D}=s_2$ then the h from Step 2a translates componentwise and then patches to a $\{\text {succ},U\}$-homomorphism.

    • otherwise there exists $t\in U^n$ with $t(j)-t(i)=S_2(i,j)$ $(\forall (i<j)\in S_1)$. Each component in $\mathfrak {D}'$ contains a $v_{i_e}\in U^\mathfrak {D}\cap D'$; shift $h\restriction _{\mathfrak {D}_e}$ s.t. $h(v_{i_e})=t(i_e)$. Now for each $v_{j_e}\ne v_{i_e}\in D_e\cap U^\mathfrak {D}$, $h(v_{j_e})=h(v_{i_e})+S_2(i_e,j_e)=t(j_e)\in U$. Do so for each component of $\mathfrak {D}'.$

      Lastly for each component in $\mathfrak {D}\smallsetminus \mathfrak {D}'$, shift h if need be.   $\square$

1.5 A.5 Details of Section 5, Fact 3 and Proposition 4

We first prove the folklore / exercise in Sect. 5, Fact 3, conceptually by mimicing the Search to Decision reduction for SAT. As in graphs, a core $\mathfrak {B}\subset _{\text {ind}}\mathfrak {A}$ is a minimal image of endomorphisms, i.e. there exists no $\sigma \in \text {End}{(\mathfrak {A})}$ s.t. $\sigma (\mathfrak {A})\subsetneq \mathfrak {B}$. For finite $\mathfrak {A}$’s, cores always exist and are unique up to isomorphism [3]. are isomorphic,¹¹ so hereinafter we talk about “the” core of $\mathfrak {A}$.

Let $\mathfrak {B}$ be the core of $\mathfrak {A}$, then inclusion $\mathfrak {B}\hookrightarrow \mathfrak {A}$ and the definitional endomorphism $\mathfrak {A}\rightarrow \mathfrak {B}$ shows homomorphic equivalence of $\mathfrak {A}$ and $\mathfrak {B}$, so it suffices to show SEARCH-CSP$(\mathfrak {A})\le ^p_T$ CSP$(\mathfrak {B})$.¹²

To make the description less cumbersome let’s define POINTED-CSP: fix arbitrary $b\in B$. POINTED-CSP$(\mathfrak {B},b)$ takes a pair $(\mathfrak {D},d)$ with $\mathfrak {D}$ a finite $\tau$-structure, $d\in D$, and returns $\texttt{1}$ iff there exists $h\in \text {Hom}_\tau (\mathfrak {D},\mathfrak {B})$ s.t. $h(d)=b$.

Claim: POINTED-CSP$(\mathfrak {B},b)\le ^p_m$ (hence $\le ^p_T$) CSP$(\mathfrak {B})$.

Proof

(Proof of Claim). Upon a POINTED-CSP$(\mathfrak {B},b)$ input $(\mathfrak {D},d)$, let $\widetilde{\mathfrak {D}}:=\dfrac{\mathfrak {D}\cup \mathfrak {B}}{b\sim d}$.¹³. This is a polynomial time construction since $|B|\sim O(1)$. I claim $(\mathfrak {D},d)\in \ \text {POINTED-CSP}(\mathfrak {B},b)\iff \widetilde{\mathfrak {D}}\in$ CSP$(\mathfrak {B})$.

• 1. $\implies$: let $h:(\mathfrak {D},d)\rightarrow (\mathfrak {B},b)$ be a basepoint-preserving homomorphism. Then $\widetilde{h}:\widetilde{\mathfrak {D}}\rightarrow \mathfrak {B}, h(x):=\left\{ \begin{array}{cc} h(x), &{} \text { if }x\in D \\ x, &{} \text { o.w.} \end{array}\right.$ is a homomorphism.

• 2. : let $h:\widetilde{\mathfrak {D}}\rightarrow \mathfrak {B}$ be a homomorphism. Then $h\vert _{\mathfrak {B}}:\mathfrak {B}\rightarrow \mathfrak {B}$ is an endomorphism of the finite core $\mathfrak {B}$. Since an endomorphism of a core is an automorphism,¹⁴ there exists $\sigma \in \text {Aut}{(\mathfrak {B})}=\text {End}{(\mathfrak {B})}$ s.t. $\sigma \circ (h\vert _{\mathfrak {B}})=1_{\mathfrak {B}}$. Now $\sigma \circ (h\vert _{\mathfrak {D}})$ is a homomorphism $\mathfrak {D}\rightarrow \mathfrak {B}$ sending d to $\sigma (h([d]))=\sigma (h([b]))=b$, which is the desired witness.

Analogously if we define a m-POINTED-CSP($\mathfrak {B},b_1,\dots ,b_m$) it also Karp-reduces to CSP$(\mathfrak {B})$, the reduction being $(\mathfrak {D},d_1,\dots ,d_m)\rightsquigarrow \widetilde{\mathfrak {D}}:=\dfrac{\mathfrak {D}\cup \mathfrak {B}}{\{d_i\sim b_i\}_{i\in [1,m]}}$.¹⁵

An algorithm that calls the CSP$(\mathfrak {B})$-oracle to solve SEARCH-CSP$(\mathfrak {A})$ therefore goes as:

Algorithm 2 finds a full homomorphism $\mathfrak {D}\rightarrow \mathfrak {B}$ $\iff \mathfrak {D}\in$ CSP$(\mathfrak {B})=$ CSP$(\mathfrak {A})$ so it solves SEARCH-CSP$(\mathfrak {A})$; furthermore it calls the CSP$(\mathfrak {B})$-oracle O(nk) times. $\implies$ is by definition, and to see , note that the existence of an $h:\mathfrak {D}\rightarrow \mathfrak {B}$ means for each $i\in [1,|D|]$ there exists some $j\in [1,|B|]$ s.t. $h(d_i)=b_j$, i.e. the existence of an accepting path.¹⁶   $\square$

Next,

Proof

(Proposition 4).

As before, decompose $\mathfrak {D}=(D,E^\mathfrak {D},U^\mathfrak {D},\textbf{0}^\mathfrak {D})$ into connected components and let the CSP$( \mathfrak {A})$ handle the non central components.

For the central component, call SEARCH-CSP( ${\mathfrak {A}^\downarrow }[{A^\downarrow }_0]$) oracle, which thanks to $|{A^\downarrow }_0|<\aleph _0$ and Fact 3 Cook reduces to CSP$( {\mathfrak {A}^\downarrow }[{A^\downarrow }_0])$. If we found an $h\in \text {Hom}_E((D,E^\mathfrak {D}),{\mathfrak {A}^\downarrow }[{A^\downarrow }_0])$, by the assumption, all $h'\in \text {Hom}_E((D,E^\mathfrak {D}),{\mathfrak {A}^\downarrow }[{A^\downarrow }_0])$ is of the form $\sigma h$ for different $\sigma$’s in $\text {End}_{E}({\mathfrak {A}^\downarrow }[{A^\downarrow }_0])$.

Note that $|{A^\downarrow }_0|<\aleph _0\implies |\text {End}_{E}({\mathfrak {A}^\downarrow }[{A^\downarrow }_0])|\le |{A^\downarrow }_0|^{|{A^\downarrow }_0|}< \aleph _0$ and $|\text {End}_{E}({\mathfrak {A}^\downarrow }[{A^\downarrow }_0])|$ is input-independent. Each $\sigma _i\in |\text {End}_{E}({\mathfrak {A}^\downarrow }[{A^\downarrow }_0])|$ is a finite domain function, whose encoding is also input-independent. For each of the O(1)-many $\sigma _i\in |\text {End}_{E}({\mathfrak {A}^\downarrow }[{A^\downarrow }_0])|$, compute $\sigma _i h$ (Since $\sigma _i$ is input-independent and ${A^\downarrow }_0$ is constant-sized, time and space cost is dominated by computing h, which is again dominated by the oracle call to CSP(${\mathfrak {A}^\downarrow }[{A^\downarrow }_0])$); then check if $\sigma _ih$ preserves $\textbf{0}^\mathfrak {D}$ in O(1) time and if $\sigma _ih$ preserves the unary by calling the U-oracle.    $\square$

B Diagrams Summarizing Main Results

1.1 B.1 For Section 4

Fig. 2.

Bounds and Equivalences of CSP$(\mathbb {Z},\text {succ},U)$

1.2 B.2 For Section 5

Fig. 3.

Results for $\mathfrak {A}:=(\mathbb {Z},E,U)$,$\mathfrak {{\mathfrak {A}^\downarrow }}:=(\mathbb {Z},E)$. Recall ${\mathfrak {A}^\downarrow }_0$ is the $\textbf{0}$-component of $({\mathfrak {A}^\downarrow },\textbf{0})$ , and ${A^\downarrow }_0$ its domain.

C Detailed Verification of Miscellaneous Claims

1.1 C.1 Full Details of Section 3, Observation 1

We give more details on the polynomial-time algorithm. Modify the greedy algorithm that determines if an input $G\in$ CSP$(\mathbb {Z},\text {succ})$. For each component $\mathfrak {D}_e\subseteq \mathfrak {D}=(D,\text {succ}^\mathfrak {D},U^\mathfrak {D})$, if $D_e\cap U^\mathfrak {D}=\emptyset$, run the CSP$(\mathbb {Z},\text {succ})$ decider on $\mathfrak {D}_e$; otherwise pick an arbitrary $d_e\in U^\mathfrak {D}\cap D_e$, then run the greedy CSP$(\mathbb {Z},\text {succ})$ algorithm on $\mathfrak {D}_e$ initiated by mapping $d_e$ to b. When the algorithm accepts, it yields a $\{\text {succ}\}$-homorphism $h_e: (D,\text {succ}^\mathfrak {D})\rightarrow (\mathbb {Z},\text {succ})$; accepts on this component only if $h_e(d'_e)\in U$ for each $d'_e\in U^\mathfrak {D}\cap D_e$. Accept $\mathfrak {D}$ only if all components accept.

If the algorithm above accepts, obviously each $h_e$ is a $\{\text {succ},U\}$-homomorphism; conversely if $\mathfrak {D}\in$ CSP$(\mathbb {Z},\text {succ},U)$, so is each component $\mathfrak {D}_e$. Hence there exists some homomorphism $h_e$ sending $d_e$ to some $x\in b+a\mathbb {Z}$. Now Aut$(\mathbb {Z},\text {succ},U)\cong \mathbb {Z}$ composes of translations by ak ($k\in \mathbb {Z}$), so there exists $h'_e\in \text {Hom}_{\text {succ},U}(\mathfrak {D}_e,(\mathbb {Z},\text {succ},U))$ sending $d_e$ to b, which is the one affording acceptance of $\mathfrak {D}_e$.

1.2 C.2 Section 3 Equation 1 Gives the L.U.B. in Karp-Order

To see that $\varinjlim \nolimits _jU_j$ defined as above is indeed the least upper bound, observe that each $U_r\le ^p_m\varinjlim \nolimits _jU_j$ via $m\mapsto mk+(r\mod k)$, and if $Q\subseteq \mathbb {Z}$ is s.t. $U_r\le ^p_mQ$ via $f_r$ for each $r\in [k]$, then $\varinjlim \nolimits _jU_j\le ^p_mQ$ via $x\mapsto f_{r}(\frac{x-(x\mod k)}{k})$ where $r=\left\{ \begin{array}{cc} x\mod k &{} \text { if } k\not \mid x \\ k &{} \text {o.w.} \end{array}\right.$

1.3 C.3 More Discussion on $\text {Diff}_d$

Proof

(details of Section 5, Observation 2).

• 1. For each component $\mathfrak {D}_e$ s.t. $\textbf{0}^\mathfrak {D}\notin \mathfrak {D}_e$, call the CSP$(\mathbb {Z},\text {succ},U)$-oracle, which reduces to the $\varinjlim \big \{U,$ CSP$(\mathbb {Z},\text {Diff}_d,U)\big \}$-oracle.

• 2. For the unique component $\mathfrak {D}_0\ni \textbf{0}^\mathfrak {D}$, run the greedy, efficient SEARCH-CSP$(\mathbb {Z},\text {Diff}_d)$ algorithm to decide if there is a (unique) $\{\text {Diff}_d,\textbf{0}\}$-homomorphism h. Reject if none found.

  Having found h, for each $u\in U^\mathfrak {D}\cap D_0$, call U-oracle on h(u). Note that $h(u)\in [-|D|,|D|]$ and the oracle call once again reduces to calling $\varinjlim \big \{U,$ CSP$(\mathbb {Z},\text {Diff}_d,U)\big \}$.

Other interesting properties of CSP$(\mathbb {Z},\text {Diff}_d,U)$ may be said. For example:

• 1. One may define eventually (mutually) d-largely gapped $\big (\underline{E(M)LG_d}\big )$ analogously to Definitions 1, 3, changing “ ” into “ ”. Keeping the definition of SR (2) unchanged, we have CSP$(\mathbb {Z},\text {Diff}_d,U)\le ^p_TU$ when ELG$_d$-SR, and CSP$(\mathbb {Z},\text {Diff}_d,\boldsymbol{U})\le ^p_T\varinjlim \boldsymbol{U}$ when EMLG$_d$-SR.

• 2. Note that $\text {Aut}(\mathbb {Z},\text {Diff}_d)\cong S_d\times (d\mathbb {Z})^d\cong S_d\times \mathbb {Z}^d$ as groups: indeed, an automorphism permutes the components and then shifts on each component by a distance in $d\mathbb {Z}$.

1.4 C.4 Figure 4 Disproves Converse of Proposition 3(1)

We show that the converse of Proposition 3(1) is not true; in fact, even $\big (\text {End}_{{E,\textbf{0}}}( {\mathfrak {A}^\downarrow }_0)=\{\textbf{1}\}\big )+$ FT PHR.¹⁷ We claim that Fig. 4 below has $\big (\text {End}_{{E,\textbf{0}}}( {\mathfrak {A}^\downarrow }_0)=\{\textbf{1}\}\big )+$ FT but no PHR: $\mathfrak {D}$ embeds in $({\mathfrak {A}^\downarrow },\textbf{0})$ in 2 ways.

Fig. 4.

Counter example ${\mathfrak {A}^\downarrow }$

We verify that Fig. 4 has $\big (\text {End}_{{E,\textbf{0}}}( {\mathfrak {A}^\downarrow }_0)=\{\textbf{1}\}\big )+$ FT yet no PHR. Without naming the constant, each vertex has in-degree 2 and out-degree 2, so any translation is an $\{E\}$-endomorphism — in fact even an automorphism, yielding FT. Furthermore, after naming $\textbf{0}^\mathfrak {A}:=0$, its predecessors and successors are fixed: let $\sigma \in \text {End}_{E,\textbf{0}}({\mathfrak {A}^\downarrow },\textbf{0})$. $\sigma (2),\sigma (1)\in \{2,1\}$ yet if $\sigma (2)=1,$ neither $\sigma (1)=1$ nor $\sigma (1)=2$ would give $E^\mathfrak {A}(\sigma (1),\sigma (2))$. It follows that $\sigma (2)=2$. Likewise, $\sigma (\pm 1)=\pm 1$ and $\sigma (-2)=-2$. For $|n|\ge 3$, $\sigma (n)=n$ follows from induction. The fact that vertices further away are fixed follow from induction, so $\text {End}_{ E,\textbf{0}}({\mathfrak {A}^\downarrow }_0)=\text {End}_{ E,\textbf{0}}({\mathfrak {A}^\downarrow },0)=\{1\}$. On the other hand, embeds into both and .

Remark 1

One might also want to note that a claim stronger than Prop. 3(1), that PHR $\implies$ End , is also false. Consider ${\mathfrak {A}^\downarrow }:=(\mathbb {Z};E,\textbf{0}:=0)$ with edges defined as

Any connected finite $\{E,\textbf{0}\}$-structure must homomorphically map to the non-negative, so a $\{E,\textbf{0}\}$-homomorphism exists iff $\mathfrak {D}$ is loopless, directed acyclic, and each vertex has a non-negative signed distance from $\textbf{0}^\mathfrak {D}$ (i.e. never “$a\rightarrow \textbf{0}^\mathfrak {D}$”). Each $\{E,\textbf{0}\}$-homomorphism from such $\mathfrak {D}$’s should it exist, must be unique, mapping each vertex to its distance from $\textbf{0}^\mathfrak {D}$. However $\text {End}_{\{E,\textbf{0}\}}{(({\mathfrak {A}^\downarrow },\textbf{0}))}$ contains swapping $-4\leftarrow -3$ with $-2\leftarrow -1$.

1.5 C.5 Definition of Connectedness in General Case (Re. Comments at Beginning of Section 5)

One may even have freedoms on the arities of the non unaries $\{E_j\}_j$. In these cases, call $x,y\in \mathbb {Z}$ “connected” if

• 1. (Base) there exists some $j\in [1,k]$ s.t. $(\dots ,x,\dots ,y,\dots )$ or $(\dots ,y,\dots ,x,\dots )\in E_j^\mathfrak {D}$. OR

• 2. $\exists z$ s.t. x, z are connected and z, y are connected.

Call $\mathfrak {D}'\subset _{\text {ind }}\mathfrak {D}$ a connected component of $\mathfrak {D}$ if for any $x\ne y\in \mathfrak {D}',$ x and y are connected; and $\forall x\in \mathfrak {D}'$, $\forall z\notin \mathfrak {D}'$, x, z are not connected. (This happens if and only if the Gaifman graph of $\mathfrak {D}'$ is a connected component of the Gaifman graph of $\mathfrak {D}$.) Vacuously, an isolated vertex (an $x\in D$ that does not appear in any $E_j^\mathfrak {D}$ with arity $\ge 2$) is a connected component.

A $\{E_j, \textbf{c}\}_{j\in [1,k]}$-structure $\mathfrak {D}$ is connected if $\mathfrak {D}$ itself is a connected component; equivalently, $\mathfrak {D}$ is not the disjoint union of two or more connected components.

A notable property relevant to later discussion: when a finite, connected $\{E_j \}_{j\in [k]}$-structure $\mathfrak {D}$ has $(E_j^\mathfrak {D}=\emptyset )_{j\in [k]}$ then $|\text {Hom}_{ \{E_j\}_{j\in [k]},\textbf{c}}(\mathfrak {D},(\mathbb {Z}, (E_j^\mathfrak {A})_j,\textbf{c}^\mathfrak {A}))|=1$ for free, because $D=\{\textbf{c}^\mathfrak {D}\}$ in this case: $\textbf{c}^\mathfrak {D}$ is its own connected component and $\mathfrak {D}$ is connected.

1.6 C.6 Verification of Corollary 4

Proof

With assumption set 1, by $U\in \texttt{P}$ and CSP$({\mathfrak {A}^\downarrow })\le ^p_m$ CSP$(\mathfrak {A})$, all other terms in the direct limit are reduced to CSP$(\mathfrak {A})$. Likewise for assumption set 2.

1.7 C.7 Remark on Further Generalizing Sect. 5

• 1. The converses of Proposition 2-type reductions, i.e. $(\mathbb {Z},E,U)\le ^p_m(\mathbb {Z},E,U,\textbf{0})$, always hold by the identity reduction (as pp-sentence).¹⁸

• 2. All arguments in Theorem 3 to Proposition 4 generalize to $(\mathbb {Z},\boldsymbol{E},\boldsymbol{U})$, i.e. with finitely many binaries and finitely many unaries; $\textbf{0}$ may be changed to $\textbf{c}$ for any $\textbf{c}\in \mathbb {Z}$. In these general cases we need a careful definition of connected components , and the arguments would be longer, but still completely analogous. For the ease of presentation, we assume our current presentation.

1.8 C.8 More Discussion on Future Directions

Section 1, we believe these are solid first steps towards characterizing the CSP of non-$\omega$-categorical binary structures. In fact, one meaningful subclass of non-$\omega$-categorical structures where $(\mathbb {Z},\text {succ},U)$ lives is those that are mutually algebraic (e.g. [12, 21]) ones, which also include all $\mathfrak {A}=(\mathbb {Z},E,U)$’s where the underlying ${\mathfrak {A}^\downarrow }$ has (universally) bounded degree. Hence one may ask:

Problem 4

Let ${\mathfrak {A}^\downarrow }=(\mathbb {Z},E)$ be of degree $d\ge 2$, i.e. $\deg _{\mathfrak {A}^\downarrow }(v)\le d$ for any $v\in \mathbb {Z}$.

• 1. Is CSP$({\mathfrak {A}^\downarrow })$ either in $\texttt{P}$ or in $\texttt{NPC}$? If not, are there meaningful dividing lines for tractability at least?

• 2. Can we fully characterize the complexity of CSP$(\mathbb {Z},E,U)$ $(U\subseteq \mathbb {Z})$ now that we further restricted the underlying E by degree?

Some facts to bear in mind: that in general it is not true that the CSP of infinite digraphs have a P-NPC dichotomy [3, 4].¹⁹ On the other hand, for any $d\ge 2$ the disjoint union of all (isomorphic types of) finite undirected graph of degree $\le 2$ has the same CSP as $K_{d+1}$, by e.g. Brook’s theorem.