In zero-sum turn-based infinite duration games played on graphs, two players, Eve and Adam, take turns in moving a token along the edges of a given (potentially infinite) directed graph, whose edges have labels from a given set of colors. This interaction goes on forever and produces an infinite sequence of colors according to which the outcome of the game is determined, using a valuation which is fixed in advance. The complexity of a strategy for either of the two players can be measured by means of how many states are required to implement it with a finite state machine. In this paper, we are interested in the question of positionality (which corresponds to the degenerate case of memory one) for Eve111Some authors use the term “half-positionality” to refer to what we will simply call “positionality”.: for which valuations is it the case that Eve can play optimally without any memory, meaning that moves depend only on the current vertex of the game, regardless of the history leading to that vertex.
Understanding memory requirements – and in particular positionality – of given valuations has been a deep and challenging endeavour initiated by Shapley (Shapley, 1953) for finite concurrent stochastic games, and then in our setting by Büchi and Landweber (Büchi and Landweber, 1969), Büchi (Büchi, 1977) and Gurevich and Harrington (Gurevich and Harrington, 1982). Among others, the seminal works of Shapley (Shapley, 1953), Ehrenfeucht and Mycielski (Ehrenfeucht and Mycielski, 1973), and later Emerson and Jutla (Emerson and Jutla, 1991), Klarlund (Klarlund, 1992), McNaughton (McNaughton, 1993) and Zielonka (Zielonka, 1998), have given us a diverse set of tools for studying these questions.
Roughly speaking, these early efforts culminated in Gimbert and Zielonka’s (Gimbert and Zielonka, 2005) characterization of bi-positionality (positionality for both players) over finite graphs on one hand, and Kopczyński’s (Kopczyński, 2006) results and conjectures on positionality on the other. In the recent years, increasingly expressive and diverse valuations have emerged from the fast-paced development of reactive synthesis, triggering more and more interest in these questions.
As we will see below, bi-positionality is by now quite well understood, and the frontiers of finite-memory determinacy are becoming clearer. However, recent approaches to finite-memory determinacy behave badly when instantiated to the case of positionality, for different reasons which are detailed below. Therefore, and walking in the footsteps of Klarlund, Kopczyński and others, we propose a generic tool for positionality, allowing for a new characterization result. Before introducing our approach, we briefly survey the state of the art, with a focus on integrating several recent and successful works from different broadly related settings.
The celebrated result of Gimbert and Zielonka (Gimbert and Zielonka, 2005) characterizes valuations which are bi-positional over finite graphs (including parity objectives, mean-payoff, energy, and discounted valuations, and many more). The characterization is most useful when stated as follows (one-to-two player lift): a valuation is bi-positional if (and only if) each player has optimal positional strategies on game-graphs which they fully control. Bi-positionality over infinite graphs is also well understood thanks to the work of Colcombet and Niwiński (Colcombet and Niwiński, 2006), who established that any prefix-independent objective which is bi-positional over arbitrary graphs is, up to renaming the colors, a parity condition (with finitely many priorities).
Finite-memory determinacy of Muller games over finite graphs was first established by Büchi and Landweber (Büchi and Landweber, 1969), and the result was extended to infinite graphs by Gurevich and Harrington (Gurevich and Harrington, 1982). Zielonka (Zielonka, 1998) was the first to investigate precise memory requirements and he introduced what Dziembowski, Jurdziński and Walukiewicz (Dziembowski et al., 1997) later called the Zielonka tree of a given Muller condition, a data structure which they used to precisely characterize the amount of memory required by optimal strategies.
Another precise characterization of finite memory requirements was given by Colcombet, Fijalkow and Horn (Colcombet et al., 2014) for generalised safety conditions over graphs of finite degree, which are those defined by excluding an arbitrary set of prefixes of colors (topologically, ). This characterization is orthogonal to the one for Muller conditions (which are prefix-independent); it provides in particular a proof of positionality for generalisations of (threshold) energy objectives, and different other results.
Le Roux, Pauly and Randour (Roux et al., 2018) identified a sufficient condition ensuring that finite memory determinacy (for both players) over finite graphs is preserved under boolean combinations. Although they encompass numerous cases from the literature, the obtained bounds are generally not tight, and thus their results instantiate badly to the case of positionality.
We mention also a recent general result of Bouyer, Le Roux and Thomasset (Bouyer et al., 2021a), in the much more general setting of (graph-less) concurrent games given by a condition : if belongs to and residuals form a well-quasi order, then it is finite-memory determined222In the concurrent setting, games are often not even determined (even when Borel). This is not an issue for considering finite-memory determinacy, which means “if a winning strategy exists, then there is one with finite memory”.. We will also rely on well-founded orders (although ours are total), but stress that our results are incomparable: to transfer the result of (Bouyer et al., 2021a) to game on graphs, one encodes the (possibly infinite) graph in the winning condition , and therefore strategies with reduced memory no longer have access to it. This gives finite memory deteminacy if the graph is finite (and if one complies with having memory bounds depending on its size), however positionality results cannot be transferred.
Chromatic and arena-independent memories
In his thesis, Kopczyński (Kopczyński, 2006) proposed to consider strategies implemented by memory-structures that depend only on the colors seen so far (rather than on the path), which he called chromatic memory – as opposed to usual chaotic memory. His motivations for studying chromatic memory are the following: first, it appears that for several (non-trivial) conditions, chromatic and chaotic memory requirements actually match; second, any -regular condition admits optimal strategies with finite chromatic memory, implemented by a deterministic (parity or Rabin) automaton recognising ; third, such strategies are arena-independent, and one may even prove (Proposition 8.9 in (Kopczyński, 2006)) that in general, there are chromatic memories of minimal size which are arena-independent. Kopczyński therefore poses the following question: does it hold that chromatic (or equivalently, arena-independent) and chaotic memory requirements match in general?
This turns out not to be the case, a (non -regular) counterexample being given by multi energy objectives, which have finite chaotic memory strategies but require infinite chromatic memory (Bouyer et al., 2020). A recent work of Casares (Casares, 2021) studies this question specifically for Muller games, for which an elegant characterization of chromatic memory is given: it coincides with the size of the minimal deterministic transition-colored Rabin automaton recognising it. Comparing with the characterization of (Dziembowski et al., 1997) via Zielonka trees reveals a gap between arena-dependent and independent memory requirements already for Muller conditions.
Arena-independent (finite) memory structures have independently been investigated recently by Bouyer, Le Roux, Oualhadj, Randour and Vandenhoven (Bouyer et al., 2020) over finite graphs. In this context, they were able to generalise the characterization of (Gimbert and Zielonka, 2005) (which corresponds to memory one), to arbitrary memory structures. As a striking consequence, the one-to-two player lift of (Gimbert and Zielonka, 2005) extends to arena-independent finite memory: if both players can play optimally with finite arena-independent memory respectively and in one-player arenas, then they can play optimally with finite arena-independent memory in general. A counterexample is also given in (Bouyer et al., 2020) for one-to-two player lifts in the case of arena-dependent finite memory.
This characterization was more recently generalised to pure arena-independent strategies in stochastic games by Bouyer, Oualhadj, Randour and Vandenhoven (Bouyer et al., 2021b), and even to concurrent games on graphs by Bordais, Bouyer and Le Roux (Bordais et al., 2021). Unfortunately, none of these result carry over well to positionality, since they inherit from (Gimbert and Zielonka, 2005) the requirement that both players rely on the same memory structure. For instance, in a Rabin game, the antagonist requires finite memory in general, and therefore the results of (Bouyer et al., 2020) cannot establish positionality. We also mention a very rencent work of Bouyer, Randour and Vandehoven (Bouyer et al., 2021c) which establishes that the existence of optimal finite chromatic memory for both players over arbitrary graphs characterizes -regularity of the objective.
Unfortunately, there appears to be no characterization similar to Gimbert and Zielonka’s for (one player) positionality. In fact, there has not been much progress in the general study of positionality since Kopczyński’s work, on which we now briefly extend.
Kopczyński’s main conjecture (Kopczyński, 2006) on positionality asserts that prefix-independent positional objectives are closed under finite unions333This immediately fails for bi-positionality; for instance, the union of two Büchi objectives is not positional for the opponent.. It can be instantiated either for positionality over arbitrary graphs, or only finite graphs, leading to two incomparable variants both of which are open, even for countable unions. An elegant counterexample to a stronger statement is presented in (Kopczyński, 2006): there are uncountable unions of Büchi conditions which are not positional over some countable graphs. One of Kopczyński’s contributions lies in introducing two classes of prefix-independent objectives, concave objectives and monotonic objectives, which are positional (over finite and arbitrary graphs, respectively) and closed under finite unions.
Monotonic objectives are those of the form , where is a (regular) language recognized by a linearly ordered deterministic automaton444The automaton is assumed to be finite, but Kopczyński points out (page 45 in (Kopczyński, 2006)) that the main results still hold whenever the state space is well-ordered and admits a maximum (stated differently, it is a non-limit ordinal). whose transitions are monotonic. Our work builds on Kopczyński’s suggestion to consider well-ordered monotonic automata; however to obtain a complete characterization we make several adjustments, most crucially we replace the automata-theoretic semantic of recognisability by the graph-theoretical universality which is more adapted to the fixpoint approach we pursue.
We introduce well-monotonic graphs, which are well-ordered graphs over which each edge relation is monotonic, and prove in a general setting that existence of universal well-monotonic graphs implies positionality. The idea of using adequate well-founded (or ordinal) measures to fold arbitrary strategies into positional ones is far from being novel: it appears in the works of Emerson and Jutla (Emerson and Jutla, 1991) (see also Walukiewicz’ presentation (Walukiewicz, 1996), and Grädel and Walukiewicz’ extensions (Grädel and Walukiewicz, 2006)), but also of Zielonka (Zielonka, 1998) (in a completely different way) for parity games, and was also formalized by Klarlund (Klarlund, 1991, 1992) in his notion of progress measures for Rabin games.
Our first contribution is rather conceptual and consists in streamlining the argument, and in particular expliciting the measuring structure as a (well-monotonic) graph. We believe that this has two advantages.
Separating the strategy-folding argument from the universality argument improves conceptual clarity. In particular, we believe that known proofs are seen in a new light, and we also extend a few known results.
Perhaps more importantly, well-monotonic graphs then appear as concrete and manageable witnesses for positionality. One can imagine many different ways of combining them. Moreover, different meaningful subclasses of well-monotonic graphs leading to as many interesting classes of positional objectives (among them, Kopczyński’s monotonic objectives) can be envisaged.
We supplement with our main technical and conceptual novelty in the form of a converse: any positional valuation which has a neutral color admits universal well-monotonic graphs. Stated differently, for such valuations, existence of universal well-monotonic graphs characterizes positionality. This is the first known characterization result for positionality (for one player).
Finally, inspired by Walukiewicz’s presentation (Walukiewicz, 1996) of Emerson and Jutla’s proof (Emerson and Jutla, 1991), we show that universality of well-monotonic graphs is preserved under finite lexicographical products of prefix-independent objectives. Thanks to our characterization, this implies that prefix-independent positional objectives with a neutral color are closed under lexicographical product. (In this scenario, the parity condition can be obtained as a lexicographical product of Büchi or of co-Büchi conditions.) We hope that similar constructions can be employed to make progress on Kopczyński’s conjecture.
Organisation of the paper
We start by formally introducing graphs, games and strategies in Section 2. Section 3 defines universality and monotonicity – the two concepts needed to state our characterization. As the object of Section 4, we then present the strategy folding technique, establishing that the existence of well-monotonic universal graphs implies positionality of a given valuation. The converse statement (in the presence of a neutral color) is derived in Section 5.
Section 6 shows that in the important special case of prefix-increasing objectives, we recover the notion of universality studied over finite graphs by Colcombet, Fijalkow, Gawrychowski and Ohlmann (Colcombet et al., 2021). We then advocate in Section 7 for the applicability of our approach by presenting constructions of well-monotonic universal graphs for many different valuations, in some cases extending existing positionality results. Finally, we introduce lexicographical products of objectives and derive the claimed closure property in Section 8.
We use to denote the set of subsets of a set , and for the set of nonempty subsets of .
In this paper, graphs are directed, edge-colored, and without sinks. Formally, given a set of colors , a -graph is given by a set of vertices , and a set of edges , such that all vertices have an outgoing edge . Note that no assumption is made in general regarding the finiteness of , or . For convenience, we write for the edge . If , we say that is a -predecessor of , and is a -successor of . It is often convenient to write , or even simply if is clear from context, instead of . It is also often the case that is fixed and clear from context, so we generally just say “graph” instead of “-graph”. The size (or cardinality) of a graph is defined to be .
A path in is a finite or infinite sequence of edges whose endpoints match, formally
which for convenience we denote by
We say that starts in or that it is a path from . By convention, the empty path starts in all vertices. The coloration of is the (finite or infinite) sequence of colors appearing on edges of .
A non-empty finite path of length is of the form and we say in this case that is a path from to , and that is the last vertex of , denoted . We sometimes make use of the notation defined by and if is a non-empty finite path, . We write
to say that is a finite path from to with coloration in the graph . We also write
to say that is an infinite path from with coloration in .
We let be the set of finite paths from in . Given a graph and a vertex , we define the unfolding of from by
Note that there is a bijection preserving the colorations between paths from in and paths from in ; in particular, colorations from in and from in are the same.
Given two graphs and , a morphism from to is a map such that for all , it holds that . Note that need not be injective. We write when is a morphism from to , and when there exists a morphism from to . A subgraph of is a graph such that and ; note that it is assumed for to be a graph, it is therefore without sinks.
We fix a set of colors . A -valuation is a map from infinite colorations to a set of values equipped with a complete linear order (a linear order which is also a complete lattice, that is, admits arbitrary suprema and infima). A -game is a tuple , where is a -graph, is the set of vertices controlled by the protagonist, and is a valuation. To help intuition, we call the protagonist Eve, and the antagonist Adam; Eve seeks to minimize the valuation whereas Adam seeks to maximize it. We let denote the complement of in . We now fix a game .
A strategy from specifies, for each path starting in and ending in , the choice of an outgoing edge from . Actually, it is more natural for us to potentially allow for several choices, which is slightly not standard but makes very little technical difference. Formally, a strategy from in is a subgraph of the unfolding with and satisfying that for all with and for all outgoing edges from , the path belongs to , and . We let denote the set of strategies from in .
The value of a strategy is the supremum valuation of an infinite coloration from in :
The value of a vertex is the infimum value of a strategy from :
A strategy from in is called optimal if . Note that there need not exist optimal strategies, as it may be that the value is reached only in the limit. Note also that we always take the point of view of Eve, the minimizer. In particular, we will make no assumption on the determinacy of the valuation; in this work, strategies for Adam are irrelevant.
A positional strategy is a strategy which makes choices depending only on the current vertex, regardless of how it was reached. Formally, a positional strategy in is a subgraph of defined over all vertices , and such that for all , all outgoing edges also belong to . Technically speaking, with respect to the definitions above, a positional strategy is not a strategy; however its unfoldings are. The value of a vertex in a positional strategy is given by
A positional strategy is optimal if for all vertices , it holds that .
A valuation is said to be positional if all games with valuation admit an optimal positional strategy. Two remarks are in order. First, note that we require positionality over arbitrary (possibly infinite) game graphs. Second, the concept we discuss is that of uniform positionality, meaning that the positional strategy should achieve an optimal value from any starting vertex.
3. Universal and monotonic graphs
We now introduce the two main concepts for our characterization of positionality, namely universality and monotonic graphs. We fix a set of colors .
Fix a valuation . We see graphs as games controlled by Adam (which have only one possible strategy, the unfolding itself). Therefore, values of vertices in a graph are given by
Given two graphs and with a morphism , since there are more colorations from in than from we have in general
We say that is -preserving if the converse inequality holds: for all , .
Given a cardinal , we say that a graph is -universal if every graph of cardinality has a -preserving morphism towards . We say that a graph is uniformly -universal if it is -universal for all cardinals .
3.2. Monotonic graphs
A -graph is monotonic if its set of vertices is equipped with a linear order which is well-behaved with respect to the edge relations, in the sense that for any and ,
if then (left-composition); and
if then (right-composition).
An example is given in Figure 1.
We say that a monotonic graph is complete if it is complete as a lattice, and for each and , admits a -predecessor. We say that a monotonic graph is well-monotonic if is a well-order, and that it is completely well-monotonic if it is both complete and well-monotonic.
The important assumption here is well-foundedness; completeness will be assumed for technical convenience. The following easy lemma shows how to turn a well-monotonic graph into a completely well-monotonic graph in general while preserving colorations.
Lemma 3.1 ().
Let be a well-monotonic graph, and define the completion of by (where ) and . Then is completely well-monotonic and such that for all , there are no more colorations from in than in .
In a completely well-monotonic graph , given a color and a vertex , the set of -predecessors of is non-empty and upwards closed (thanks to left-composition), and moreover the map is monotonic (thanks to right-composition). We say that the family of monotonic maps over is the min-predecessor table of .
The data of a completely well-monotonic graph corresponds exactly to a (non-limit) ordinal vertex set , together with a family of monotonic maps over describing the min-predecessor table.
4. Structure implies Positionality
In this section, we show the following result.
Theorem 4.1 ().
Let be a valuation such that for all cardinals , there exists a completely well-monotonic -universal graph. Then is positional.
Our proof is inspired by those of Emerson and Jutla (Emerson and Jutla, 1991) and Klarlund (Klarlund, 1992), respectively for parity games and Rabin games; it relies on using progress measures to fold arbitrary strategies into positional ones.
4.1. Progress measures
Fix a set of colors , a -game where , and a completely well-monotonic -graph . We let denote the min-predecessor tables in , which are monotonic. We consider the operator given by
Note that the is well-defined thanks to well-foundedness of ; however the might not be met (if has infinite branching). Since the ’s are monotonic, is a monotonic operator over the complete lattice , ordered coordinatewise. A map is a progress measure if it is a prefixpoint of , that is, if . By the Knaster-Tarski Theorem, there exists a least progress measure.
4.2. Proof of Theorem 4.1
We break the proof into two parts. First, we show how to define a progress measure from an arbitrary strategy, then we show how to obtain a positional strategy from a progress measure. Both steps preserve the value, therefore the positional strategy corresponding to the least progress measure is optimal.
We now fix a valuation , and assume that for all cardinals there exists a completely well-monotonic -universal graph. We also fix a game , and a completely well-monotonic -universal graph , for some cardinal satisfying for all ’s.
From strategies to progress measures.
Consider a strategy from , and pick a -preserving morphism from to . In particular, we have
We now let be defined by
Note that vertices which are not reached in the strategy are mapped to , the maximal element in . For other vertices however, the infimum defining is a minimum thanks to well-orderedness, which is crucial for the result below.
Lemma 4.2 ().
The map is a progress measure satisfying
Since and we have by definition that
therefore (by right-composition) any coloration from in is also a coloration from in , hence .
Let . To prove that is a progress measure, we aim to show that . If , there is nothing to prove. Otherwise, the infimum defining is a minimum by well-foundedness, thus there exists a path belonging to and satisfying . There are two (similar) cases according to the player controlling .
If , then has a successor in (since is a graph). Then we have
where follows from the fact that with and therefore by definition of , and because is monotonic; and follows from the fact that is a graph morphism and thus .
If , then in if and only if with in . Thus we now obtain
where and are just like in the first item.∎
From progress measures to positional strategies.
We now consider a progress measure . For all , we have
We thus define a positional strategy by setting, for each , the set of outgoing edges to be the ones which meet the above minimum.
Lemma 4.3 ().
Let be a finite path in . There is a path with coloration from to in .
We prove the result by induction on the length of . The result is clear for paths of length . Assume that has length and that the result is known for shorter paths. We write and by induction we have a path in . We show that .
If then by definition of it follows that therefore .
If , then we have
therefore . ∎
5. Positionality implies Structure
This section focuses on establishing our main technical novelty which is stated as follows. (Neutral colors are defined just below.)
Theorem 5.1 ().
Let be a positional valuation admitting a strongly neutral color, and let be a -graph. There exists a -graph which is completely well-monotonic with a -preserving morphism .
We obtain a converse to Theorem 4.1 as a consequence.
Corollary 5.2 ().
Let be a positional valuation admitting a strongly neutral color. For all cardinals , there exists a completely well-monotonic -universal graph.
Proof of Corollary 5.2.
Let be a cardinal, and let be the disjoint union of all -graphs of cardinal , up to isomorphism. Note that it is -universal. Theorem 5.1 gives a graph which is completely monotonic and has a -preserving morphism ; now is -universal by composition of -preserving morphisms. ∎
Before proving Theorem 4.1, we define neutral colors.
A color is neutral with respect to a valuation if for all , and for all of the form , where is an arbitrary sequence of integers, we have
A color is eventually good for Eve if for all we have
A color is strongly neutral if it is both neutral and eventually good for Eve.
Most well-studied valuations have a strongly neutral colour; this will be the case for all examples studied in Section 7. In fact, it is not known whether, given a positional -valuation , its unique extension to a -valuation (where ) for which is strongly neutral, is itself positional in general. We actually conjecture this to be the case; under this conjecture, our characterization is complete.
We now fix a positional valuation , with a strongly neutral color , and a graph . Our proof consists of the two following steps:
add many -edges to while preserving ; then
add even more edges by closing around -edges (this is made formal below), and quotient by -equivalence.
For the second step to produce a completely well-monotonic graph, we need to guarantee that there are sufficiently many -edges which were added in the first step. We start by the second step; in particular, we formalize what “sufficiently many” means. The first step is more involved and exploits positionality of .
We say that a graph has sufficiently many -edges if is well-founded, that is,
Lemma 5.3 ().
If has sufficiently many -edges then there exists a completely well-monotonic graph with a -preserving morphism .
We first define the -closure of by
and claim that the identity over defines a -preserving morphism from to . Indeed, for all and for all paths in , there is a path in with . By neutrality of we have , and thus
Note that in , satisfies left and right composition with all colors, and in particular it is transitive (by taking ). It is moreover well-founded (and thus also total) and reflexive thanks to the assumption of the Lemma. However, it is not antisymmetric, which is why we now quotient with respect to -equivalence.
Formally, we define over by
which is an equivalence relation. Note that vertices which are -equivalent have the same incoming and outgoing edges in (since is -closed), therefore the graph over given by
is well-defined, and moreover colorations from in are the same as colorations from in . Hence the projection defines a preserving morphism from to . Now is a well-founded order satisfying left and right composition in , therefore is well-monotonic.
Finally, we take to be the completion of (see Lemma 3.1), which is completely well-monotonic, and (by composition) has a -preserving morphism from . ∎
Note that the second step has not made use of positionality of ; it is exploited below.
We now show that sufficiently many edges can be added to a graph while preserving , thanks to its positionality. We consider the game given by with , , and
An example is given in Figure 2.
When playing in , Adam follows a path in , with the additional possibility, at any point, to switch from a vertex to an Eve-vertex containing . It is then left to Eve to choose a successor from , which can be any vertex in . A natural choice is to go back to , which guarantees a small value thanks to strong neutrality of .
Lemma 5.4 ().
For all , we have .
Let . We consider the strategy from described above. Formally, is defined by setting the only outgoing edge of to be . Then infinite paths from in either have colorations of the form , where is a coloration of an infinite path from in , or of the form , where is a coloration of a finite path from in . We conclude thanks to strong neutrality of that
We now exploit positionality of . Observe that a positional strategy in corresponds to the choice of (at least) a successor for each nonempty subset of vertices.
Lemma 5.5 ().
Let be an optimal positional strategy in , and let be the graph defined by
The identity defines a -preserving morphism from to . Moreover, has sufficiently many -edges.
By optimality of , and thanks to Lemma 5.4, we have for all that . Consider an infinite path in ; it is of the form
where for each , is a path in and there exists such that and . Therefore, there is a path of the form
in . By neutrality of we have and hence
thus the identity is -preserving from to .
Finally, each non-empty has an -successor in , which satisfies that each has an -edge towards in . Stated differently, has sufficiently many -edges. ∎
6. Prefix-increasing objectives
In this section, we show that our notion of universality instantiates to that introduced by Colcombet and Fijalkow (and studied over finite graphs), in the case of prefix-increasing objectives.
We say that a valuation is prefix-increasing (resp. prefix-decreasing) if adding a prefix increases (resp. decreases) the value: for any and , we have (resp. ). If a valuation is both prefix-increasing and prefix-decreasing, we say that it is prefix-invariant. We have the following technical lemma.
Lemma 6.1 ().
Assume that is prefix-increasing and consider a graph . If two vertices and satisfy then there is no edge in from to .
By contradiction, let be an edge in and pick a path from with . Then is a path from and we have
which is a contradiction since is a path from in . ∎
We say that is an objective if
is the ordered pair. In this case, we also say that is qualitative. From the point of Eve, is interpreted as winning, whereas is losing. Following the usual convention, we identify a qualitative valuation with the set of infinite words which are winning for Eve. We say that a vertex (in a graph) satisfies if all colorations from belong to ; this amounts to saying that has value . We also say that a graph satisfies if all its vertices satisfy .
We now consider the case of a prefix-increasing objective . We consider a completely well-monotonic graph which we call , with vertex set . We let be its restriction to the set of vertices which satisfy ; by definition, satisfies . The above lemma states in this case that there are no edges in from to .
By left-composition in , is a downward-closed subset of , thus is a well-ordering over . Therefore is a well-monotonic graph; it is however not complete in general. The three monotonic graphs from the statement below are depicted in Figure 3.
Lemma 6.2 ().
(Universality for prefix-increasing ) Let be a cardinal. The following are equivalent.
embeds all graphs of cardinality satisfying .
The three monotonic graphs from the statement above are depicted in Figure 3.
We show that (iii) (ii) (i) (iii) in this order.
Given a graph of cardinality , we let denote the set of vertices which satisfy . By Lemma 6.1, there is no edge in from to , hence the restriction of to is a graph. Since there are all edges from to in , a morphism extends to a morphism by setting for . It is -preserving by definition: if satisfies then thus which satisfies . This gives the first implication.
For of cardinality , if is a -preserving morphism, then maps to and its complement to . Now the map which coincides with the identity over and maps to the maximal element of is also -preserving since there are no edges leaving in , and it is a morphism since has all -loops in . We conclude with the second implication by composition of -preserving morphism.
For the third implication, it suffices to see that if satisfies then a -preserving morphism in embeds in . ∎
Therefore, the notion of being universal in the qualitative prefix-increasing case corresponds to the one of Colcombet, Fijalkow, Gawrychowski