Separating LREC from LFP

LREC= is an extension of first-order logic with a logarithmic recursion operator. It was introduced by Grohe et al. and shown to capture the complexity class L over trees and interval graphs. It does not capture L in general as it is contained in FPC - fixed-point logic with counting. We show that this containment is strict. In particular, we show that the path systems problem, a classic P-complete problem which is definable in LFP - fixed-point logic - is not definable in LREC= This shows that the logarithmic recursion mechanism is provably weaker than general least fixed points. The proof is based on a novel Spoiler-Duplicator game tailored for this logic.

Authors

• 16 publications
• 1 publication
• Capturing Logarithmic Space and Polynomial Time on Chordal Claw-Free Graphs

We show that the class of chordal claw-free graphs admits LREC=-definabl...
02/28/2018 ∙ by Berit Grußien, et al. ∙ 0

• Temporal Constraint Satisfaction Problems in Fixed-Point Logic

Finite-domain constraint satisfaction problems are either solvable by Da...
02/21/2020 ∙ by Manuel Bodirsky, et al. ∙ 0

• Logarithmic-Bounded Second-Order Quantifiers and Limited Nondeterminism

We add logarithmic-bounded second-order quantifiers to the inflationary ...
12/09/2019 ∙ by Kexu Wang, et al. ∙ 0

Whether it be in normal form games, or in fair allocations, or in voter ...
07/22/2019 ∙ by Ramit Das, et al. ∙ 0

• Inapproximability of Unique Games in Fixed-Point Logic with Counting

We study the extent to which it is possible to approximate the optimal v...
04/09/2021 ∙ by Jamie Tucker-Foltz, et al. ∙ 0

• Generalized Absorptive Polynomials and Provenance Semantics for Fixed-Point Logic

Semiring provenance is a successful approach to provide detailed informa...
10/17/2019 ∙ by Katrin M. Dannert, et al. ∙ 0

• Nonclassical truth with classical strength. A proof-theoretic analysis of compositional truth over HYPE

Questions concerning the proof-theoretic strength of classical versus no...
07/14/2020 ∙ by Martin Fischer, et al. ∙ 0

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1 Introduction

The founding result of the subject of descriptive complexity is Fagin’s characterization of as the class of properties definable in existential second-order logic [6]. Similar characterizations for complexity classes below are not known and remain an active area of investigation. Much attention has been devoted to the question of whether there is a logic for , and this is often said to be the central open question of descriptive complexity. Perhaps less well known but still wide open is the question of whether there is a logic giving a descriptive characterization of —the class of properties of finite structures decidable deterministically in logarithmic space.

Immerman established logical characterizations of and on ordered structures. The former is given by the properties definable in LFP—the extension of first-order logic with a least fixed-point operator (a result obtained independently by Vardi)—and the latter by DTC—the extension of first-order logic with a deterministic transitive closure operator. These logics are known to be strictly weaker than the corresponding complexity classes in the absence of a built-in order (see [15] for details). Moreover, on unordered structures, it is known that the expressive power of DTC is strictly weaker than that of LFP [7], whereas the corresponding question for ordered structures is equivalent to the separation of from .

Since the simplest properties separating LFP and DTC from the complexity classes and respectively are counting properties, there has also been much interest in the extensions of these logics with counting mechanisms. The logic FPC—fixed-point logic with counting—is widely studied (see [3] for a brief introduction). Though it is known to be strictly weaker than by a celebrated result of Cai, Fürer and Immerman [2], it gives a robust definition of the class of problems in which are symmetrically solvable [1] and captures on a wide range of graph classes. Most significantly, Grohe has proved that any polynomial-time property of graphs that excludes a fixed graph as a minor is definable in FPC [12]. Results showing that FPC captures on a class of structures (including Grohe’s theorem) are typically established by showing that a polynomial-time canonical labelling algorithm on the class can be implemented in FPC. Such results were established for specific classes of graphs such as trees [14] and graphs of bounded tree-width [9] before Grohe’s theorem which supersedes them all.

The situation for logics capturing is less clear-cut, even for classes of structures where it is known that logarithmic-space canonical labelling algorithms are possible. For instance, Etessami and Immerman [5] showed that the extension of DTC with counting fails to capture even on trees. What this suggests is that the weakness of DTC is not just in the lack of a means of counting but that the recursion mechanism embodied in the deterministic transitive closure operator is too weak. An interesting suggestion to remedy this is in the logic LREC introduced in [10], which incorporates a rich recursion mechanism which can still be evaluated within logarithmic space. To be precise, the paper introduces two versions of the logic, one called LREC and the second . While the first is shown to be sufficient to capture the complexity class on trees, it is the latter that is the more robust logic. In particular, properly extends LREC in expressive power, is closed under first-order interpretations and (unlike LREC) can express undirected reachability. Moreover, has also been shown to capture on other interesting classes of structures [13].

It is known that does not capture on all graphs. In particular, is included in FPC and the CFI construction that separates FPC from

can be suitably padded to give a separation of

from  [8]. While Grohe et al. proved that LREC is properly contained in FPC (for instance, it cannot express reachability on undirected graphs), the question was left open for the stronger logic . We settle this question in the present paper. That is, we prove that the expressive power of is strictly weaker than FPC. Indeed, we show that the path systems problem (PSP), a natural -complete problem that is expressible in LFP (even without counting) is not expressible in . The result is established by first defining an Ehrenfeucht-style pebble game for the -recursion quantifier used in the logic and then deploying it on an intricate class of instances of PSP. Describing the winning Duplicator strategy in the game is challenging and takes up the bulk of the paper.

2 Preliminaries

We assume that the reader is familiar with the definitions of the basic logics used in finite model theory, in particular first-order logic FO, fixed-point logic LFP and their extension swith coungint, FOC and FPC respectively. These definitions can be found in standard textbooks [4, 16]. We consider vocabularies with relation and constant symbols, which we call relational vocabularies for short. Formulas in a vocabulary are interepreted in finite -structures. We often use Fraktur letters for structures and the corresponding Roman letter to denote the universe of the structure.

In the case of logics with counting, such as FOC and FPC, the interpretation also involves a number domain. The logic allows for two kinds of variables (and more generally, two kinds of terms) which range over elements and numbers respectively. Thus, we consider a -structure to be extended with a domain of non-negative integers up to the cardinality of . Element variables are interpreted by elements of and number variables by elements of the number domain. We write to denote the number domain associated with the structure . This can be identified with the set and we assume this is disjoint from the set . The logics include a counting operator and we assume a normal form where every occurrence of the counting operator is in a formula of the form where is a number variable. This formula is to be read as saying that the number of elements satisfying is .

Any formula in any of the logics defines a query. If is a formula in vocabulary and is an -tuple of element variables such that all free variables of are among , then defines an -ary query. This is a map taking each -structure to an -ary relation on the universe of containing the tuples such that . If and so is a sentence, we call the query a Boolean query. A Boolean query is often identified with a class of -structures. Note that the query is not completely specified by the formula alone, but rather by the formula along with the tuple of variables . In particular the order on the variables is given by the tuple.

To allow for formulas which may also have free number variables, we consider numerical queries. An -ary numerical query is a map that takes any -structures to a relation and is invariant under isomorphisms. The last condition means that if is an isomorphism from to , then if, and only if, . Such queries arise quite naturally. As an example, the function that takes a graph to the number of connected components in is a -ary numerical query. The function taking a graph to the set of pairs where is a vertex in and the number of vertices in the connected component of is a -ary numerical query.

Suppose is a formula of a logic with counting, such as FOC or FPC, with free variables among , where contains element variables and number variables. Without loss of generality we assume all element variables appear in before the number variables. Interpreted in a structure , the formula defines a relation of mixed type: a subset of . We treat the tuples of numbers as coding positive integers in a natural way. So, suppose and therefore . For a tuple we write denote the number

 m2∑i=1(n+1)i−1ri.

In this way, we can see as defining an -ary numerical query which takes to the set .

2.1 Generalized Quantifiers and Operators

In the next subsection we define the logics LREC and which were introduced by Grohe et al. [11]. They were originally defined as an extension of first-order logic with a new kind of recursion operator called -recursion, designed to be computable in logspace. We present an equivalent formulation through a variation of generalized quantifiers in the style of Lindström. To do this, we first briefly review interpretations and generalized quantifiers. In what follows, fix a logic , which can be any of the logics FO, FOC, LFP or FPC. Whenever we refer to formulas, this is to be read as formulas of .

Given two relational vocabularies and and a positive integer , where contains no constant symbols, an interpretation of in of dimension is a sequence of formulas

 (δ(→x,→w),ε(→x,→y,→w),φR1(→z1,→w),…,φRs(→zs,→w)),

where and are tuples of variables of length , and each is a tuple of variables of length where is the arity of the relation symbol . The variables are the parameter variables. Note that all tuples of variables may contain both element variables and number variables.

Let be a structure and be an interpretation for the parameter variables . The interpretation  associates a -structure  to if there is a map  from  to the universe  of  such that: (i)  is surjective onto ; (ii)  if, and only if, ; and (iii)  if, and only if, . Note that an interpretation  associates a -structure with  only if  defines an equivalence relation on  that is a congruence with respect to the relations defined by the formulaa . In such cases, however,  is uniquely defined up to isomorphism and we write .

There are many kinds of interpretation that are defined in the literature. The ones we have defined here are fairly generous, in that they allow for relativization, vectorization and quotienting. The first of these means that the universe of is defined as a subset of rather than the whole set. The formula is the relativizing formula. Vecotorization refers to the fact that can be greater than and so can map -structures to structures that are polynomially larger, and quotienting refers to the fact that the universe is obtained by quotienting with the congruence relation defined by .

Let be an isomorphism-closed class of -structures (or equivalently a Boolean query). A standard way of extending any logic to obtain a minimal extension in which the Boolean query can be expressed is to adjoin to the Lindström quantifier , corresponding to (see [4, Chapt. 12]). This allows us to write formulas of the form where is an interpretation as above. The quantifier binds the variables appearing in so the free varialbles of are . The formula is true in a -structure with an intepretation for the free variables if, and only if, is in .

We generalize such quantifiers to queries that are not necessarily Boolean. In general, consider an -ary numerical query over -structures. We define , the extension of the logic with an operator for as follows. Let be an interpretation of in of dimension . Let be a tuple of variables of length and a tuple of number terms. Then,

 [QK→x,→y,→z1,…,→zsI](→u,→s)

is a formula with free variables . This formula is true in a -structure , with an interpretation of for , for and for if, and only if,

 ([→a]ε,⟨→r⟩)∈K(I(A,→p)).

Here, denotes the -tuple of elements of obtained by taking the equivalence classes of the -tuples that make up under the equivalence relation defined by .

2.2 Lrec=

In this subsection we define , the logic in which we prove the inexpressibility result. We define the logic as an extension of FOC by means of a generalized operator in the sense of Section 2.1 above. This definition is superficially different from that of Grohe et al. [10] where it is defined as a restricted fixed-point operator, but the two definitions are easily seen to be equivalent.

We first define a unary numeric query on a class of labelled graphs. These are directed graphs together with a labelling that gives for each vertex a set of numbers . For each such we define . To do this, we first introduce some notation. For any binary relation , and , let denote the set and denote . Then is defined by recursion on by the following rule:

 (u,ℓ)∈χ(G,C):⟺ℓ≥0 and |{v∈uE∣(v,⌊(ℓ−1)/|Ev|⌋)∈χ(G,C)}|∈C(u).

The logic LREC is the closure of FOC with a generalized operator for computing the numeric query . Note that to get this formally, we have to extend the idea of interpretation we had earlier to be able to produce a labelled graph. In a logic with couting, this is easily achieved as the interpretation will allow for formulas defining relations of mixed element and number type.

To define the logic , we considered labelled semi-graphs. A semi-graph is a structure , where is a set of nodes, and and are two binary relations on . Again, we consider a semi-graph together with a labelling such that for each , . Let be the symmetric reflexive transitive closure of and denote the equivalence class of under . Write for set of such equivalence classes, i.e. the quotient of under this equivalence relation. We write for the directed graph with vertex set and edge set

 [E]:={([a],[b])∣(a′,b′)∈E for some a′∈[a] and b′∈[b]}.

Write for the labelling of given by . We can then define the numeric query on labelled semi-graphs by the condition:

 ^χ(G,C)=χ([G],^C).

The logic is then defined as the closure of FOC under a generalized operator for the numeric query . For simplicity, we only allow applications of the operator to interpretations without relativization of quotienting. We explian below why this is no loss of generality. Indeed, it is also easily checked that this restricted definition corresponds exactly to the original definition of the logic given by Grohe et al.

The syntax of the logic can then be defined as follows.

Definition 2.1.

For a vocabulary , the set of formulas of is defined by extending formula formation rules by the following rule. Suppose , and are formulas where , and are tuples of variables all of length ; is a tuple of number variables of length and is a tuple of number variables of length , then

 φ:=[lrec→u,→v,→pφE,φ∼,φC](→w,→r)

is an formula. The free variables of the formula are those in and , along with the free variables in and that are not among and and those of that are not among and .

To define the semantics of the logic, consider the formula interpreted in a -structure . Let the semigraph be given by the interpretation on and be the labelling defined by . Then the formula is satisfied in with and interpreting and respectively if, and only if, .

Note that this amounts to extending FOC with a generalized operator for the numeric query . We have not allowed for relatavization or quotienting in the interpretations in our definition and we now explain why this involves no loss of generality. The definition of over the semi-graph is defined by taking over the quotient of with respect to the reflexive, symmetric and transitive closure of . Thus, we can replace a definable congruence in an interpretation by combining it with the definition of without changing the semantics. By the same token, we can replace any relativizing formula by incorporating it in the definition of the binary relation and ensuring that only tuples that satisfy are involved in this binary relation. It is easily seen from the definition of that for any vertex that has no -neighbours, either is in and thus all pairs are in for all , or 0 is not in and thus no such pairs are in .

We define two parameters by which we measure the size of an formula, rank and iteration-degree. As a base case, the rank and iteration-degree of any atomic formula is 0. If has rank and iteration-degree , then , and all have rank and iteration degree Finally, if we have the formula

 φ:=[lrec→u,→v,→pφE,φ∼,φC](→w,→r)

where the length of is and is , then the rank of , denoted , is equal to

 max[2c+rank(φ∼),2c+rank(φE),c+d+rank(φC)]

and if has length then the iteration-degree of , denoted , is equal to

 max[q,deg(φ∼),deg(φE),deg(φC)].

Note that the rank and iteration-degree values may be distinct from each other, and the iteration-degree need not be larger than that of sub-formulas, regardless of the form of . This will be important when we introduce the game for the logic.

2.3 The Path Systems Problem

In this subsection we define the path systems problem (). Moreover we will define a specific subclass of instances which we will be using to prove that is inexpressible in .

The problem is defined as a class of structures in a vocabulary with a ternary relation , unary relation , and constant . Given a universe and a relation , we define the upward closure of any set as the smallest superset of such that for , if is in and is in , then is in .

The decision problem then consists of those structures for which is in the upwards closure of .

As we will be using to prove a separation between and LFP, we verify that is definable in LFP. This is well known. For completeness, we give the defining sentence.

 φ:=[lfpX,uS(u)∨(∃v,w.X(v)∧X(w)∧R(v,w,u))](t).

3 Lrec= Game

In order to prove the inexpressibility result for , we introduce a Spoiler-Duplicator game. This game does not exactly characterize the expressive power of the logic, but it provides a sufficient condition for indistinguishability of structures.

The game we define is an extension of the classic Ehrenfeucht-Fraïssé game for first-order logic played on a pair of structures and , where Spoiler aims to demonstrate a difference between the two structures and Duplicator aims to demonstrate that they are not distinguishable. Before introducing it formally, we make some observations. Because our logic includes counting, the games are based on bijection games, which characterise first-order logic with counting. Thus, our games have two kinds of moves—bijection moves, which account for counting and ordinary quantifers, and what we call graph moves which account for the lrec operator.

In a graph move, Spoiler chooses an interpretation which defines a labelled semi-graph from each of and . An auxilliary game is then played on the pair of semi-graphs in which Spoiler aims to show that the two semi-graphs are distinguished by the query while Duplicator attempts to hide the difference between them. At any point in this auxilliary game, Spoiler can choose to revert to the main game, for instance if a position is reached from where Spoiler can win the ordinary bijection game.

It should be noted that in requiring Spoiler to provide an explicit interpretation in the graph game, we build in definability in the logic into the rules of the game. In that sense, it does not provide an independent characterization of definability. Nonetheless, it does give a sufficient criterion for proving undefinability. Similarly, Theorem 3.2 below only gives one direction of the connection between the game and the logic. That is, it shows that if there is a formula distinguishing and , it yields a winning strategy for Spoiler. We do not claim, and do not need, the other direction.

A final remark is that we have, for simplicity, only defined the game for the special case where and are structures over the same universe. Again, this is sufficient for our purpose as the structures we construct on which we play the game satisfy this condition.

The game is parameterized by the two measures of size of formulas we introduced in Section 2.2: the rank and iteration degree. To define the game, we first introduce some notation for partial maps.

For any sets and with and , and function , we say is a partial function between and , which we can denote by . We say that a function extends if . Let denote the set where is defined. For partial function with for , let be the partial function defined on with if is in and otherwise.

3.1 Games

We now give the formal definition of the game and prove its adequacy.

Definition 3.1.

The -step, -degree game over vocabulary is played by two players, Spoiler and Duplicator on two -structures and of same universe of size . At any stage of the game, the game position consists of a partial injection where for all constants in , and the domain of has at most elements.

Let be the domain of , and fix an ordering on it so that is the tuple . As long as , Spoiler can play either an extension move or a graph move. These are defined as follows.

• Extension move: If Spoiler selects this move then the duplicator begins by choosing a bijection that extends . The spoiler then picks some . The resulting position is .

• Graph move: In this move Spoiler begins by selecting , and queries , and where and are tuples of variables of length . Both queries have iteration-degree at most and rank at most . Let be , be the semi-graph defined in by the interpretation of , and be the corresponding semi-graph obtained from . The two players now play a game on these structures through a series of rounds. At each round , the position of this game consists of a tuple , a partial injection with domain of size at most , and a value . Initially, Spoiler chooses and and is set to the empty map.

If , then the graph move ends and the main game continues from the new position . Also, Spoiler may choose to end the graph move and continue the main game from the position . Otherwise we continue to move which proceeds as follows

1. Duplicator begins by choosing a partial bijection with the property that where .

2. For each with , Duplicator chooses an injection satisfying the following conditions for all . Here, for any , we write to denote the set of elements of that occur in the tuple .

1. if, and only if, ;

2. ; and

3. if, and only if, .

If Duplicator cannot choose such a set of partial bijections, then it loses the game.

3. Spoiler chooses some and we let and .

Spoiler wins if at any point is not a partial isomorphism between and . Duplicator wins if reaches and Spoiler has not won.

We can now show how establishing how Duplicator winning strategies in the game can be used to show inexpressibility results for .

Theorem 3.2.

Suppose and are two -structures with the same universe , is a partial injection with for all constants in and that enumerates the domain of . If Duplicator has a winning strategy in the -step, -degree game over and , then and agree on all formulas with rank at most and iteration-degree at most .

Proof.

We prove the contra-positive, so assume that some -formula with rank at most and iteration-degree at most is true in and false in . If then does not induce a partial isomorphism between and , since otherwise the quantifier-free formula would distinguish from . Suppose then that . If is of the form , or , then Spoiler plays the extension move. Since is true in and false in , the number of elements satisfying in the two structures is different. Hence, for any bijection extending that Duplicator might pick, there is some some such that and do not agree on . That is to say, either or is a formula true in the former but false in the latter. Either of these is a formula of rank at most and so if Spoiler chooses and extends to it wins by the induction hypothesis.

Assume next that is of the form

 [lrec→x,→y,→pφE,φ∼,φC](→w,→r)

where the lengths of and are both andthe length of is . To simplify notation, let denote and denote . Let be , and for , let be the semi-graph obtained by the interpretation in and the labelling defined on it by . For any , we write for the set of elements that occur in and for the set of those elements of that are in .

Spoiler then chooses the starting node to be the interpretation of , and to be the value where is the interpretation in of . Note that by our choice of , it is true that and . We prove by induction on that if if, and only if, then Spoiler has a winning strategy in the game move. Assume without loss of generality that is in . For , let be the number

 |{v∈u[FS]∣(v,⌊(ℓi−1)/|[FS]v|⌋)∈χ(GS,CS)}|.

Then,either or but . So, if we must have the latter case, where Spoiler can simply reset to and proceed by playing the game on the structures and where . Here we also have the base case — if then . If, instead, then Spoiler proceeds to the next step of the iteration. If Duplicator is able to come up with a valid set of partial bijections in step 3 of the iteration, there must be some with if and only if , where (which is also equal to according to the conditions Duplicator’s choice must satisfy). Thus, by the induction hypothesis, Spoiler has a winning strategy if it picks to be . ∎

3.2 Structures

Here we describe the particular instances of the path systems problem for which we construct Duplicator winning strategies in the game we have just defined, and outline the strategy.

Consider a tree to be defined as a directed graph where edges are oriented from a parent to its children. The structures we consider are obtained by taking the product of a complete binary tree with a large cyclic group of prime order . The unary relation then encodes an assignment of values in to the leaves of the tree and the ternary relation is chosen so that determining whether the target is in the upward closure of amounts to summing these values along the tree. We give a formal definition for future reference.

Definition 3.3.

For a positive integer , let be the complete binary tree of height , the set of its leaves and its root. For any prime , function and element define the structure to be the instance of with

1. universe

Note that for each node there is a unique value of for which is in the upward closure of . To be precise, is the sum (modulo ) of all the values of for leaves below in the tree . In particular is a positive instance of if, and only if, . What we aim to show is that two structures and with are indistinguishable by formulas of rank and degree as long as and are large enough with respect to and .

Intuitively it is clear that a fixed-point definition of the upwards closure of , such as given by the LFP formula will determine in (i.e. the height of the tree) iterations whether is in the closure of . What makes this difficult for LREC is that at each element there are distinct pairs of elements for which . The number of paths multiplies giving distinct ways of reaching , and by an appropriate choice of and , we can ensure that this is not bounded by a polynomial in the number of elements of , which is . Moreover, we cannot eliminate the multiplicity of paths by taking a suitable quotient. Of course, this only shows that the obvious inductive method of computing the upward closure of cannot be implemented in . To give a full proof, we have to consider all other ways that this might be defined, and that is the role of the game.

In the winning strategy we describe, Duplicator plays particular bijections which we now describe. Let and be two instances of obtained as described above from a tree and prime , with the same set and different values of . Note, in particular, that and have the same universe .

For any set . A function induces a bijection given by . We extend this to a bijection on the set given by for all and , by letting it be the identity on all elements of . In the Duplicator strategies we describe, all bijections played are of this form. Thus, we usually describe them just by specifying the function which we call the offset function. We abuse terminology somewhat and say that is a partial isomorphism from to to mean that induces a partial isomorphism.

Roughly speaking, Duplicator’s winning strategy is to play offset functions which are zero at the leaves of the tree and offset by the difference between the values of in and at the root. Spoiler has to try and expose this inconsistency by building a path between the root and the leaves. We show that Duplicator can maintain a height (depending on the parameters and ) below which it plays offsets of zero, without the inconsistency being exposed. Of course, Duplicator has to respond to graph moves in the game, so we have to consider paths in the interpreted semigraphs, where each node may involve elements from many different heights in the tree. This is what makes describing the Duplicator winning strategy challenging. In the next section we develop the tools for describing it.

4 Duplicator Winning Pebblings and Extender functions

The Duplicator strategies we describe in the next section for the LREC game rely on certain combinatorial properties of the complete binary tree and certain offset functions . In this section we develop some combinatorial properties of such trees and functions that allow us to effectively describe the strategies.

Let be a complete (directed) binary tree with leaves. The height of a node , denoted is the distance of to a leaf (since is complete, this is the same for all leaves reachable from ). Thus, if is a leaf its height is and if is the root of , its height is .

Let be the set of triples such that and and let . In other words is the collection of unordered sets of three elements consisting of a node of along with its two children. We say that a three element set is related if . We call an element of a related triple.

Say a set is closed if for every related triple , if then . For every , there is a unique minimal closed set such that . We call the closure of .

Let denote the minimum height of any element of and denote the maximum height of any element of .

Proof.

Consider the sequence of sets given by and . Then . A simple induction on shows that for all , establishing the claim. ∎

For any pair of vertices , there is a unique undirected path such that , and for all with , we have or . Say that a set is connected whenever the undirected path from to is contained in .

A connected component of a closed set is a maximal closed connected subset of . It is clear that every closed set is a disjoint union of closed connected components.

For , we say a set encloses if every path from to a leaf goes through an element of . We say that minimally encloses if encloses and no proper subset of encloses .

If is a closed connected set, there is a unique element with . Moreover, there is a set which minimally encloses and such that is exactly the set of elements such that is on the path from to for some . We call the frontier of . Note that this allows the trivial case when . In all cases, we have . We write for .

Proposition 4.2.

If is a closed connected set with root and frontier then

 |F|>height(X).
Proof.

Let . The proof proceeds by induction on . If , then and the inequality is satisfied. Suppose then that , and so there is some element in whose height is less than that of . Since is connected this implies that some child of is in and since is closed this implies both children of are in . Let the two children of be and . Then, we have where and are closed connected sets with roots and respectively. Let and be the respective frontiers of and and note both of these are non-empty. Since and , we have and so by induction hypothesis . Since is non-empty, we have

 |F|=|F1|+|F2|>height(s1)+1−min-h(X1)≥height% (r)−min-h(X)=height(X)

as required. ∎

For a closed set , we say a function is consistent if whenever are such that and , then . If , we say that a function extends if . The following is a useful characterization of consistent functions on closed sets.

Proposition 4.3.

For a closed set , is consistent if, and only if, for every and such that minimally encloses , we have .

Proof.

The direction from right to left is immediate, since for any with , we have that minimally encloses . Thus, by assumption as required.

In the other direction, assume that is consistent and suppose minimally encloses . Let and we proceed by induction on . If then and is clearly true. Suppose then the claim is true for all with and let be a set minimally enclosing with . Let and be the two children of and and the subsets of that minimally enclose and respectively. Then . Here the first equality holds from the fact that and form a partition of , the second by induction hypothesis and the third by the consistency of . ∎

For any (not necessarily closed) set , we say a function is consistent if there is a consistent