On amenability of constraint satisfaction problems

03/29/2020
by   Michal R. Przybylek, et al.
University of Warsaw
0

Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern model theory. The aim of this paper is threefold. (1) We give a simple purely-logical proof of the claim and show that the advanced results from topology and model theory are not needed; (2) we introduce an intrinsic characterisation of the statement "definable CSP has a solution iff it has a definable solution" and investigate it in general intuitionistic set theories (3) we show that the results from modern model theory are indeed needed, but for the implication reversed: we prove that "definable CSP has a solution iff it has a definable solution" holds over a countable structure if and only if the automorphism group of the structure is extremely amenable.

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1. Introduction

Nowadays, there is no longer any question that computer-aided solutions to real-world problems are critical for the industry. Even relatively small problems can have high complexity, what makes them intractable for human beings. Very many real-world decision problems of high complexity can be abstractly specified as constraint satisfaction problems e.g. hardware verification and diagnosis: (Clarke et al., 2003), (Gotlieb, 2012), automated planning and scheduling (Do and Kambhampati, 2001), (Fox and Sadeh-Koniecpol, 1990), temporal and spatial reasoning (Renz and Nebel, 2007), (Bodirsky and Chen, 2007), air traffic managment (Allignol et al., 2012),… to name a few. A constraint satisfaction problem (CSP) can be abstractly defined as a triple , where:

  • is the domain

  • is the set of variables

  • is a set of constraints of the form , where and

A solution to this problem is an assignment that satisfies all constraints in , i.e.: for every we have that holds. Classical and best explored variant of CSP is finite CSP — i.e. the set of variables, the set of constraints and the domain of the variables are all finite. Although the computational cost of finite CSP is high (i.e. the general problem is NP-complete), it can be solved in a finite time by a machine111For a general reference on solving classical CSP see (Rossi et al., 2006).

Unfortunately, when it comes to problems concerning behaviours of autonomous systems, the classical variant is too restrictive. Such problems can be naturally specified as CSP with infinite sets of variables (corresponding to the states of a system) and infinite sets of constraints (corresponding to the transitions between the states of a system). In recent years, we have witnessed a giant progress in solving infinite variants of CSP. The authors of (Bodirsky and Nešetřil, 2006) (see also a survey article (Bodirsky, 2008)) applied algebraic and model-theoretic tools to analyze CSP over infinite domains. This research inspired the Warsaw Logical Group to investigate, so called, locally finite CSP — i.e. CSP over finite domain, whose variables and constraints form a definable infinite set of finite arities (see (Klin et al., 2015) and (Ochremiak, 2016)). They showed that that CSP defined in the first-order theory of rational numbers with their natural ordering can be solved effectively222To be more precise, they worked in the maximal tight extension of the theory, see (Przybylek, 2020) for more details.. The key technical observation was a property of definability in rational numbers, which we reformulate as the following axiom.

Axiom 1 (DEF-CSP).

A definable CSP over a finite domain has a definable solution if and only if it has a solution.

It was further observed in (Bodirsky et al., 2013) that infinite CSP — i.e. CSP whose domains, variables and constraints form infinite definable sets reduce to locally finite CSP (an explicit reduction is given in Section 4 of (Klin et al., 2016)). For this reason, without loss of generality, we can focus on CSP over finite domains. An example of such CSP is the problem of 3-colorability of graphs.

Example 1.0 (3-colorability of an infinite graph).

Consider the following infinite graph definable over natural numbers with equality:

One may wander if this infinite graph is 3-colorable. Figure 1 gives the negative answer by exhibiting a finite subgraph, which is not 3-colorable. This problem fits into the framework of CSP as follows: the domain consists of three constants , the set of variables is the set of vertices of the graph, and the set of constraints is given as , where is the inequality relation on . Observe, that every set definable in , can be treated as a set definable in the rational numbers with their natural ordering. Therefore, we can use the machinery of (Klin et al., 2015) to solve such problems effectively.

Figure 1. Counterexample to 3-colorability.

SET PASSW

START

AUTH TRY 1

AUTH TRY 2

AUTH TRY 3

GRANT AUTH

?

?

?

?

?

,

EXIT AUTH

CHNG PASSW

Figure 2. A register machine that models access control to some parts of the system.

An important source of infinite graphs come from finite memory machines in the sense of Kaminski, Michael and Francez (Kaminski and Francez, 1994).

Example 1.0 (Access-control register machine).

Figure 2 represents a register machine with one register . The machine starts in state “SET PASSW”, where it awaits for the user to provide a password . This password is then stored in register , and the machine enters state “START”. Inside the blue rectangle the machine can perform actions that do not require authentication, whereas the actions that require authentication are presented inside the red rectangle. The red rectangle can be entered by the state “GRANT AUTH”, which can be accessed from one of three authentication states. In order to authorise, the machine moves to state “AUTH TRY 1”, where it gets input from the user. If the input is the same as the value previously stored in register , then the machine enters state “GRANT AUTH”. Otherwise, it moves to state “AUTH TRY 2” and repeats the procedure. Upon second unsuccessful authorisation, the machine moves to state “AUTH TRY 3”. But if the user provides a wrong password when the machine is in state “AUTH TRY 3”, the register is erased (replaced with a value that is outside of the user’s alphabet) — preventing the machine to reach any of the states from the red rectangle. Inside the red rectangle any action that requires authentication can be performed. For example, the user may request the change of the password. Observe that in contrast to finite automata, the graph of possible configurations in a register machine is infinite. Nonetheless, we can formulate many properties of such graphs as CSP problems over natural numbers with equality and solve them effectively.

Remark 1.1 ().

Every constraint satisfaction problem can be presented as a pair of relational structures , over the same relational signature . This signature consists of a pair for every relation from a constraint . The interpretation of symbol in is , and the interpretation in is the relation itself. Moreover, a solution to the CSP is a homomorphism from to .

To understand these results, we have to recall some basic concepts from model and set theory. We shall do this in Section 2. The aim of this paper is to reverse the theorem stated as Axiom DEF-CSP and give the full characterisation of set theories with atoms where Axiom DEF-CSP holds. But we shall do much more: in Section 3 we reformulate Axiom DEF-CSP as an intrinsic Axiom of any Boolean topos (Axiom CSP) and show that it is equivalent to another, well-known, axiom: Boolean prime ideal theorem. Then in Subsection 3.1 we show that Axiom CSP holds in for every finite if and only if the automorphism group of is extremely amenable. By the transfer principle we conclude that this is equivalent to Axiom DEF-CSP in . In Section 4 we investigate Axiom CSP in non-Boolean toposes pointing out many obstacles to the equivalence between Boolean prime ideal theorem and possible formulations of Axiom CSP.

2. Set theory with atoms

For any structure we can build a von Neumann-like hierarchy of sets with elements from (Mostowski, 1939), (Halbeisen, 2017). The elements of will be thought of as “atoms”.

Definition 2.0 (The cumulative hierarchy of sets with atoms).

Let be an algebraic structure with universe . Consider the following sets defined by transfinite recursion:

  • if is a limit ordinal

Then the cumulative hierarchy of sets with atoms is defined as .

Observe, that the universe carries a natural action of the automorphism group of structure — it is just applied pointwise to the atoms of a set. If is a set with atoms then by its set-wise stabiliser we shall mean the set: ; and by its point-wise stabiliser the set: . Moreover, for every , these sets inherit a group structure from .

There is an important sub-hierarchy of the cumulative hierarchy of sets with atoms , which consists of “symmetric sets” only. To define this hierarchy, we have to equip with a structure of a topological group.

Definition 2.0 (Symmetric set).

A set is symmetric if the set-wise stabilisers of all of its descendants is an open set (an open subgroup of ), i.e. for every we have that: is open in , where is the reflexive-transitive closure of the membership relation .

Of a special interest is the topology on inherited from the product topology on . We shall call this topology the canonical topology on . In this topology, a subgroup of is open if there is a finite such that: , i.e.: group contains a pointwise stabiliser of some finite set of atoms.

Definition 2.0 (Sets with atoms).

The sub-hierarchy of that consists of symmetric sets according to the canonical topology on will be denoted by .

Here are some standard examples of sets with atoms.

Example 2.0 (The basic Fraenkel-Mostowski model).

Let be the structure of natural numbers with equality. We call the basic Fraenkel-Mostowski model of set theory with atoms. Observe that is the group of all bijections (permutations) on . The following are examples of sets in :

  • all sets without atoms , e.g. 

  • all finite subsets of , e.g. 

  • all cofinite subsets of , e.g. 

Here are examples of sets in which are not symmetric:

  • the set of all functions from to

Example 2.0 (The ordered Fraenkel-Mostowski model).

Let be the structure of rational numbers with their natural ordering. We call the ordered Fraenkel-Mostowski model of set theory with atoms. Observe that is the group of all order-preserving bijections on .All symmetric sets from Example 2.4 are symmetric sets in when is replaced by . Here are some further symmetric sets:

Observe that the group is actually the group of automorphism of structure extended with constants , i.e.: . Then a set is symmetric if and only if there is a finite such that and the canonical action of topological group on discrete set is continuous. A symmetric set is called -equivariant (or equivariant in case ) if . Therefore, the (non-full) subcategory of on -equivariant sets and -equivariant functions is equivalent to the category of continuous actions of the topological group on discrete sets.

Example 2.0 (Equivariant sets).

In the basic Fraenkel-Mostowski model:

  • all sets without atoms are equivariant

  • all finite subsets are -equivariant

  • all finite subsets are -equivariant

  • are equivariant

Definition 2.0 (Definable set).

We shall say that an -equivariant set is definable if its canonical action has only finitely many orbits, i.e. if the relation has finitely many equivalence classes.

For an open subgroup of let us denote by the quotient set . This set carries a natural continuous action of , i.e. for , we have . All transitive (i.e. single orbit) actions of on discrete sets are essentialy of this form (see for example Chapter III, Section 9 of (MacLane and Moerdijk, 2012)). Therefore, equivariant definable sets are essentially finite unions of sets of the form . Moreover, if structure is -categorical, then equivariant definable sets are the same as sets definable in the first order theory of extended with elimination of imaginaries (Przybylek, 2020).

Definition 2.0 (Ramsey property).

A structure has a Ramsey property if for every open subgroup of , every function and every finite set there is such that is constant on , i.e. there exists such that for all we have .

The authors of (Klin et al., 2015) working in the ordered Fraenkel-Mostowski model , showed that an equivarian definable constraint satisfaction problem has a solution if and only if it has an equivariant definable solution. A careful inspection of their proof shows that this result can be strengthen to all equivariant sets. The proof is based their results on a recently discovered result in topological dynamic (Pestov, 1998). We shall show that this advanced result is not needed at all. Before that, let us recall a very old problem about the independence of the Axiom of Choice from other axioms.

Definition 2.0 (Ideal).

Let be a Boolean algebra. An ideal in is a proper subobject satisfying the following conditions:

  • if then

  • if then for every such that we have that

Definition 2.0 (Prime ideal).

Let be an ideal in . We say that is prime if for every either or .

The Boolean Prime Ideal Theorem (BPIT) states that every ideal in Boolean algebra can be extended to a prime ideal. It is a routine to check that BPIT is follows from the Axiom of Choice (Jech, 2008), (Howard and Rubin, 1998). It was a long-standing open problem whether the reverse implication holds as well. In 1964 Halpern (Halpern, 1964) used a model of ZFA over the rational numbers with the canonical ordering (nowadays called the ordered FM model ) to prove that the Axiom of Choice is not a consequence of BPIT in set theory with atoms. That is, he showd that in the Axiom of Choice fails badly, but BPIT holds. This result was later amplified in (Halpern and Lévy, 1971) to give the first proof that the Axiom of Choice is not a consequence of BPIT in ZF (without atoms).

Remark 2.1 ().

An ideal in can be represented by a homomorphism to a Boolean algebra , i.e. . A prime ideal is an ideal that can be represented by a homomorphism to equipped with the usual Boolean algebra structure. Therefore, an ideal in can be extended to a prime ideal iff has a prime ideal . In this case, . This means, that BPIT is equivalent to the statement that every non-trivial Boolean algebra has a prime ideal. We shall use this characterisation in Section 3.

It is the result of Halpern that we use to prove that in the cumulative hierarchy the following holds: “an -equivariant CSP has a solution if and only if it has an -equivariant solution”. This may be formulated as ZFA-Axiom CSP.

Axiom 2 (ZFA-Axiom CSP).

An -equivariant CSP has an -equivariant solution if and only if it has a solution.

But, in fact, we do more. First, we reformulate ZFA-Axiom CSP as an intrinsic property of a topos and call it Axiom CSP. Then, we show that in Boolean toposes Axiom CSP is actually equivalent to BPIT. In particular, for every set of atoms we have that satisfies Axiom CSP if and only if it satisfies BPIT. This will give (1) and (2) from the abstract, with the one caveat: equivariance is not an intrinsic property of , therefore we have to state Axiom CSP in every , and then by the transfer principle (see (Przybylek, 2020)) recover the desired property.

A careful inspection of the proof of Halpern (Halpern, 1964) shows that the crucial property of is that has the Ramsey property. This was further explored in (Johnstone, 1984) and in full details in (Blass, 1986). Moreover, Theorem 2 of (Blass, 1986) states that the Ramsey property of is equivalent to BPIT in . Therefore, Ramsey property of is also equivalent to Axiom CSP in . This is, however, not enough from the reason mentioned in the above: the literal translation of BPIT to says that every symmetric ideal on a symmetric Boolean algebra can be extended to a symmetric prime ideal. This statement is weaker than: “every -equivariant ideal on an -equivariant Boolean algebra can be extended to an -equivariant prime ideal”. Fortunately, inspection of the proof (Blass, 1986) shows that the constructed prime ideal is, in fact, -equivariant.

In 2005 Kechris, Pestov and Todorcevic in their famous work on topological dynamic (Kechris et al., 2005) showed that for countable single-sorted structures the Ramsey property for is equivalent to extreme amenability of .

Definition 2.0 (Extremely amenable group).

A topological group is called extremely amenable if its every action on a non-empty compact Hausdorff space has a fixed point.

Therefore, for countable single-sorted structures BPIT in is equivalent to the extreme amenability of . This was first observed by Andreas Blass in 2011 in (Blass, 2011). Furthermore, Proposition 4.7 in (Kechris et al., 2005) says that the class of such structures coincides with the class of structures that arise as the Fraisse limit of a Fraisse order class with the Ramsey property.

From the perspective of effective computation in set with atoms, structure have to be countable, thus the restriction in the above equivalence to countable structures only is not severe. Moreover, the Fraisse limit (over a relational signature) is always -categorical, a property crucial for the termination of certain while-programs (see (Przybylek, 2020) for more discussion).

3. The axiom in Boolean toposes

In this section we shall work in the internal language of a Boolean topos. A reader who is not familiar with the notion of the internal language may read the proofs as taking place in ZFA minus the axiom of extensionality333A set in a non-well-pointed topos may have more content than mere elements (e.g. -equivariant sets with atoms). We shall be extra careful when defining set-theoretic concepts, such as finiteness, or a prime ideal. Although, in Boolean toposes many different definitions of such concepts coincide, this would not be the case for non-Boolean toposes studied in the next section.

Definition 3.0 (Kuratowski finiteness).

Let be a set. By we shall mean the sub-join-semilatice of the powerset generated by singletons and the empty set. A set is Kuratowski-finite if it is the top element in .

For the rest of this section we shall just write finite set for Kuratowski-finite set. The chief idea behind the above definition is that since a non-empty finite set can be constructed from singletons by taking binary unions, we have a certain induction principle. Let us assume that: (base of the induction) holds for singletons, and (step of the induction) whenever holds for and then holds for , then (conclusion) holds for . For example, we can show that the Axiom of Choice internally holds for finite sets. To see this, recall the usual reformulation of AC for finite sets: every surjection onto a finite set has a section , i.e.: . Let us assume that is a surjection. Then for every finite , the function , where , is also a surjection. This can be proven by induction over . If is the empty set, or a singleton, then the claim clearly holds. Therefore, let us assume the claim holds for finite , and show that it also holds for . Since the topos is Boolean, without the loss of generality, we may assume that and are disjoint. The function decomposes on disjoint and with . Because the Cartesian product of two surjections is a surjection, we may infer that is a surjection, what completes the step of the induction. Therefore, if is a surjection then for every finite we have that is a surjection. By setting , we obtain that is a surjection and so for every there exists such that . In particular, for there exists such that . But, , what completes the proof.

Definition 3.0 (Finitary relation).

For sets we shall call the set of finitary relations from to . A finitary relation is a partial function if the following holds: . We shell denote the set of finitary partial functions from to by .

In a Boolean topos a subset of a finite set is finite, therefore if and are finite, then a finitary relation from to is just a relation from to .

Definition 3.0 (Finitary homomorphism).

Let and be two relational structures over a common signature . A relational homomorphism from to is a relation that preserves all relations , i.e.:

A finitary homomorphism is a finitary partial function which is also a relational homomorphism. The set of all finitary homomorphisms from to will be denoted by .

Let us observe that there is a morphism that assigns to a finitary homomorphism its domain .

Definition 3.0 (Jointly-total homomorphisms).

Let and be two relational structures over a common signature . We shall say that a set of finitary homomorphisms is jointly total if every finite is a subdomain of a finitary homomorphisms from , i.e.: .

Now, we are ready to state Axiom CSP in Boolean toposes.

Axiom 3 (CSP).

For every relational signature and a pair of structures and over such that is a finite cardinal, the following are equivalent:

  • there exists a homomorphism from to

  • the set of finitary homomorphisms is jointly total

In the below we shall show that Axiom CSP in Boolean toposes is equivalent to Boolean prime ideal theorem (BPIT). Let us recall the terminology first.

Definition 3.0 (Boolean algebra).

An algebra is a structure , where is a constant, is a binary operation, and is an unary operation. Consider relation defined as: . We say that is a Boolean algebra if the following holds:

  • is a partial order on with finite joins given by and the greatest element

  • for every we have that:

If is a Boolean algebra, then is its internal true value, and operation is the internal conjunction. Other operations in a Boolean algebra can be defined in the usual way:

  • for the false value

  • for the internal disjunction

Axiom 4 (BPIT).

For every non-trivial Boolean algebra there is a homomorphism to the initial Boolean algebra .

The constraint satisfaction problem is defined over relational structures. Therefore, to fit into the framework of CSP we should treat a Boolean algebra as if it was defined over a relational signature, with an unary predicate , ternary predicate and binary predicate . The axioms should express that there exists unique that satisfy and that and are functional relations.

Theorem 3.6 (Axiom CSP implies BPIT).

Axiom CSP implies BPIT in Boolean toposes.

Proof.

Let be a Boolean algebra. By Axiom CSP, it suffices to show that the set of finitary homomorphisms is jointly total, i.e. for every finite in there exists a partial homomorphism . We can assume that is closed under Boolean-algebra operations and still finite. The reason for that is that if is finite then in a Boolean topos is finite as well (it coincides with ). Because we have shown that the AC holds for finite sets, the standard proof of Zorn’s Lemma can be carried over to our setting to show that has a maximal ideal, therefore (using again Boolean logic of the topos) has a prime ideal. ∎

Theorem 3.7 (BPIT implies Axiom CSP).

Axiom BPIT implies Axiom CSP in Boolean toposes.

Proof.

Let us assume that and are structures over relational signature . Furthermore, assume that is a finite cardinal and the set of finitary homomorphisms is jointly total. We shall treat as a set of propositional variables. Consider the following subsets of propositions , where is treated as the free Boolean algebra on :

Consider the following set of propositions: . Let us say that two propositions from are equivalent if there is a finite such that every valuation satisfying satisfies . Then divided by this equivalence relation is again a Boolean algebra with the usual operations. We want to show that is non-trivial, i.e. . Because every finite subset involves only finitely many variables , the set is finite. In fact, can be rewritten as the union of:

Since is finite, by the assumption, there exists a finitary homomorphism with , which induces a valuation . By the definition of the constraints, this valuation makes satisfiable. Therefore, every finite is satisfiable, and so is non-trivial. By Axiom BPIT, there is a prime ideal , which composed with the canonical embedding gives a prime ideal on . By the definition maps propositions from to . Consider the restriction of to variables . By propositions valuation is total and by propositions it is single-valued. Moreover, by propositions the valuation dos not violate any constraints. Therefore, is a homomorphism from to . ∎

3.1. Characterisation theorems

This subsection states our main characterisation theorems. We shall begin with a simple purely-logical proof that ZFA-Axiom CSP holds in the ordered Fraenkel-Mostowski model of set theory with atoms. Observe, that we rely only on the old combinatorial result of Halpern (Halpern, 1964).

Theorem 3.8 (ZFA-Axiom CSP in ).

ZFA-Axiom CSP holds in .

Proof.

A careful inspection of the proof of Halpern (Halpern, 1964) shows that if a Boolean algebra in is -equivariant than it has an -equivariant prime ideal. Therefore, BPIT holds in . By Theorem 3.7, Axiom CSP holds in . Therefore, ZFA-Axiom CSP holds in . ∎

The proof of the next theorem is similar.

Theorem 3.9 (ZFA-Axiom CSP in ).

Let be a countable structure. ZFA-Axiom CSP holds in if and only if the automorphism group of is extremely amenable.

Proof.

A careful inspection of Theorem 2 of (Blass, 1986) states that the Ramsey property of is equivalent to the property that every -equivariant Boolean algebra in has an -equivariant prime ideal. Therefore, Ramsey property of is also equivalent to ZFA-Axiom CSP in . And by the result of Kechris, Pestov and Todorcevic (Kechris et al., 2005), ZFA-Axiom CSP in is equivalent to extreme amenability of the automorphism group of . ∎

We shall now consider a weaker versions of Axiom CSP and show that in continuous sets over a localic group it is equivalent to Axiom CSP.

Definition 3.0 (Compact object).

An object of a cocomplete category is called compact if its co-representation preserves filtered colimits of monomorphisms.

Axiom 5 (Compact CSP).

For every relational signature and a pair of structures and over such that is a finite cardinal and is compact, the following are equivalent:

  • there exists a homomorphism from to

  • the set of finitary homomorphisms is jointly total

An object in can be regarded as a relational structure , where is a binary relation on . Then a function is a homomorphism iff it is equivariant.

Theorem 3.11 (Compact CSP implies Axiom CSP in continuous sets).

Let be a localic group and be the topos of its continuous actions on . Then Compact CSP holds in iff Axiom CSP holds in .

Proof.

An object in can be represented as a disjoint union of its orbits . By the definition of compactness for every finite set of orbits the object is compact.

Let us assume that every finite subset of has a solution, by Axiom CSP for compact objects every has an equivariant solution. Moreover, if then every equivariant solution of can be restricted to an equivariant solution of . Observe that can be regarded as a classical structure over a signature extended by relations . Then by Axiom CSP for , the classical structure over this extended signature has a solution. But, by definition of , such solution must be equivariant. ∎

We can summarize the above characterisations in the next theorem.

Theorem 3.12 (Characterisation theorem).

Let be a countable structure. Then the following are equivalent:

  1. is the Fraisse limit of a Fraisse order class with the Ramsey property

  2. is extremely amenable

  3. ZFA-Axiom CSP holds in

  4. Axiom DEF-CSP holds in

  5. Axiom CSP holds in

  6. Axiom CSP holds in for every finite

Proof.

is Proposition 4.7 in (Kechris et al., 2005). is the subject of Theorem 3.9. is the consequence of Theorem 3.11. is trivial, and follows from Theorem 3.6 and the main theorem of (Blass, 2011). is trivial. ∎

4. The case of non-Boolean toposes

When we move to non-Boolean toposes, we have to be extra careful when stating classical definitions and axioms, because in constructive mathematics classically equivalent statements may be far different. Fortunately for us, the concept of Boolean algebra, ideal and prime ideal move smoothly to the intuitionistic setting with one caveat: not every maximal ideal in a Boolean algebra has to be prime.

On the other hand, Axiom CSP is much more difficult to handle in the intuitionistic setting. Actually, we have several different variants of Axiom CSP depending on our interpretation of “finiteness” and admissible relational structures. Therefore, we should not expect that Axiom CSP is equivalent to BPIT in constructive mathematics, because BPIT does not involve any notion of finiteness and there is not much concern about admissibility of Boolean algebra operations (however, we could take this into account). In general, the stronger the notion of “finiteness” and “admissibility” is, the stronger Axiom CSP we obtain.

Let us discuss some possible definitions for an admissible structure :

  1. Only complemented relations are admissible. That is, subobjects such that there exists a subobject with the property that and .

  2. is decidable. That is, the sobobject that correspond to the equality predicate is complemented. Because, we assume that equality is always presented in the signature, decidability of is subsumed by the previous point.

  3. All relations are admissible.

In the next subsections we discuss Axiom CSP with respect to two internal notions of “finiteness”: Kuratowski finiteness from Definition 3.1 and Kuratowski subfiniteness (i.e. being a subobject of a Kuratowski finite object).

4.1. Kuratowski finiteness is too strong

Consider Sierpienski topos . It is a routine to check that BPIT holds in , but Axiom CSP does not hold even in case “finiteness” is interpreted as Kuratowski finiteness and only complemented relations are admissible. For a counterexample consider structures from Figure 3. The structure on the right side is the terminal object equipped with the empty unary relation . The structure on the left side is the only non-trivial subobject of equipped with the full unary relation . There is a unique morphism from to , but it is not a homomorphism, since it does not preserve the unary relation, i.e. . On the other hand, the only Kuratowski finite subobject of is and the object of homomorphisms is isomorphic to .

This example shows that Axiom CSP with Kuratowski finiteness is too strong to be provable from BPIT, and too strong in general. If we weaken Axiom CSP by weakening the notion of finiteness to Kuratowski subfiniteness then Axiom CSP will hold even if all relations are admissible. The reason is that a structure in can be encoded as a structure in with one additional relation encoding the graph of function , i.e.  and one unary relation to distinguish domain from the codomain, i.e. . Then for every finite substructure of there is a finite substructure corresponding to a Kuratowski subfinite substructure of . Therefore, Axiom CSP holds in by the Axiom CSP in .

4.2. Kuratowski subfiniteness is too weak

Consider topos . Figure 4 shows an example of a Boolean algebra, which does not have a prime ideal. Moreover, this example explicitly shows why we cannot carry over our proof of Theorem 3.6 to constructive mathematics — the Boolean algebra under consideration is Kuratowski finite, what means that in not every finite Boolean algebra has a prime ideal. On the other hand, Axiom CSP with Kuratowski finite subobjects fails and with Kuratowski subfinite subobjects holds for the same reasons as in the Sierpienski topos . Therefore, example from Figure 4 shows that Axiom CSP with Kuratowski subfiniteness is too weak to prove BPIT.

hfflfflfflfflChOOChfflfflC
Figure 3. Axiom CSP fails in for Kuratowski finiteness. The structures are equipped with a single unary relation that holds on blue elements only.
 h[blue]//C