Orbit-finite dimensional vector spaces, and orbit-finite systems of linear equations

Here is some naive questions that came to one’s mind when considering vector spaces where the dimension is not finite but orbit-finite: under a reasonable assumption on atoms , say -categoricity,

— is it true that every vector subspace of an orbit-finite dimensional vector space is orbit-finite dimensional?

— is it true that the set of solutions of an orbit-finite  system of linear equations is orbit-finite dimensional?

Automorphism groups of definable structures

We consider equality atoms , i.e., is a countable set with the equality relation. Recall that by definable set we mean a structure which can be defined (in a possibly nested fashion) from using finite unions, finite tuples, and set-builder expressions with first-order formulas ranging over , e.g. is definable using the parameter . A definable relational structure is a relational structure  where and each relation is a definable set. Up to isomorphism, definable structures are the same as structures which interpret in , using first-order interpretations (this correspondence preserves parameters), but sometimes using definable sets makes constructions easier.

The problem described in the previous post is to determine isomorphism between two definable structures. In this post, we describe this problem in terms of automorphism groups. This post is based on discussions with Manuel Bodirsky and Antoine Mottet.

System of equations over sets of integers

Consider system of equations of the form

where variables are interpreted as sets of integers, and are expressions built from variables , constants sets , addition (interpreted element-wise as , union ,  and a restricted form of intersection. Namely, we allow expressions of the form  with and expression and a constant denoting a semilinear subset of .

(The study of such systems of equations stems from the analysis of pushdown automata over timed atoms [1], where equations in the form above are used to represent the delays of the timestamps of stack symbols between the time of push and the time of the corresponding pop.)

A solution is an assignment of a subset of integers to each variable . We are interested in the emptiness problem: is empty in the least solution of a given system of equations as above? For example, natural numbers are the least solution to and even integers are the least solution to . Finite sets of constants like can be succinctly expressed by small equations using their binary expansion.

Note that the use of intersection in  is severely restricted. If intersections of the form for two variables are allowed, then the emptiness problem is equivalent to the emptiness problem of conjunctive grammars, and the latter problem is undecidable  [2].

On the other hand, emptiness is in PTIME if no intersection is allowed: By interpreting the commutative as the noncommutative language concatenation  (which preserves emptiness in the absence of intersections), we reduce to a context-free grammar—whose emptiness problem is solvable in PTIME.

Not all decision problems need be computationally simple, even in the absence of intersections. The -membership problem, which asks whether  (i.e., a word with the same number of ‘s and ‘s in the noncommutative view) is in the least solution, is NP-complete. This is an interesting example where parsing is harder than emptiness—it is usually the other way around, e.g., for conjunctive grammars emptiness is undecidable, while parsing is decidable (even in PTIME [2]).

Emptiness for equations where only intersections with finite are allowed, is NP-complete too. This can be shown by a simple iterative algorithm computing coarser and coarser under-approximations of the least solution, using at each step an oracle for the -membership problem for intersection-less equations.

The general case of intersections with semilinear is open. The emptiness problem in this case is equivalent to the emptiness problem for the class of timed poshdown automata [1]. We conjecture that the problem decidable in this case, and that, moreover, the least solution is semilinear.

REFERENCES

[1] L. Clemente and S. Lasota, “Timed Pushdown Automata”, LICS’15.

[2] A. Jez and A. Okhotin, “Conjunctive grammars over a unary alphabet: Undecidability and unbounded growth,” Theory Comput. Syst., vol. 46, no. 1, pp. 27–58, 2010.

Orbit-finite 0-simple semigroups

We ask the question of how the Rees-Suskevitch theorem should be generalised to provide representations of orbit-finite 0-simple semigroups. We believe that orbit-finite groupoids will be useful for this.

Orbit-finite groupoids

Orbit-finite groups are finite, and therefore not very interesting from the perspective of sets with atoms. It turns out that orbit-finite groupoids form interesting structures. They will be useful in the study of orbit-finite semigroups in the following post.

Can orbit-finite semigroups be straightened?

A straight set is an equivariant set which is equivariantly isomorphic to a disjiont union of sets of the form . A straight semigroup is an equivariant semigroup whose universe is a straight set. We raise the following:

Question 1: is every equivariant, orbit-finite semigroup an image of some orbit-finite, straight semigroup under an equivariant mapping?

We show some simple preliminary observations towards the above question.

Orbit-finite CSPs

Consider the following decision problem (for classical Turing machines). Let be a fixed, finite signature.

Input: Two orbit-finite relational structures , over .

Decide: is there a homomorphism from to ?

Question 1. Is the above problem decidable?

Observe that we are not asking for the existence of an equivariant homomorphism. If this were the case, the problem would be clearly decidable, since there are only finitely many equivariant homomorphisms between orbit-finite sets, and they can be effectively scanned. In this problem, the homomorphisms do not even have to be finitely supported! Below is an example where a homomorphism exists, but no finitely supported homomorphism exists.

Example. Consider the equality atoms. Let be the graph whose vertices are pairs of distinct atoms, and in which edges join a pair with its inverse. Let be the -clique. Then maps to , but no finitely supported homomorphism exists.

Following the idea of this post, in order to solve this problem, it might be useful to move to the totally ordered atoms .

Question 2. In the totally ordered atoms, if there is a homomorphism between orbit-finite structures , , is there one which is equivariant?

A positive answer to Question 2 would imply a positive answer to Question 1.

Edit. After reading Bartek’s comment below, I noticed that Question 2 has an even more negative answer. Consider the structures and (orders reversed). There is a homomorphism from to , namely the mapping , but there is none which is finitely supported.

Decidability results based on a WQO in homogeneous atoms

There are several decision problems for certain computation models with atoms that are decidable when atoms admit certain well quasi order (WQO),  and undecidable otherwise. We recall the problems and formulate few questions related to WQOs.