LATTICE EXIT MODELS S. GILL WILLIAMSON ABSTRACT. We discuss a class of problems wic we call lattice exit models. At one level, tese problems provide undergraduate level exercises in labeling te vertices of graps (e.g., dept first searc). At anoter level (teorems about large scale regularities of labels) tey provide concrete geometric examples of ZFC independence. We note some combinatorial and algoritmic implications.. INTRODUCTION We expand on some of te geometric and combinatorial concepts contained in te foundational work of H. Friedman ( [Fri97], [Fri98]). Let N be te set of nonnegative integers and k. For z = (n,..., n k ) N k, max{n i i =,..., k} will be denoted by max(z). Define min(z) similarly. Definition. (Downward directed grap). Let G = (N k, Θ) (vertex set N k, edge set Θ) be a directed grap. If every (x, y) of Θ satisfies max(x) > max(y) ten we call G a downward directed lattice grap. All lattice graps tat we consider will be downward directed. Definition. (Vertex induced subgrap G D ). For D N k let G D = (D, Θ D ) be te subgrap of G wit vertex set D and edge set Θ D = {(x, y) (x, y) Θ, x, y D}. We call G D te subgrap of G induced by D. Definition. (Pat and terminal pat in G D ). For t, a sequence of distinct vertices of G D, (x, x,..., x t ), is a pat of lengt t in G D if (x i, x i+ ) Θ D, i =,..., t. For x D, (x) as a pat of lengt. Te pat (x, x,..., x t ) is terminal if tere is no pat of te form Department of Computer Science and Engineering, University of California San Diego; ttp://cseweb.ucsd.edu/~gill/. Keywords: lattice exit models, ZFC independence, subset sum problem, order type equivalence, regressive regularity.
LATTICE EXIT MODELS 9 8 7 0 0 7 8 9 FIGURE. Downward directed, k = (x, x,..., x t, x t+ ). We say x is a terminal vertex of G D if te pat (x) is a terminal pat in G D. Definition. (Cubes and Cartesian powers in N k ). Te set E E k, were E i N, E i = p, i =,..., k, are k-cubes of lengt p. If E i = E, i =,..., k, ten tis cube is E k := k E, te kt Cartesian power of E. Definition. (p D and significant labels). For finite D N k, let G D = (D, Θ D ). Let P D (z) be te set of all x wit a pat in G D from z to x (including te pat (z)). Define p D by p D (z) = min({min(x) x P D (z)}). We call p D te total pat label function. Te set {z p D (z) < min(z)} is te set of vertices wit significant labels. Te set {p D (z) p D (z) < min(z)} is te set of significant labels for p D. Definition. (ˆt D and significant labels). For finite D N k, let G D = (D, Θ D ). Let T D (z) be te set of all last vertices of terminal pats (x, x,..., x t ) were z = x. Define ˆt D by ˆt D (z) = max(z) if (z) terminal, else ˆt D (z) = min({min(x) x T D (z)})
LATTICE EXIT MODELS We call ˆt D te terminal pat label function. Te set {z ˆt D (z) < min(z)} is te set of vertices wit significant labels. Te set {ˆt D (z) ˆt D (z) < min(z)} is te set of significant labels for ˆt D. In definition.8 we define a variation on p D of definition. called ˆp D were ˆp D (z) = max(z) if (z) terminal. Tis convention and ˆt D (z) = max(z) if (z) terminal are used to uniquely distinguis terminal vertices wen te graps are downward (definition.). We next give some standard grap teory terminology. Definition.7. Let z D, G D = (D, Θ D ). Te adjacent vertices of z, GD z, are defined by GD z = {x (z, x) Θ D}. Non-terminal adjacent vertices of z, NT z, are defined by NT z = {x x GD z, (x) not terminal}. Te terminal adjacent vertices of z, T z, are defined by T z = GD z \ NT z. Definition.8 (Recursive definitions ˆp D, ˆt D ). Let z D, G D = (D, Θ D ). We define ˆt D (z) = max(z) if (z) is terminal. Else ˆt D (z) is te minimum over te set {ˆt D (x) x NT z } {min(x) x T z }. We define ˆp D (z) = max(z) if (z) is terminal. Else ˆp D (z) is te minimum over te set { ˆp D (x) x NT z } {min(x) x T z } {min(z)}. Note: ˆt D (z) = max(z) iff (z) is terminal, and ˆp D (z) = max(z) iff (z) is terminal. Lattice Exit Story (some intuition): We consider G D wit terminal label function ˆt D (case k = ). Tink of te digrap G D as a complex of caves (vertices) and tunnels (downward directed). Eac cave as a label on its wall giving its coordinates in N. Te value of ˆt D (z) is also written on te wall of cave z. An explorer is lowered into cave z and tasked wit finding a terminal cave x, accessible from z, tat as sortest distance to te boundary of te lattice (equal to ˆt D (z) and called te lattice exit distance ). Specifically, if ˆt D (z) min(z) ten no exploration is needed; cave z is closest to te boundary. If ˆt D (z) < min(z) ten te explorer finds a terminal pat z,..., x wit ˆt D (z) = min(x). Te pat (z,..., x) is ten written on te wall of cave z. Te set {z ˆt D (z) < min(z)} gives te caves were getting out of z gets one closer to te boundary. Te set {ˆt D (z) ˆt D (z) < min(z)} represents te distances obtained by suc explorations (i.e., significant labels for ˆt D ).
LATTICE EXIT MODELS 9 8 7 ^t D(v)= ^t D(w)= ^ t D(z)= p ^ D(w)= p ^ D(z)= z w 0 0 7 8 9 v FIGURE. Terminal vs. pat labels, k =. REGULARITIES COMFORTABLY IN ZFC We discuss te large scale regularties of ˆp D (z). We start wit a version of Ramsey s Teorem ([GRS90], p.). First we define order equivalence of k-tuples. Definition. (Equivalent ordered k-tuples). Two k-tuples in N k, x = (n,..., n k ) and y = (m,..., m k ), are order equivalent tuples (ot) if {(i, j) n i < n j } = {(i, j) m i < m j } and {(i, j) n i = n j } = {(i, j) m i = m j }. Note tat ot is an equivalence relation on N k. Te standard SDR (system of distinct representatives) for te ot equivalence relation is gotten by replacing x = (n,..., n k ) by ρ(x) := (ρ Sx (n ),..., ρ Sx (n k )) were ρ Sx (n j ) is te rank of n j in S x = {n,..., n k } (e.g, x = (, 8,,, 8), S x = {,, 8}, ρ(x) = (0,,, 0, )). Te number of equivalence classes is k j= σ(k, j) were σ(k, j) is te number of surjections from a k set to a j set. Teorem. (Ramsey s teorem version). If f : N r X, Im(f) = {f(z) z N r } finite, ten tere exists infinite H = { 0,,...} N s.t. f is constant on te order equivalence classes of H r. Definition. (X notation). Define X (statement) = 0 if statement false, and X (statement) = if statement true.
LATTICE EXIT MODELS 9 8 7 [] [] [] [] [] 8 [] [] [] [] [] 0 0 7 8 9 FIGURE. [ ˆp D (z)] (in brackets) and ˆt D (z) by dept first searc ((,), (,0), (,)) ((7,9), (,), (,)) ((7,9), (,0)) 9 () (0) 8 7 () (0) 0 0 7 8 9 FIGURE. Basic idea for proof of teorem. Teorem. ( ˆp D large scale regularity structure). Let G = (N k, Θ). Tere exists an infinite H = { 0,,...} N suc tat for all z D = H k
LATTICE EXIT MODELS eiter ˆp D (z) = 0 < min(z) or ˆp D (z) min(z). Tus, ˆp D as at most one significant label: { ˆp D (z) ˆp D (z) < min(z)}. Proof. Recall definition. of te function ˆp D (z). Use Ramsey (. wit r = k) on N k : w = (n,... n k, m,... m k ) N k (x, y) N k N k, x = (n,... n k ), y = (m,... m k ). Let f (w) = X ((x, y) Θ), so tat Im( f ) = {0, }. By Ramsey s teorem, tere is an infinite H N suc tat f is constant on te order type equivalence classes of H k H k H k. We sow tat ˆp D (z) as at most one significant label on D = H k. Let z = x,..., x t be a sortest pat in G D from z to a vertex min(x t ) = ˆp D (z). We claim min(x t ) = 0. Oterwise, replace every minimum coordinate of x t by 0 to obtain ˆx t and note tat (x t, x t ) and (x t, ˆx t ) ave te same ot in H k. Tus, (x t, ˆx t ) Θ and z = x,..., ˆx t is a sortest pat required.. BASIC DEFINITIONS AND THEOREMS Definition. (decreasing sets of functions). Let f and g be functions wit domains contained in N k and ranges in N. Define f g by () domain( f ) domain(g) and () for all x domain( f ), f (x) g(x). A set S of suc functions is decreasing if for all f, g S wit domain( f ) domain(g), f g. Definition. (regressive value). Let X N k and f : X Y N. An integer n is a regressive value of f on X if tere exist x suc tat f (x) = n < min(x). Definition. (field of a function and reflexive functions). For A N k define field(a) to be te set of all coordinates of elements of A. A function f is reflexive in N k if domain( f ) N k and range( f ) field(domain( f )). Definition. (te set of functions T(k) ). T(k) denotes all reflexive functions wit finite domain: domain( f ) <. Definition. (full and jump free). Let Q T(k) denote a collection of reflexive functions in N k wose domains are finite subsets of N k. () full: Q is a full family of functions on N k if for every finite subset D N k tere is at least one function f in Q wose domain is D. () jump free: For D N k and x D define D x = {z z D, max(z) < max(x)}. Suppose tat for all f A and f B in Q, were f A as domain A and f B as domain B, te conditions
LATTICE EXIT MODELS 7 x A B, A x B x, and f A (y) = f B (y) for all y A x imply tat f A (x) f B (x). Ten Q will be called a jump free family of functions on N k. Definition. (Regressively regular over E). Let k, D N k, D finite, f : D N. We say f is regressively regular over E, E k D, if for eac ot eiter () or (): () decreasing mins: For all x, y E k of order type ot, f (x) = f (y) < min(e) () non decreasing mins: For all x E k of order type ot f (x) min(x). Teorem.7 (Decreasing class). Let k, p and S T(k) be a full and decreasing family of functions. Ten some f S as at most k k regressive values on some Cartesian power E k domain( f ), E = p. In fact, tere exists E A N, E = p and f S, domain( f ) = A k suc tat f is regressively regular over E. Teorem.8 (Jump free teorem ([Fri97], [Fri98])). Let p, k and S T(k) be a full and jump free family. Ten some f S as at most k k regressive values on some E k domain( f ), E = p. In fact, some f S is regressively regular over some E of cardinality p. z B A x a subset of B x x x=(x,x ) A A x B x max(x) z FIGURE. Basic jump free condition. We use ZFC for te axioms of set teory, Zermelo-Frankel plus te axiom of coice (see Wikipedia). Te jump free teorem can be proved in ZFC + ( n)( n-subtle cardinal) but not in ( n-subtle cardinal) for any fixed n (assuming tis teory is consistent). A proof is in Section of [Fri97], Applications of Large Cardinals to Grap Teory, October, 997, No. of Downloadable Manuscripts. Te decreasing class teorem is proved in Section of [Fri97] using tecniques witin ZFC (Ramsey teory in particular). Te jump free teorem is used to study lattice posets in [RW99] (Appendix A defines n-subtle cardinals). Te
LATTICE EXIT MODELS 8 B x = + A x = x=(,9) 9 ^ t A (y) = ^ t B (y) y є A x = > 8 x є A U B A x Bx U ^ (x) ^ (x) t A t B 7 0 0 7 8 9 FIGURE. Dased edges not allowed by jump free. functions ˆp D define a full and decreasing class and tus ave large scale regularities of te form specified in Teorem.7. Te functions ˆt D form a full but not decreasing class. We will use te jump-free teorem to describe te large scale regularities of tese functions.. LARGE SCALE REGULARITIES OF MORE COMPLEX LATTICE EXIT MODELS Lemma. ({ˆt D } full, reflexive, jump free). Take S = {ˆt D D N k, D < } (see.). Ten S is full, reflexive, and jump free. Proof. See figures and. Full and reflexive is immediate. Let ˆt A and ˆt B satisfy te conditions of f A and f B in definition.. Note tat by definition, x / A x or B x. If (x) is terminal in A ten ˆt A (x) = max(x) ˆt B (x) by te downward condition on G. Else, let (x,..., y) be a terminal pat in G A. Ten ˆt B (y) = ˆt A (y) = max(y) implies (x,..., y) is a terminal pat in G B. Tus, ˆt A (x) ˆt B (x) as was to be sown. Teorem. (Jump free teorem for ˆt D ). Let S = {ˆt D D N k, D < } and let p, k. Ten some f S as at most k k regressive values on
LATTICE EXIT MODELS 9 9 8 7 E={,,,8} ^t D 0 0 7 8 9 FIGURE 7. Regressive regularity of ˆt D on E = {,,, 8} some E k domain( f ), E = p. In fact, some f S is regressively regular over some E of cardinality p. Proof. Follows from lemma. and te jump free teorem.8. See figure 7 for an example of regressive regularity. Following Friedman [Fri97]: Definition. (Partial selection). A function F wit domain a subset of X and range a subset of Y will be called a partial function from X to Y (denoted by F : X Y). If z X but z is not in te domain of F, we say F is not defined at z. Let r. A partial function F : N k (N k N) r N will be called a partial selection function if wenever F(x, ((y, n ), (y, n ),... (y r, n r ))) is defined we ave F(x, ((y, n ), (y, n ),... (y r, n r ))) = n i for some i r. Next we generalize te ˆt D to a function ŝ D. We refer to te former function as te terminal vertex model and to te latter as te committee model. Definition. (ŝ D for G D ). Let r, z D, G D = (D, Θ D ), D finite, G z D = {x (z, x) Θ D}. Let F : N k (N k N) r N be a partial selection function. We define ŝ D (z) recursively as follows. Let Φ D z := {F[z, (y, n ), (y, n ),..., (y r, n r )], y i G z D}
LATTICE EXIT MODELS 0 be te set of defined values of F were n i = ŝ D (y i ) if Φ D y i = and n i = min(y i ) if Φ D y i =. If Φ D z =, define ŝ D (z) = max(z). If Φ D z =, define ŝ D (z) be te minimum over Φ D z. NOTE: If Φz D = ten an induction on max(z) sows ŝ D (z) < max(z). Recall tat (G, Θ) is downward. Tus, Φz D = iff ŝ D (z) = max(z). Teorem. (Large scale regularities for ŝ D ). Let r, p, k. S = {ŝ D D N k, D < }. Ten some f S as at most k k regressive values over some E k domain( f ), E = p. In fact, some f S is regressively regular over some E of cardinality p. Proof. Recall.8. Let S = {ŝ D D N k, D < }. S is obviously full and reflexive. We sow S is jump free. We sow for all ŝ A and ŝ B in S, te conditions x A B, A x B x, and ŝ A (y) = ŝ B (y) for all y A x imply tat ŝ A (x) ŝ B (x). (i.e., S is jump free). If Φx A = ten ŝ A (x) = max(x) ŝ B (x). Let n = F[x, (y, n ), (y, n ),... (y r, n r )] Φx A were n i = ŝ A (y i ) if ŝ A (y i ) < max(y i ) and n i = min(y i ) if ŝ A (y i ) = max(y i ). But ŝ A (y i ) = ŝ B (y i ), i =,..., r, implies n Φx B and tus Φx A Φx B and ŝ A (x) = min(φx A ) min(φx B ) = ŝ B (x). F[x, ((,), )), ((,8), ), ((8,7), 7)] = C F[x, ((,8), )), ((8,7), 7)] = 7 C x=(7, ) () F[x, ((,8), )), ((, 7), )] = C (,) 0 Φ D x ={,, 7} sˆ D (x) = Φ 9 D C z ={} sˆ D (z) = max(z) C 7 e.g., z=(8,7) max(z)=8 8 D is all points sown: E={7,} and E x E= (,8) {(7,7), (7,), (,7),,)} 7 7 (7,7) (8,7)(8) (,7) (7) C E={7,}E={7,} (,) () () (7) E D (8) D () (9) F () () 0 0 7 8 9 0 FIGURE 8. An example of ŝ D
LATTICE EXIT MODELS As an example of computing ŝ D, consider figure 8. Te values of te terminal vertices were Φ A x = are sown in parenteses, left to rigt: (), (), (), (), (), (8), (8), (9). Tese numbers are max((a, b)) for eac terminal vertex (a, b). We assume we ave a partial selection functions of te form F : N (N N) r N (r =, ere). To compute ŝ D (x) for x = (7, ) tere are tree defined values: F[x, ((, ), ), ((, 8), ), ((8, 7), 7)] =, F[x, ((, 8), ), ((8, 7), 7)] = 7, F[x, ((, 8), ), ((, 7), )] =. Intuitively, we tink of tese as (ordered) committees reporting values to te boss, x = (7, ). Te first committee, C, consists of subordinates, (, ), (, 8), (8, 7) reporting respectively,, 7. Te committee decides to report (indicated by C in figure 8). Te recursive construction starts wit terminal vertices reporting teir minimal coordinates. But, te value reported by eac committee is not, in general, te actual minimum of te reports of te individual members. Neverteless, te boss, x = (7, ) always takes te minimum of te values reported to im by te committees. In tis case te values reported by te committees are, 7, te boss takes (i.e., ŝ D (x) = for te boss, x = (7, )). Observe in figure 8 tat te values in parenteses, (), (), (), (), (), (8), (8), (9), don t figure into te recursive construction of ŝ D. Tey immediately pass teir minimum values on to te computation:,,,,,, 7,. Tis leads to te following generalization of definition.. Definition. ( ρ D for G D). Let r, k, z D, D finite, G D = (D, Θ D ). Let F : N k (N k N) r N be a partial selection function. Let ρ D : D N be suc tat min(x) ρ D (x). We define ρ D recursively on max. Let Φ D z := {F[z, (y, n ), (y, n ),..., (y r, n r )], y i G z D} be te set of defined values of F were n i = ρ D (y i) if Φy D i =, and n i = min(y i ) if Φy D i =. If Φz D =, define ρ D (z) to be te minimum over Φz D. If Φz D =, define ρ D (z) = ρ D(z). Note tat ρ D need not be reflexive on D. Lemma.7. Let E be of cardinality p. Ten ŝ D is regressively regular over E iff ρ D is regressively regular over E. In fact, ρ D (x) = ŝ D(x) < max(x) if Φ D x =. Proof. Let x, y E k. From te recursive definitions. and., te sets Φx D are te same for bot ρ D (x) and ŝ D(x). Tus, w = ŝ D (x) = ŝ D (y) < min(e) iff w = ρ D (x) = ρ D (y) < min(e) as tese relations imply bot Φx D = and Φy D =. Likewise, if Φx D = ten max(x) > ŝ D (x) =
LATTICE EXIT MODELS ρ D (x) min(x). If ΦD x =, by definition ρ D (x) = ρ D(x) min(x) and ŝ D (x) = max(x) min(x). Teorem.8 (Regressive regularity of ρ D ). Let r, p, k. Let G = (N k, Θ) be downward directed. Let S = { ρ D D Nk, D < }. Ten some f S as at most k k regressive values on some E k domain( f ) = D, E = p. In fact, some f S is regressively regular over some E of cardinality p. Proof. Follows from lemma.7 wic sows tat te sets, E, E = p, over wic ρ D is regressively regular don t depend on te function ρ D as defined. In fact, ρ D (x) = ŝ D(x) < max(x) if Φx D =. If Φx D = ten te values ρ D (x) = ρ D(x) are only constrained by te condition ρ D (x) min(x). Teorem. wit max(x) replaced by min(x) wen Φx D = as been sown by Friedman to be independent of ZFC (same large cardinals as te jump free teorem). See Teorem. troug Teorem. [Fri97]. Tus, a special case of teorem.8 (ρ D = min) is independent of ZFC. Lemma.7 sows tat teorem.8 for any ρ D results in exactly te same sets E of regressive regularity as teorem.. Hence, teorem.8 provides a family of ZFC independent jump free type teorems parameterized by te ρ D. Figure 9 sows an example of a regressively regular ρ D function over a set E. Te coices of F (definition.) aren t unique. We use ρ := ρ D. For z = (x, y) we define ρ(z) = x + y, labeling only E wit tese.. COMBINATORIAL FORMULATIONS We start by defining some canonical systems of distinct representatives (SDRs). Definition. (SDRs). Let SUR(k, j) be te surjective maps {0,..., k } to {0,..., j }. Let F kp = { f f : {0,... k } {0,..., p }}. Define OT(k, p) := p j=sur(k, j). OT(k, p) is te canonical SDR for te order type equivalence relation on F kp. Let E = {e 0,..., e p } be a subset of N, e 0 < < e p. Let E k = {e f f F kp } were e f = (e f (0),..., e f (p ) ). Define OT(k, p, E) = {e f f OT(k, p)}
LATTICE EXIT MODELS 0 For z E of ot (0, ), ρ D (z) = < min(e). For ot (0, 0), (, 0), ρ D (z) min(z). 9 8 7 8 ρ(s, t) = s + t 8 0 0 7 8 9 0 Construct ρ D regressively regular over E = {, 7, }. D is E plus circled vertices. Terminals: (, ), (, ), (, ), (, ), (, ), (, ), (9, ), (9, ), (7, 7), (9, 7), (, 7), (, ). We make up bosses and committees to define ρ D to be regressively regular over E. For boss (, 7) we assume committees {((, ), ), ((, ), ))} and {((, ), ), ((, ), )}, reporting values and. Te boss (, 7) takes te minimum,. Tus, ρ D ((, 7)) =. F((, 7), ((, ), ), ((, ), )) =, F((, 7), ((, ), ), ((, ), )) = implies D ((, 7)) =. Defining F((, ), ((, ), ), ((, ), )) = implies ρ D (, ) =. Defining F((, 8), ((, ), ), ((, ), )) = implies ρ D ((, 8)) =. F((, ), ((, 8), )) = F((7, ), ((, 8), )) = gives ρ D ((, )) = and ρ D ((7, )) =. F((, ), ((9, ), ), ((9, ), ), ((9, 7), 7)) = implies ρ D ((, )) = F((7, ), ((, ), )) = implies ρ D ((7, )) =. For terminal (s, t) define ρ(s, t) = s + t. FIGURE 9. Computing example of ρ D to be te canonical SDR for order type equivalence on E k. Define E k f = {e g g F kp, g f } to be te equivalence class in E k associated wit f OT(k, p).
LATTICE EXIT MODELS Referring to figure 0, E 0 = {(e, e, e 0 ), (e, e, e 0 ), (e, e e 0 ), (e, e, e )}. OTs {0,, } {0,,, } 000 00 00 00 00 00 00 00 00 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 FIGURE 0. Canonical order type array T entries in F, Definition. ( Canonical order type array T). Define a canonical order type array, T, wit entries in F kp, as follows. Te first column of T is te vector T () = ( f, f,... f m ) were te f i are te elements of OT(k, p). Te sets SUR(k, j) are listed by j wit elements of eac set in lexicograpic order. Te row, T (i), i =,..., m, is T (i) = ( f i, f i,..., f ini ), te lexicograpically ordered equivalence class associated wit f i. Te entries T(i, j) = f ij were i = π( f i ), j = π( f ij ), are te positions of f i and f ij in te lists of column T () and row T (i) respectively. Definition. (TE k and gtk E ). Let T be a canonical order type array over F kp, E = {e 0,..., e p } N. Replacing eac f in T by e f (definition.) gives an array wic we denote by X = TE k. Tus, X(i, j) = e f ij. If domain(g) E k ten Y = gte k as Y(i, j) = g(e f ij ) = g(x(i, j)). Consider figure 9 were E = {, 7, }. T = 00 0 0 0 0 T E = Using ρ D as in figure 9 we get e 00 e e e 0 e 0 e = e 0 e 0 e ρ D (, ) ρ D (7, 7) ρ D (, ) ρ D T E = ρ D (, 7) ρ D (, ) ρ D (7, ) = ρ D (7, ) ρ D (, ) ρ D (, wic displays te regressive regularity over E. (, ) (7, 7) (, ) (, 7) (, ) (7, ) (7, ) (, ) (, 7) 8 8
LATTICE EXIT MODELS Note tat te pair of arrays, (X, Y), were X = T E and Y = ρ D T E, X = (, ) (7, 7) (, ) (, 7) (, ) (7, ) (7, ) (, ) (, 7) completely specifies te function ρ D. and Y = 8 8 We use te notation of definition., lemma.7 and teorem.8. Definition. (D capped by E k D). For D N k, let max(d) be te maximum over max(z), z D. Let setmax(d) = {z z D, max(z) = max(d)}. If setmax(d) = setmax(e k ), we say tat D is capped by E k D wit te cap defined to be setmax(e k ). Definition. (Canonical capped bi-array). Let T be a canonical order type array over F kp, E = {e 0,..., e p } N. Te pair (X, Y) = (TE k, ρ D Tk E ) is a canonical bi-array representation of te function ρ D. Let D be capped by E k D. Te pair (X, Y) = (TE k, ρ D Tk E ) is a canonical capped bi-array representation of te function ρ D. We write (X, Y) p = (TE k, ρ D Tk E ) to indicate E = p. See figure 7 for an example were D is capped by E k D. Te downward condition on G implies it is always possible to coose D to satisfy tis condition witout canging te function (X, Y) = (T k E, ρ D Tk E ). Suc a D, capped by E k, contains a description of E k exposed in te cap, setmax(e k ). Note te following invariants of (X, Y) = (T k E, ρ D Tk E ). Any pair (X σ, Y σ ), were te rows are permuted by σ also represents te function ρ D and preserves te elements of te first column. Given a permutation τ on {,..., n i } wit τ() = te elements of rows X (i) and Y (i), i =,... m i, can be replaced by X(i, τ()),..., X(i, τ(i)) and Y(i, τ()),..., Y(i, τ(i)), preserving te elements of te first column wile still representing te function ρ D. We use te notation of definition., lemma.7 and teorem.8. Definition. (subsets of E k ). Let f be regressively regular over E N, E = p (definition.). Define subsets E k L := {x x Ek, f (x) < min(e)}, E k U := {x x Ek, f (x) min(x)}. For f = ρ D, Ek L = {x x Ek, ρ D (x) < min(e)}. Ek U is furter partitioned E k = = {x x Ek U, Φ D(x) = } and E k = {x x Ek U, Φ D(x) = }.
LATTICE EXIT MODELS Note tat bot EL k and Ek U, wen nonempty, are unions of elements (blocks) of OT(k, p, E) (definition.). By definition, te regressively regular f is constant on te blocks of te order type equivalence classes contained in EL k. Recall tat ρ D is restricted to E k in te recursive construction of ρ D and can be canged on tis set as long as te condition ρ D (z) min(z) olds. Suc canges in ρ D leave te sets of definition. invariant. As an example, consider figure 9. Tere, k =, p = wit E = {, 7, }; E is indicated by small squares, D by squares plus circles. Te SDR for order type equivalence on F, is OT(, ) = {(0, 0), (0, ), (, 0)}, OT(,, E) = {(e (0,0), e (0,), e (,0) } = {(e 0, e 0 ), (e 0, e ), (e, e 0 )} = {(, ), (, 7), (7, )}. We ave E k = {e f f F, } (figure 9.) EL k = {(, 7), (, ), (7, )} and, in tis case, te order equivalence class E k (0,) = Ek L. In tis example, E k U = Ek (0,0) Ek (,0) wit E k = diag(ek ) {(, 7)} and E k = = Ek L {(, ), (7, )} were diag(e k ) = {e 00, e, e } = {(, ), (7, 7), (, )}. It is easy to see in general tat EL k Ek = and tat eiter diag(ek ) E k or diag(e k ) EL k.. USING THE FLEXIBILITY OF ρ D Definition. (Regressive regularity for (X, Y)). A canonical (capped) bi-array (X, Y) = (T k E, ρ D Tk E ) is regressively regular if for eac row Y (i), i =,..., m, eiter or Y (i) (j) min(x (i) (j)), j n i Y (i) (s) = Y (i) (t) < min(e), s, t n i. Teorem. (Version of teorem.8 for bi-arrays). Let r, p, k. Let G = (N k, Θ) be downward directed. Tere is a canonically capped bi-array (X, Y) = (TE k, ρ D Tk E ), E = p, suc tat {Y(i, j) Y(i, j) < min(x(i, j))} < k k. In fact, tere exists a regressively regular canonical capped bi-array (X, Y) = (TE k, ρ D Tk E ), E = p. Proof. Restatement of teorem.8 for canonical capped bi-arrays.
LATTICE EXIT MODELS 7 Lemma. (Coosing ρ D ). Let r, p, k. Let G = (N k, Θ) be downward directed. Let (X, Y) = (TE k, ρ D Tk E ), E = p, be a regressively regular canonical capped bi-array. Assume EL k is nonempty and diag(ek ) E k. Let e 0 = (e 0, e 0,..., e 0 ) E k. Ten ρ D can be cosen suc tat () ρ D (z) + ρ D (e 0 ) = ( Ek L + )e 0. z EL k Proof. We can use te notation of eiter teorem. or teorem.8. We use te latter. Recall tat E k L = {z ρ D (z) < e 0} were e 0 = min(e). Setting S = z E k L ρ D (z) we ave S < Ek L e 0. But diag(e k ) E k implies ρ D (e 0 ) = ρ D(e 0 ) wic can be assigned any value ρ D(e 0 ) min(e 0 ) = e 0. Tus, assign ρ D (e 0 ) = e 0 + ( EL k e 0 S) > e 0 since EL k =. Tus () is satisfied. As an aside, in te notation of teorem. we can calculate E k L e 0 as follows: () E k L e 0 = e 0 m i= X (Y(i, ) < e 0 )n i. Also, () z E k L ρ D (z) = m i= X (Y(i, ) < e 0 ) n i j= Y(i, j). We engage in a tougt experiment by using teorem. and lemma. to construct a class of sequences of instances to te classical subset sum problem. We assume tat for eac z diag(n k ), te set of partial selection functions (see.) of te form F[z, (y, n ),...], z diag(n k ), is empty. Tis restricted diagonal condition guarantees tat, r, Φ D z = {F[z, (y, n ), (y, n ),..., (y r, n r )], y i G z D} is empty for z diag(e k ). Tis implies diag(e k ) E k as in lemma.. Let r, p, k. Let G = (N k, Θ) be downward directed and diagonally restricted. Let (X, Y) = (TE k, ρ D Tk E ), E = p =,,..., be a sequence of regressively regular canonical capped bi-arrays. Wit eac bi-array we associate te multiset M p = {Y(i, j) i m, j n i, Y(i, ) < e 0 } {Y(, ),... Y(, p)}. Assume first tat EL k =. For eac suc M p let t p = ( EL k + )e 0 as in lemma.. As in lemma., take Y(, ) = ρ D (e 0 ) as specified in lemma.. Lemma. states tat for tis instance tere is a solution to te subset sum problem. Coose Y(, ),..., Y(, p) suc tat tis solution is unique.
LATTICE EXIT MODELS 8 If E k L =, coose Y(, ),..., Y(, p) and t p suc tat tere is no solution. Tus, te instances to te subset sum problems just described ave solutions if and only if EL k =. Tis condition can be verified by inspecting te first column, Y () and comparing it wit te first column X () (because of regressive regularity). Te number of comparisons is m < k k, k fixed. To summarize, we ave defined a class of instances to te subset sum problem, parameterized by r, k and G = (N k, Θ), a downward directed and diagonally restricted grap. Te parameter p goes to infinity to measure te size of te instances. We fix k and vary r and G. Our procedure for cecking te solutions for tese instances is localized to te first columns of X and Y and tus bounded by k k. Te existence of tese instances and teir solution as been demonstrated by using a corollary to a teorem independent of ZFC (teorem.). No oter proof of existence is known to us. Te corollary we used is teorem. wit downward directed grap G = (N k, Θ) replaced by downward directed, diagonally restricted grap G = (N k, Θ). We conjecture tat tis corollary is also independent of ZFC. In tis case, te only proof of te existence of tese instances and teir solution would use a ZFC independent teorem. Acknowledgments: Te autor tanks Professors Jeff Remmel and Sam Buss (University of California San Diego, Department of Matematics) and Professor Emeritus Rod Canfield (University of Georgia, Department of Computer Science) for teir elpful comments and suggestions. REFERENCES [Fri97] Harvey Friedman. Applications of large cardinals to grap teory. Tecnical report, Department of Matematics, Oio State University, 997. [Fri98] Harvey Friedman. Finite functions and te necessary use of large cardinals. Ann. of Mat., 8:80 89, 998. [GRS90] R. L. Graam, B. L. Rotscild, and J. H. Spencer. Ramsey Teory nd Ed. Jon Wiley, New York, 990. [RW99] Jeffrey B. Remmel and S. Gill Williamson. Large-scale regularities of lattice embeddings of posets. Order, : 0, 999.