Extensional Uniformity for Boolean Circuits

Extensional Uniformit y for Bo olean Circuits ∗ Pierre McKenzie 1 , Michael Thomas 2 , and Herib ert V ollmer 2 1 D´ ep. d’informatique et de rec herche op´ erationnelle, Universit ´ e de Montr´ eal, C.P . 6128, succ. Centre-Ville, Montr ´ eal (Qu´ eb ec), H3C 3J7 Canada mckenzie@i ro.umontr eal.ca 2 Institut f¨ ur Theoretisc h e Informatik, Leibniz Universit¨ at Hannov er, App elstr. 4, 30167 Hannov er, Germany {thomas, vol lmer}@thi. uni- hannover.de Abstract Imp osing an extensional uniformit y condition on a non-uni- form circuit complexity class C means simply intersecting C with a uni- form class L . By contra st, the u sual intensional uniformity conditions require th at a resource-b ounded machine b e able t o exhibit th e circuits in the circuit family d efining C . W e sa y that ( C , L ) has the Uniformity Duality Pr op erty if the extensionally uniform class C ∩ L can b e captured intensio nally by means of adding so-called L -numer ic al pr e dic ates to the first-order descriptive complexity apparatus describing the connection language of the circuit family definin g C . This pap er exhibits p ositiv e instances and negative instances of the Uni- formit y Duality Prop erty . Keywords. Boolean circuits, uniformity , descriptive complexity 1 In t ro duction A family { C n } n ≥ 1 of Bo olean cir c uits is uniform if the wa y in which C n +1 can differ from C n is restr icted. Generally , unifo r mit y is impo sed by requiring that some form o f a resource-b ounded c o nstructor on input n b e able to fully or partially descr ibe C n (see [8,21,1,5,14] o r r e fer to [24] for an overview). Circuit- based lang uage classe s c a n then b e compar ed with classes that are ba sed on a finite c o mputing mechanism such as a T uring machine. Recall t he gist of descr iptiv e complexity . Consider the set of w ords w ∈ { a, b } ⋆ having no b at a n even p o sition. This lang uage is describ ed by the FO[ <, Even ] formula ¬∃ i  Even ( i ) ∧ P b ( i )  . In such a first-o r der formula, the v a riables r ange ov er p ositions in w , a predicate P σ for σ ∈ { a, b } ho lds at i iff w i = σ , and a numeric al predica te, such as the obvious 1 - ary Even predica te here, holds a t its arguments iff these arguments fulfill the s p ecific relatio n. The following viewp oin t has emerg ed [5 ,3,6] ov er tw o decades: when a cir cuit- b ase d langu age class is char acterize d using fi rst-or der descriptive c omplexity, the cir cuit uniformity c onditions spring up in the lo gic in the form of r estrictions on the set of numeric al pr e dic ates al lowe d . ∗ Supp orted in part by DFG VO 630/ 6-1, by the NSERC of Canada and by the (Qu´ eb ec) FQRNT. As a well studied example [12,5], FO[ <, + , × ] = DLOGTIME-uniform A C 0 ( non-uniform AC 0 = FO[ ar b ], where the la tter class is the cla ss of languages definable by firs t- o rder formulae entitled to arb itra ry numerical predicates (we use a lo gic a nd the set of lang uages it captures interc hangeably when this brings no co nfusion). In a related v e in but with a different emphasis, Straubing [23] presents a bea utiful account of the relationship b etw een automata theory , forma l lo gic and (non-uniform) circuit complexity . Straubing concludes by expr essing the prov en fact that AC 0 ( A CC 0 and the c e lebrated conjectures that AC 0 [ q ] ( ACC 0 and that ACC 0 ( NC 1 as ins ta nces of the following conjecture conce rning the class REG o f r egular languag es: Q [ arb ] ∩ REG = Q [ reg ] . (1) In Straubing’s instances, Q is an appropriate s e t o f quantifiers c hosen fr om {∃} ∪ {∃ ( q,r ) : 0 ≤ r < q } and reg is the set of r e g ular nu merical pr edicates, that is, the set of those n umerical predica tes of ar bitrary arity definable in a formal sens e by finite automata. W e stress the po int of v ie w that intersecting {∃} [ arb ] = F O[ arb ] with REG to form FO[ arb ] ∩ REG in conjecture (1) amounts to imp osing unifor mit y on the non-unifor m cla ss FO[ arb ]. And once again, im- po sing uniformity has the effect o f restricting the numerical pr edicates: it is a prov en fact that F O[ arb ] ∩ RE G = FO[ reg ], and conjecture (1) express es the hop e tha t this phenomenon ex tends from {∃} to other Q , which would determine m uch o f the internal s tr ucture of NC 1 . W e ask: 1. Does the duality b etw een uniformity in a cir c uit-based cla ss and numerical predicates in its logica l characterizatio n extend b ey o nd NC 1 ? 2. What w ould play the r ole o f the regular n umerical predicates in s uc h a duality? 3. Could such a dua lit y help unders tanding classes such as the con text-free languages in AC 0 ? T o tackle the first q uestion, we note that int ersecting with RE G is just one out o f many p ossible wa ys in which one ca n “ impose uniformity”. Indeed, if L is any unifor m lang uage cla ss, one can r eplace Q [ arb ] ∩ REG by Q [ arb ] ∩ L to get another unifor m sub class of Q [ arb ]. F or example, co nsider any “formal la nguage class” (in the lo ose terminolo gy used b y La nge when discussing languag e theory versus complexity theo r y [14]), such as the class CFL of c on tex t- fr ee lang ua ges. Undoubtedly , CFL is a uniform class o f la nguages. Therefore, the cla ss Q [ arb ] ∩ CFL is another uniform class well worth comparing with Q [ < , +] or Q [ <, + , × ]. Of cours e , FO[ arb ] ∩ CFL is no ne o ther than the p o orly understo od class AC 0 ∩ CFL, and when Q is a quantifier given by some word problem o f a nonsolv able group, (F O+ { Q } )[ a rb ] ∩ CFL is the po orly unders too d c la ss NC 1 ∩ CFL alluded to 20 years ag o [1 1]. The pr e sen t pap er thus considers classes Q [ arb ] ∩ L for v ario us Q and L . T o explain its title, we note that the cons tructor-based approa c h defines uniform classes by sp ecifying their prop erties: such definitions a re intensional definitions. By co n tra st, v iewing Q [ arb ] ∩ REG as a unifor m class amo un ts to a n extensional 2 definition, namely o ne that selects the members of Q [ arb ] that will collectively form the uniform class. I n this pap e r we set up the extensional uniformity frame- work a nd we study classes Q [ arb ] ∩ L for Q ⊇ {∃} . Certainly , the uniform class L will deter mine the class of n umerical predicates we hav e to use when try ing to capture Q [ arb ] ∩L , as Straubing do es for L = RE G , as an in tensionally unifor m class. A co n tributio n of this pa per is to pr o vide a meaningful definition fo r the set L N of L -n u meric al pr e dic ates . Informally , L N is the set of relatio ns ov er the natural num b ers that ar e defina ble in the sense of Stra ubing [23, Section I II.2 ] by a languag e over a single to n a lphabet drawn from L . When L is RE G, the L -numerical predicates are precis ely Str aubing’s regular num erical predicates. Fix a set Q o f monoidal o r g roupoida l quantifiers in the sense of [5,24,16]. (As pro to t ypica l exa mples, the reader unfamiliar with suc h quantifiers may think of the us ual exis ten tial and universal quantifiers, of Str aubing’s “there exis t r mo dulo q ” quantifiers, or o f threshold quantifiers such as “there e xist a ma jo r it y” or “there exist at least t ”). W e prop ose the Uniformity Duality Pr op erty for ( Q , L ) a s a natural gener alization of conjecture (1): Uniformit y Duality Prop ert y for ( Q , L ) : Q [ arb ] ∩ L = Q [ <, L N ] ∩ L . Barring ton, Immerman and Str aubing [5] ha ve shown that Q [ arb ] equa ls A C 0 [ Q ], that is, non-uniform A C 0 with Q gates . Behle and Lange [6] have shown that Q [ <, L N ] equals FO[ <, L N ]-uniform AC 0 [ Q ], that is, uniform AC 0 [ Q ] where the direc t connection language o f the circuit families can b e describ ed by means of the logic FO[ <, L N ]. Hence the Uniformity Duality Pr operty can b e restated in circuit co mplexit y- theoretic terms as follows: Uniformit y Duality Prop ert y for ( Q , L ), 2nd form : A C 0 [ Q ] ∩ L = F O[ <, L N ]-uniform AC 0 [ Q ] ∩ L . By definition, Q [ arb ] ∩ L ⊇ Q [ <, L N ] ∩ L . The critical question is whether the reverse inclusio n holds. Intuitiv ely , the Unifor mit y Duality Prop erty states tha t the “extensio nal uniformity induced by intersecting Q [ arb ] with L ” is a strong enough restr iction imp osed on Q [ arb ] to permit expressing the unifor m c la ss using the L -numerical predicates , or in other words: the extensiona l uniformity given by in tersecting the no n- uniform class with L co incides with the intensional uniformity condition given by fir st-order log ic with L -numerical predica tes. F ur- ther motiv ation for this definition of Q [ <, L N ] ∩ L is as follows: – when constructor s serve to define uniform c lasses, they hav e acc ess to input lengths but no t to the inputs themselves; a co n venient logical analog to this is to us e the unary alphab et la nguages fr om L as a ba sis for defining the extra numerical predicates – if the closure pr operties o f L differ from the closure prop erties of Q [ ar b ], then Q [ arb ] ∩ L = Q [ <, L N ] may fail trivially (this o ccurs for example when L = CFL and Q = {∃} since the non-context-free langua ge { a n b n c n : n ≥ 0 } is 3 easily se en to b elong to Q [ <, L N ] by closure under int ersection o f the la tter ); hence in tersecting Q [ <, L N ] with L b efore co mparing it with Q [ arb ] ∩ L is necessary to obtain a reasona ble gener alization of Straubing ’s conjecture fo r classes L that a re not Bo olean-closed. W e now state o ur results, class ifie d, lo osely , as foundatio na l observ a tions (F) or technical statements (T). W e let L b e any class of languag es. (F) By design, the Uniformity Dualit y Pr operty for ( Q , REG) is precisely Str au- bing’s conjecture (1) , hence its conjectured v alidit y holds the key to the int ernal structure of NC 1 . (F) The Uniformity Duality Pro perty for ( {∃} , NEUTRAL) is precisely the Cr ane Beach Conjecture [4]; here, NEUTRAL is the cla ss of lang uages L that have a neutr a l letter, i.e., a letter e that may b e a rbitrarily inserted into or deleted from words without changing member ship in L . The Cra ne Beach conjecture, stating that any neutra l letter la nguage in AC 0 = FO[ ar b ] can b e express ed in FO[ < ], was motiv ated by attempts to develop a purely automata-theo retic pro of that Parity , a neutral letter languag e, is not in AC 0 . The Cr ane Beach Conjecture was ultimately refuted [4], but several of its v ar ian ts have b een studied. Thus [4 ]: – the Uniformity Duality Pr operty for ( {∃} , NE UTRAL ) fails – the Uniformity Duality Pr operty for ( {∃} , NE UTRAL ∩ REG) holds – the Unifor mit y Duality Prop erty for ( { ∃} , NEUTRAL ∩ { tw o-letter lan- guages } ) holds. (T) O ur definition for the s et L N of L -numerical predicates parallels Straubing’s definition of regula r numerical predicates. F o r kernel-clos ed language clas ses L that are closed under ho momorphisms, inv erse homomorphisms and in- tersection with a regular languag e , we furthermore characterize L N as the set o f predicates express ible as one generalize d unary L - q uan tifier applied to an FO[ < ]-form ula. (In tuitiv ely , L -numerical predicates are thos e pr edicates definable in fir st-order logic with one “or a cle call” to a langua ge from L .) (T) W e characterize the numerical pr edicates that surr ound the co n text-fr ee lan- guages: firs t-order com binations of CFL N suffice to capture all semilinear predicates over N ; in particular, FO[ <, +] = FO[DCFL N ] = FO[BC(CFL) N ], where DCFL denotes the deterministic context-free languag es and BC(CFL) is the Bo olean clo sure of CFL. (T) W e deduce that, despite the fa ct that FO[BC(CFL) N ] contains all the s emi- linear relations, the Uniformity Dualit y Pr operty fails for ( {∃} , L ) in ea ch o f the following ca ses: – L = CFL – L = VPL, the “visibly pushdown languag es” recently introduce d b y [2] – L = Bo olean clo sure of the deterministic c o n text-fr ee language s – L = Bo olean clo sure of the linear co n text-fre e lang ua ges – L = Bo olean clo sure of the context-free languages. The cr ux of the justifications of these nega tiv e results is a pro of that the com- plement of the “Immerma n languag e”, used in disproving the Crane Be a c h Conjecture, is co ntext-free. 4 (T) At the opp osite end of the spec trum, while it is clear that the Uniformity Dualit y Pro p erty holds for the set o f a ll lang ua ges and any Q , we show tha t the Uniformity Dualit y Pro perty alre ady holds for ( Q , L ) whenever Q is a set of gr oupoidal quant ifiers and L = NTIME( n ) L ; th us it holds for, e. g., the rudimentary la nguages, DSP ACE( n ), CSL and PSP ACE . The rest of this pap er is o rganized as follows. Section 2 contains preliminaries. Section 3 defines the L -numerical predica tes and introduces the Uniformity Du- ality P roper ty formally . The context-free n umerical predicates are inv estigated in Section 4, and the duality prop ert y for classes of con text-free langua g es is considered in Section 5. Section 6 shows that the duality pr operty holds when L is “la rge enough”. Section 7 co ncludes with a summary and a discussion. 2 Preliminaries 2.1 Complexity Theory W e ass ume familiarity with standard notions in formal lang uages, auto ma ta and complexity theory . When dealing with cir c uit complexit y classes, all reference s will be made to the non-uniform versions unless otherwise stated. Thus AC 0 refers o f the Bo olean functions computed by co nstan t-depth p olynomial-s ize unbounded-fan- in {∨ , ∧ , ¬} -cir cuits. And DLOGTIME- uniform AC 0 refers to the set o f those functions in AC 0 computable by a circuit family having a direct connectio n lan- guage decida ble in time O (log n ) on a deterministic T ur ing machine (cf. [5 ,24]). 2.2 First-Order Logi c Let N b e the natura l num b ers { 1 , 2 , 3 , . . . } and let N 0 = N ∪ { 0 } . A signatur e σ is a finite s e t o f relatio n symbols with fixe d a rit y and co nstan t s y m b ols. A σ -structure A = hU A , σ A i consists of a se t U A , called universe , a nd a set σ A that con tains a n interpr et atio n R A ⊆ ( U A ) k for eac h k -ary relation s ym b ol R ∈ σ . W e fix the interpretations of the “sta ndard” numerical predicates < , + , × , etc. to their natur al interpretations. B y Bit we will denote the bina r y rela tion { ( x, i ) ∈ N 2 : bit i in the binary repr esen tation of x is 1 } . F or logics ov er str ings with alphab et Σ , we will use signa tur es extending σ Σ = { P a : a ∈ Σ } and ident ify w = w 1 · · · w n ∈ Σ ⋆ with A w = h{ 1 , . . . , n } , σ A w }i , where P A w a = { i ∈ N : w i = a } for all a ∈ Σ . W e will not distinguish b etw een a r elation symbol and its interpretation, when the meaning is cle a r from the context. Let Q b e a set o f (firs t-order) qua n tifier s . W e deno te by Q [ σ ] the set o f first- order fo r m ula e ov er σ using quantifiers from Q only . The set of all Q [ σ ]-formulae will be refer red to as the lo gic Q [ σ ]. In c ase Q = {∃} ( Q = {∃} ∪ Q ′ } ), we will also write FO[ σ ] (FO+ Q ′ [ σ ], resp ectively). When discussing logics over strings, we will o mit the relation symbols from σ Σ . Say that a language L ⊆ Σ ⋆ is definable in a logic Q [ σ ] if there e x ists a Q [ σ ]-formula ϕ s uch that A w | = ϕ ⇐ ⇒ w ∈ L for all w ∈ Σ ⋆ , and say that a 5 relation R ⊆ N n is definable by a Q [ σ ]-formula if there exists a formula ϕ with free v ar iables x 1 , . . . , x n that defines R for all sufficient ly larg e initial s egmen t of N , i. e., if h{ 1 , . . . , m } , σ i | = ϕ ( c 1 , . . . , c n ) ⇐ ⇒ ( c 1 , . . . , c n ) ∈ R fo r all m ≥ c max , where c max = max { c 1 , . . . , c n } [22, Section 3.1]. By abuse of notation, we will write L ∈ Q [ σ ] (o r R ∈ Q [ σ ]) to express tha t a lang uage L (a relation R , r esp.) is definable by a Q [ σ ]-for mula and us e a lo g ic and the set of la nguages a nd rela tions it de fines in terchangeably . 3 The Uniformit y Dualit y Prop ert y In order to generalize conjecture (1), we prop ose Definition 3 .2 as a simple gen- eralization of the r e gular numerical predica tes defined using V -structur e s b y Straubing [23, Section I I I.2]. Definition 3 .1. L et V n = { x 1 , . . . , x n } b e a nonempty set of variables and let Σ b e a finite alphab et. A V n -structure is a se qu enc e w = ( a 1 , V 1 ) · · · ( a m , V m ) ∈ ( Σ × P ( V n )) ⋆ such that a 1 , . . . , a m ∈ Σ and the n onempty sets among V 1 , . . . , V m form a p ar- tition of V n (the undersc or e distinguishes V n -structu r es fr om or dinary st r ings). Define Γ n = { 0 } × P ( V n ) . We say that a V n -structu r e w is unary if w ∈ Γ ⋆ n , i. e., if a 1 · · · a n is define d over the singleton alphab et { 0 } ; in that c ase, we de- fine the k er nel of w , k er n ( w ) , as t he maximal pr efix of w that do es not end with (0 , ∅ ) ; t o signify that x i ∈ V c i for al l 1 ≤ i ≤ n , we also write kern( w ) as [ x 1 = c 1 , . . . , x n = c n ] and we let w N stand for ( c 1 , . . . , c n ) . We define Struc n as the language of al l such wor ds in Γ ⋆ n that ar e unary V n -structu r es and let Struc = S n> 0 Struc n . An y set L of unary V n -structures naturally prescrib es a relatio n ov er the natural num b ers. Hence, a set of such L presc ribes a set of r elations, or numerical predicates, ov er N . Definition 3 .2. L et L ⊆ Γ ⋆ n b e a unary V n -languag e , t ha t is, a set of unary V n -structu r es. L et L N = { w N : w ∈ L } denote the r elation over N n define d by L . Then the L -numerical predicates ar e define d as L N = { L N : L ∈ L and L ⊆ Struc } . We say that a language L is k ernel-closed if, for every w ∈ L , kern( w ) ∈ L . W e further s ay that a language class L is k ernel-closed if, for every L ∈ L t her e exists an L ′ ∈ L su ch that L N = L ′ N and L ′ is kernel-close d. R emark 3.3. A unary V n -languag e L defines a unique numerical relation L N . Conv er sely , if tw o unary V n -structures v and w define the sa me tuple v N = w N , then one of the tw o is o btained from the other by padding on the r ig h t with the letter (0 , ∅ ), i.e., v ∈ w (0 , ∅ ) ⋆ or w ∈ v (0 , ∅ ) ⋆ . Hence a numerical r elation R uniquely determines kern( L ) = { k er n ( w ) : w ∈ L } for a ny lang uage L such that R = L N . 6 W e p oint out the following facts, where w e write ≡ q r for the unar y predicate { x : x ≡ r mo d q } . Prop osition 3.4. L et APE R and NEUTRAL denote the set of ap erio dic lan- guages and the set of languages having a neutr al letter r esp e ctively. Then 1. APER N = FO[ < ] , 2. REG N = (A C 0 ∩ REG) N = FO[ <, {≡ q r : 0 ≤ r < q } ] = reg , and 3. NEUTRAL N ⊆ FO[ < ] . Pr o of. F or par t 1, let L ∈ APER w ith L ⊆ Struc n . Define L ′ = kern( L ) · (0 , ∅ ) ⋆ . W e claim that L ′ is a perio dic. T o see this, note that for a n y lang uage K , any monoid rec o gnizing L a lso r ecognizes LK − 1 = { v ∈ Γ ⋆ n : v u ∈ L for some u ∈ K } [2 0, P ropo sition 2.5]. Hence L [(0 , ∅ ) ⋆ ] − 1 is ap erio dic. But k ern( L ) ⊆ L [(0 , ∅ ) ⋆ ] − 1 . So L ′ equals L [(0 , ∅ ) ⋆ ] − 1 · (0 , ∅ ) ⋆ and is indee d a perio dic. Hence there exists a formula ϕ ∈ FO[ < ] such that L ( ϕ ) = L ′ [19]. Let ψ ( x 1 , . . . , x n ) b e obtained from ϕ by repla cing each P (0 ,V ) ( x ), V ⊆ V n , with V x i ∈ V x = x i ∧ V x i / ∈ V x 6 = x i . Then for all w = [ x 1 = c 1 , . . . , x n = c n ](0 , ∅ ) j and all m ≥ max { c i : 1 ≤ i ≤ n } , A w | = ϕ ⇐ ⇒ h{ 1 , . . . , m } , < A i | = ψ ( c 1 , . . . , c n ) . Let ~ x and ~ c abbreviate x 1 , . . . , x n and c 1 , . . . , c n , resp ectiv ely , and let c max = max { c i : 1 ≤ i ≤ n } . Then ~ c ∈ L N = ⇒ ∃ i : [ x 1 = c 1 , . . . , x n = c n ](0 , ∅ ) i ∈ L = ⇒ [ x 1 = c 1 , . . . , x n = c n ] ∈ kern ( L ) = ⇒ ∀ i : [ x 1 = c 1 , . . . , x n = c n ](0 , ∅ ) i ∈ L ′ = ⇒ ∀ i : A [ x 1 = c 1 ,...,x n = c n ](0 , ∅ ) i | = ϕ = ⇒ ∀ m ≥ c max : h{ 1 , . . . , m } , <, = , ~ c i | = ψ ( ~ x ) and ~ c / ∈ L N = ⇒ ∀ i : [ x 1 = c 1 , . . . , x n = c n ](0 , ∅ ) i / ∈ L = ⇒ [ x 1 = c 1 , . . . , x n = c n ] / ∈ kern( L ) = ⇒ ∀ i : [ x 1 = c 1 , . . . , x n = c n ](0 , ∅ ) i / ∈ L ′ = ⇒ ∀ i : A [ x 1 = c 1 ,...,x n = c n ](0 , ∅ ) i 6| = ϕ = ⇒ ∀ m ≥ c max : h{ 1 , . . . , m } , <, = , ~ c i 6| = ψ ( ~ x ) Hence L N = { ~ c ∈ N n : ( ∀ m ≥ c max )[ h{ 1 , . . . , m } , <, = , ~ c i | = ψ ( ~ x )] } . But ψ ( ~ x ) ∈ FO[ < ] and therefor e L N ∈ FO [ < ]. ˆ A F or the other inclusio n, let R ∈ N n ∩ FO[ < ] via for m ula ψ ( ~ x ). Define ϕ ≡ ∃ x 1 · · · ∃ x n  ψ ( ~ x ) ∧ χ ( ~ x )  , where χ ( ~ x ) ≡ ^ 1 ≤ i ≤ n _ V ∈ P ( V n ) , x i ∈ V  P (0 ,V ) ( x i ) ∧ ∀ z  _ V ′ ∈ P ( V n ) , x i ∈ V ′  P (0 ,V ′ ) ( z ) ↔ z = x i    . 7 The purp ose o f χ ( ~ x ) is to bind the v a riables in ~ x to their resp ectiv e v alues in a corres p onding V n -structure: for a string w = (0 , V 1 ) · · · (0 , V m ) ∈ Γ ⋆ n and a tuple ~ c , A w | = χ ( ~ c ) holds iff m ≥ max { c i : 1 ≤ i ≤ n } , w is a unary V n -structure, and for a ll 1 ≤ i ≤ n , x i ∈ V c i . Alike the ab ov e equiv alence, we obtain ~ c ∈ R ⇐ ⇒ ∀ m ≥ c max : h{ 1 , . . . , m } , <, = , ~ c i | = ψ ( ~ x ) ⇐ ⇒ ∀ i : A [ x 1 = c 1 ,...,x n = c n ](0 , ∅ ) i | = ϕ. Let L b e the unary V n -languag e { w ∈ Struc n : A w | = ϕ } . Then R = L N =  L ( ϕ )  N with ϕ ∈ FO[ < ]. Thu s M L is finite and a perio dic and L N ∈ AP ER N . Part 2 follows a nalogously from [2 3, Theorems I II.1.1 a nd I II.2.1 ]. F or Part 3, let R ∈ NEUTRAL N . Then R = L N for so me neutral letter language L ⊆ Struc n . Assume that the neutra l letter of L is (0 , ∅ ), other w is e L ⊆ Struc n implies that L is empty . Since we ca n insert or delete (0 , ∅ ) in any word at will, L is fully deter mined b y the (0 , ∅ )-free words it contains, that is, L = [ ( V 1 ,V 2 ,...,V k ) parti tions V n and (0 ,V 1 )(0 ,V 2 ) ··· (0 ,V k ) ∈ L (0 , ∅ ) ⋆ (0 , V 1 )(0 , ∅ ) ⋆ (0 , V 2 ) · · · (0 , ∅ ) ⋆ (0 , V k )(0 , ∅ ) ⋆ . This is a finite union of regula r ap erio dic languages . Hence L is regula r ap erio dic, and L N is in FO[ < ] by Part 1. Having discussed the L - numerical predicates, we can sta te the prop ert y ex- pressing the dual facets of uniformity , namely , intersecting with a n a priori uni- form cla ss o n the one hand, and a dding the corr e s ponding numerical predicates to first-or der logics on the other. Prop ert y 3.5 (Uniformit y Duality for ( Q , L )). L et Q b e a set of quantifiers and let L b e a language class, then Q [ arb ] ∩ L = Q [ <, L N ] ∩ L . As Q [ arb ] = AC 0 [ Q ] [5] and Q [ <, L N ] = FO[ <, L N ]-uniform AC 0 [ Q ] [6], the ab o v e prop erty eq uiv alently states that A C 0 [ Q ] ∩ L = FO[ <, L N ]-uniform AC 0 [ Q ] ∩ L . As a co nsequence of P ropo sition 3.4 (1– 2), the Uniformit y Duality Pro perty is eq uiv alent to the instances of the Str aubing conjectures obtained by setting Q and L as w e exp ect, for example Q ⊆ {∃} ∪ {∃ ( q,r ) : 0 ≤ r < q } a nd L = RE G yield exactly (1). Similarly , as a consequence of Prop osition 3.4 (3), the Unifor- mit y Duality P r operty is eq uiv alent to the Cr ane Beach Conjecture if FO[ < ] ⊆ L . Prop erty 3.5 is thus false when Q = { ∃} a nd L is the set NEUTRAL o f all neu- tral letter language s . F or so me other c la sses, the Crane Beach Conjecture and th us P roper t y 3.5 hold: consider for example the case L = REG ∩ NEUTRAL [4], or the cas e Q = {∃} and L ⊆ NEUTRAL ∩ FO[+]. Accordingly the Unifor- mit y Dualit y Pro perty b oth genera lizes the co njectures of Straubing e t al. and captures the intuition underlying the Crane Beach Conjecture. Enc o uraged by this unification, we will take a closer lo ok at the Uniformity Dua lity in the case of first-or der logic and context-free langua ges in the next section. 8 In the r est of this section, we present an alternative characterization of L N using F O[ < ]-transforma tio ns and unary Lindstr¨ om quantifiers. This is further justification for our definitio n of L -numerical predica tes. The reader unfamiliar with this topic may skip to the end of Section 3. Digressio n: Nume rical Predicates and Generalized Quan tifiers Generalized or Lindstr ¨ om qua ntifiers provide a very general yet c oheren t a p- proach to extending the descriptive complexity of first-o rder log ics [17]. Since we only deal with unary Lindstr¨ om qua ntifiers ov er str ings, we will restric t o ur definition to this ca se. Definition 3 .6. L et ∆ = { a 1 , . . . , a t } b e an alp hab et, ϕ 1 , . . . , ϕ t − 1 b e FO [ < ] - formulae, e ach with k + 1 fr e e varia bles x 1 , x 2 , . . . , x k , y , and let ~ x abbr eviate x 1 , x 2 , . . . , x k . F urther, let Struct ( σ ) denote the set of finite structur es A = hU A , σ A i over σ . Then ϕ 1 , . . . , ϕ t − 1 define an F O[ < ]-transformatio n [ ϕ 1 ( ~ x ) , . . . , ϕ t − 1 ( ~ x )] : str uct ( { <, x 1 , . . . , x k } ) → ∆ ⋆ as fol lows: L et A ∈ struct ( { <, x 1 , . . . , x k } ) , x A i = c i ∈ U A , 1 ≤ i ≤ k , and s = |U A | , t hen [ ϕ 1 ( ~ x ) , . . . , ϕ t − 1 ( ~ x )]( A ) = v 1 · · · v s ∈ ∆ ⋆ , wher e v i =      a 1 , if A | = ϕ 1 ( c 1 , . . . , c k , i ) , a j , if A | = ϕ j ( c 1 , . . . , c k , i ) ∧ V j − 1 l =1 ¬ ϕ l ( c 1 , . . . , c k , i ) , 1 < j < t, a t , if A | = V t − 1 l =1 ¬ ϕ l ( c 1 , . . . , c k , i ) . A language L ⊆ ∆ ⋆ and an FO[ < ] -tr ansformation [ ϕ 1 ( ~ x ) , . . . , ϕ t − 1 ( ~ x )] now nat- ur al ly define a (unar y) Lindstr¨ om qua ntifier Q un L via A | = Q un L y [ ϕ 1 ( ~ x, y ) , . . . , ϕ t − 1 ( ~ x, y )] ⇐ ⇒ [ ϕ 1 ( ~ x ) , . . . , ϕ t − 1 ( ~ x )]( A ) ∈ L. Final ly, t he set of r elations definable by formulae Q un L y [ ϕ 1 ( ~ x, y ) , . . . , ϕ t − 1 ( ~ x, y )] , wher e L ∈ L and ϕ 1 , . . . , ϕ t − 1 ∈ FO [ < ] , wil l b e denote d by Q un L F O[ < ] . The notation [ ϕ 1 ( ~ x ) , . . . , ϕ t − 1 ( ~ x )] is chosen to distinguish the v aria bles in ~ x from y ; the v a riables in ~ x ar e interpreted by A whereas y is utilized in the transformatio n. Theorem 3.7 . L et L b e a kernel-close d language class which is close d under ho- momorphisms, inverse homomorphisms and interse ction with r e gular languages, then L N = Q un L F O[ < ] ; that is, t he L -n umeric al pr e dic ates c orr esp ond to the pr e dic ates definable using a u nary Lindstr ¨ om quantifier over L and an FO[ < ] - tr ansformation. Pr o of. F or the inclus ion fr om left to right, consider a rela tion L N ⊆ N n in L N and let V n = { x 1 , . . . , x n } . Define the F O[ < ]-transformatio n [ ϕ 1 ( ~ x ) , . . . , ϕ 2 n − 1 ( ~ x )] from N n to unary V -structures a s follows. F or 1 ≤ i < 2 n , let ϕ i ≡ ^ j ∈ V i ( y = x j ) ∧ ^ j / ∈ V i ( y 6 = x j ) , 9 where V i denotes the i th subset of V in the natural subset ordering , and as- so ciate the letter (0 , V i ) with ϕ i . L e t ~ c = ( c 1 , . . . c n ) ∈ N n and denote by A the structur e ( { 1 , . . . , l } , σ ) with l = max i { c i } and { <, x 1 , . . . , x n } ⊆ σ . Then [ ϕ 1 ( ~ x ) , . . . , ϕ 2 n − 1 ( ~ x )] ma ps A to the una ry V n -structure v 1 · · · v l . Then ~ c ∈ L N = ⇒ ∃ j ∃ w = [ x 1 = c 1 , . . . , x n = c n ] : w (0 , ∅ ) j ∈ L = ⇒ [ x 1 = c 1 , . . . , x n = c n ] ∈ L (since L is kernel-closed) = ⇒ [ ϕ 1 ( ~ x ) , . . . , ϕ 2 n − 1 ( ~ x )]( A ) ∈ L = ⇒ A | = Q un L y [ ϕ 1 ( ~ x, y ) , . . . , ϕ 2 n − 1 ( ~ x, y )] , and the re v er se implica tions hold for any unary V n -languag e L . F or the opp osite inclusion, let R ⊆ N n ∩ Q un L F O[ < ] via the transformatio n [ ϕ 1 ( ~ x ) , . . . , ϕ k − 1 ( ~ x )] and the langua ge L ⊆ ∆ ⋆ ∩ L , | ∆ | = k . W e must ex hibit a unary V n -languag e A ∈ L such that R = A N , i. e., R = h (kern( A )), wher e h : { kern( w ) : w is a unar y V n -structure } → N n [ x 1 = c 1 , . . . , x n = c n ] 7→ ( c 1 , . . . , c n ) . Our pro of is similar to the pro of of Niv at’s Theor em, see [1 6, Theorem 2.4]. Define B ⊆  Γ n × ∆  ⋆ to consist of all words  u 1 v 1  · · ·  u l v l  such that u = u 1 · · · u l is the kernel of a unary V n -structure a nd [ ϕ 1 ( ~ x ) , . . . , ϕ k − 1 ( ~ x )] maps u N to v = v 1 · · · v l . Define the le ngth-preserving ho momorphisms f :  u v  7→ u and g :  u v  7→ v . Then R = ( h ◦ k ern ◦ f )  B ∩ g − 1 ( L )  . W e cla im that B is re g ular. Theor em 3.7 then follows from the clo sure prop- erties of L by s etting A = f  B ∩ g − 1 ( L )  F or 1 ≤ i ≤ k − 1, let ϕ ′ i ( y ) b e defined as ϕ ′ i ( y ) ≡ ∃ x 1 · · · ∃ x n ( ϕ i ( ~ x, y ) ∧ ψ ( ~ x ) ∧ π ( ~ x )), where ψ ( ~ x ) a nd π ( ~ x ) bind the v aria bles in ~ x to their resp ective v alues in a corres p onding unary V n -structure and a sserts that each v ar ia ble o ccurs exa c tly once; tha t is , ψ ( ~ x ) ≡ ^ 1 ≤ i ≤ n _ d ∈ ∆ _ V ∈ P ( V n ) , x i ∈ V P ( (0 ,V ) d ) ( x i ) , π ( ~ x ) ≡ ^ 1 ≤ i ≤ n ∀ z   _ d ∈ ∆ _ V ∈ P ( V n ) , x i ∈ V P ( (0 ,V ) d ) ( z )  ↔ z = x i  . Now let χ j ( z ) ≡ W V ∈ P ( V n ) P ( (0 ,V ) d j ) ( z ) , 1 ≤ j ≤ k , where d j is the j th letter in ∆ . Then a s tring  u 1 v 1  · · ·  u l v l  ∈  Γ n × ∆  ⋆ is in B if a nd only if u = u 1 · · · u l is the kernel of a unar y V n -structure and ∀ z  k − 1 ^ i =1   ϕ ′ i ( z ) ∧ i − 1 ^ l =1 ¬ ϕ ′ l ( z )  ↔ χ i ( z )  ∧  k − 1 ^ l =1 ¬ ϕ ′ l ( z ) ↔ χ k ( z )   holds on ~ z = u N , where the empty conjunction is defined to b e true. Concluding, B ∈ FO[ < ] ⊂ RE G . W e stress that the a b ov e result provides a logica l characterization of the L - nu merical pr edicates for all kernel-closed classes L forming a co ne , viz. a cla s s 10 of language s L clo sed under homomorphisms, in verse ho momorphisms a nd in- tersection with r egular languag e s [10]. As the closure under these o perations is equiv alent to the closur e under ra tional trans ductions (i. e., transductions p er- formed by finite automata [7]), we obta in: Corollary 3 .8. L et L b e kernel-close d and close d under r ational tr ansduct io ns, then L N = Q un L F O[ < ] . 4 Characterizing the Con text-F ree N um erical Predicates In order to examine whether the Uniformity Duality Prop ert y for first-order logics holds in the case o f context-free languag es, we firs t need to consider the counterpart of the regula r n umerical pre dicates, that is, CFL N . Our r esults in this sectio n will relate CFL N to addition w. r. t. to first-or der combinations, a nd are ba sed upon a result b y Ginsburg [9]. Ginsburg show ed that the n umber of rep etitions per fragment in bounded context-free languages co rresp onds to a subset of the s emilinear sets. F or a star t, no te that addition is definable in DCFL N . Lemma 4 .1. A ddition is definable in DCFL N . Pr o of. Let V 3 = { x 1 , x 2 , x 3 } and L + be a unary V n -languag e defining addition, that is, L N + = { ( x 1 , x 2 , x 3 ) : x 1 + x 2 = x 3 } . Then L + is recog niz e d by the following deterministic PDA P = ( Q, Γ n , ∆, δ, q 0 , ⊥ , { q acc } ), wher e Q = { q 0 , q acc } , ∆ = {⊥ , 0 } and δ is defined as follows: δ ( z 0 , (0 , ∅ ) , γ ) = ( z 0 , 0 γ ) δ ( z y , (0 , ∅ ) , γ ) = ( z y , γ ) δ ( z 0 , (0 , { x } ) , γ ) = ( z x , 0 γ ) δ ( z y , (0 , { x } ) , γ ) = ( z xy , γ ) δ ( z 0 , (0 , { y } ) , γ ) = ( z y , 0 γ ) δ ( z xy , (0 , ∅ ) , 0) = ( z xy , ε ) δ ( z x , (0 , ∅ ) , γ ) = ( z x , 0 γ ) δ ( z xy , (0 , { z } ) , 0) = ( z z , ε ) δ ( z x , (0 , { y } ) , γ ) = ( z xy , γ ) δ ( z z , ε , ⊥ ) = ( z acc , ⊥ ) where γ ∈ ∆ . Next, we restate the res ult of Ginsbur g in order to prepar e ground for the examination of the con text-free numerical pr edicates. In the following, let w ⋆ abbreviate { w } ⋆ and say that a la nguage L ⊆ Σ ⋆ is b oun de d if there exists a n n ∈ N a nd w 1 , . . . , w n ∈ Σ + such that L ⊆ w ⋆ 1 · · · w ⋆ n . Definition 4 .2. A set R ⊆ N n 0 is str atified if 1. e ach element in R has at m ost t wo non-zero c o or dinates, 2. ther e ar e no inte gers i , j, k , l and x = ( x 1 , . . . , x n ) , x ′ = ( x ′ 1 , . . . , x ′ n ) in R such that 1 ≤ i < j < k < l ≤ n and x i x ′ j x k x ′ l 6 = 0 . Mor e over, a set S ⊆ N n is said to b e stratified semilinear if it is expr essible as a finite union of line ar sets, e ach with a str atifie d set of p erio ds; that is, S = S m i =1 { ~ α i 0 + P n i j =1 k · ~ α ij : k ∈ N 0 } , wher e ~ α i 0 ∈ N n , ~ α ij ∈ N n 0 , 1 ≤ j ≤ n i , 1 ≤ i ≤ m , and e ach P i = { ~ α ij : 1 ≤ j ≤ n i } is str atifie d. 11 Theorem 4.3 ([9, Theorem 5. 4.2]). L et Σ b e an alphab et and L ⊆ w ⋆ 1 · · · w ⋆ n b e b ounde d by w 1 , . . . , w n ∈ Σ + . Then L is c ontext - fr e e if and only if the set E ( L ) =  ( e 1 , . . . , e n ) ∈ N n 0 : w e 1 1 . . . w e n n ∈ L  is a str atifie d semiline ar set. Theorem 4.3 relates the b o unded context-free langua ges to a stric t subset of the semilinear s e ts. The semilinear sets a re exa ctly tho se s ets defina ble by FO [+]- formulae. Ther e a re how e ver sets in FO [+] that are undefinable in CFL N : e. g., if R = { ( x, 2 x, 3 x ) : x ∈ N } was definable in CFL N then { a n b n c n : n ∈ N } ∈ CFL. Hence, FO [+] ca n not b e captured by CFL N alone. Y et, addition is definable in CFL N , ther efore we will in the following inv e s tigate the relationship betw een first- order lo gic with addition, FO[+], and the Bo olean closur e of CFL, BC(CFL) . Lemma 4 .4. L et R ⊆ N n and let R = L N for language L . Then L is b oun de d. Pr o of. Let R ⊆ N n and L suc h that R = L N . Let L ⊆ Γ ⋆ n , where V n = { x 1 , . . . , x n } is a se t of v ar iables. E v ery unary V n -structure w ∈ L of a given order type t ∈ T belongs to the langua ge L t = (0 , ∅ ) ⋆ (0 , V 1 ) (0 , ∅ ) ⋆ (0 , V 2 ) · · · (0 , V k ) (0 , ∅ ) ⋆ , where ( V 1 , V 2 , . . . , V k ) is an ordered pa r tition of V n . The n um ber p o f s uc h or dered partitions dep ends on n ; in particula r, p is finite. Hence L ( L ⋆ 1 L ⋆ 2 · · · L ⋆ t p , wher e t 1 , t 2 , . . . , t p exhaust the p ossible t ypes . Since each letter (0 , V i ) b elongs to the language Q V ⊆V n (0 , V ) ⋆ , it follows that L ⊆ Q 1 ≤ i ≤ p  Q V ⊆V n (0 , V ) ⋆  2 n +1 . Lemma 4 .5. Al l b ounde d c ontext- fr e e languages ar e definable in FO[+] . Pr o of. Since L is bounded, there exis t w 1 , . . . , w n ∈ Σ + such that L ⊆ w ⋆ 1 . . . w ⋆ n . By Theo rem 4.3, it holds that the set E ( L ) = { ( e 1 , . . . , e n ) : w e 1 1 . . . w e n n ∈ L } is str atified semilinear. It follows by semilinear it y alone that, for ~ e = ( e 1 , . . . , e n ), E ( L ) = S m i =1 { ~ e : ~ e = ~ α i 0 + P n i j =1 k · ~ α ij , k ∈ N 0 } , where ~ α i 0 ∈ N n , ~ α ij ∈ N n 0 for all 1 ≤ j ≤ n i , 1 ≤ i ≤ m . As a consequence, L is defined via the formula ϕ = ∃ e 1 · · · ∃ e n  ϕ w 1 ,...,w n ( e 1 , . . . , e n ) ∧ ϕ lin ( e 1 , . . . , e n )  , where ϕ w 1 ,...,w n ( e 1 , . . . , e n ) chec ks w he ther the input is of the form w e 1 1 · · · w e n n and ϕ lin ( e 1 , . . . , e n ) ≡ m _ i =1  ∃ a i, 1 · · · ∃ a i,n i n ^ k =1  e k = α i 0 k + n i X j =1 a ij α ij k   . In particular, ϕ w e 1 1 ...w e n n ∈ FO[+]. Concluding, L ∈ F O[+]. Lemma 4 .6. L et V n = { x 1 , . . . , x n } and let L b e a u nary V n -language. Then L ∈ F O[+] implies L N ∈ FO[+] . 12 Pr o of. Since L ∈ F O[+], there exists a formula ϕ ∈ FO [+] suc h that for all w = (0 , V 1 ) · · · (0 , V m ) ∈ Γ ⋆ n , h{ 1 , . . . , m } , <, + , P i | = ϕ ⇐ ⇒ w ∈ L , where P = { P (0 ,V ) : V ∈ P ( V n ) } and P (0 ,V ) ( z ) is true if and only if z ∈ V , for 1 ≤ z ≤ m . Let ~ y = ( y 1 , . . . , y n ). W e construct the formula ϕ ′ ( ~ y ) from ϕ b y replacing, for ea c h v ariable z and each V ∈ P ( V n ), the pr edicate P (0 ,V ) ( z ) with V y i ∈ V z = y i ∧ V y i / ∈ V z 6 = y i . Then h{ 1 , . . . , m } , <, + i | = ϕ ′ ( ~ y ) ⇐ ⇒ h{ 1 , . . . , m } , < , + , P i | = ϕ. Hence the formula ϕ ′ witnesses L N ∈ FO[+]. Theorem 4.7 . BC(CFL N ) ⊆ BC(CFL) N ⊆ FO [+] . Pr o of. The inclusio n B C (CFL N ) ⊆ BC(CFL) N follows from (1.) the fact that for every unary V n -languag e L ∈ CFL, there exists an equiv alent kernel-closed una ry V n -languag e L ′ ∈ C FL ; (2.) L N 1 ∩ L N 2 = ( L 1 ∩ L 2 ) N and L N 1 ∪ L N 2 = ( L 1 ∪ L 2 ) N for kernel-closed la nguages L 1 , L 2 ; and (3.) the obser v atio n that for a kernel-close d L ∈ CFL with R = L N , the language L ′ = L (0 , ∅ ) ⋆ ∈ CFL also verifies R = L ′ N , so tha t R = ( L ′ ∩ Struc n ) N ∈ BC(CFL) N . It re ma ins to show that BC(CFL) N ⊆ FO[+]. Denote by coCFL set o f lan- guages who se complemen t is in CFL, i. e ., coCFL = { L : L ∈ CFL } . Let R ⊆ BC(CFL) N ∩ N n and V n = { x 1 , . . . , x n } . F urther, let L be so me unar y V n -languag e suc h that R = L N ; w. l. o . g. L = S m i =1 T k i j =1 L ij where L ij ∈ CFL ∪ coCFL. It holds that L is a unary V n -languag e if a nd only if T k i j =1 L ij is a unary V n -languag e for all 1 ≤ i ≤ m . Hence, we need to show that L N i is FO[+]- definable, for any unary V n -languag e L i = T k i j =1 L ij with L ij ∈ C FL ∪ coCFL. Note that Struc n is b ounded and definable in F O[ < ] ⊂ RE G. Then L i = ( L i 1 ∩ Struc n ) ∩ · · · ∩ ( L ik ∩ Struc n ) and each ( L ij ∩ Struc n ), 1 ≤ j ≤ k , is bo unded (Lemma 4.4 ). W e have to distinguish the following tw o ca ses: Case 1: L ij ∈ CFL. Then L ij ∩ Struc n ∈ CFL. Thus Lemma 4 .5 implies L j ∩ Struc n ∈ FO [+]. Case 2: L ij ∈ coCFL. As L ij ∩ Struc n is b ounded, it can be written a s L ij ∩ Struc n = { w e 1 1 · · · w e n n : ( e 1 , . . . , e n ) ∈ X } for words w 1 , . . . , w n and some relation X ⊆ N n 0 . Hence, L ij ∩ Struc n = L ij ∪ Struc n = ( L ij ∩ Struc n ) ∪ Struc n where L ij ∩ Struc n is the intersection of a co ntext-free langua g e with a reg- ular languag e and therefor e co n text-fr ee. F urther note that L ij ∩ Struc n = { w e 1 1 · · · w e n n ∈ Struc n : ( e 1 , . . . , e n ) / ∈ X } is bounded. F rom Lemma 4 .5 it now follows that L ij ∩ Struc n ∈ FO[+]. Thus, finally , L ij ∩ Struc n = ( L ij ∩ Struc n ) ∪ Struc n ∈ FO[+]. Summarizing, L i = ( L i 1 ∩ Struc n ) ∩ · · · ∩ ( L ik ∩ Struc n ) is definable in F O[+] using the conjunction of the defining formulae. Since L i is a unary V n -languag e by a ssumption, Lemma 4.6 implies the claim. 13 That is, the r elations defina ble in the Bo olean c lo sure of the context-free unary V n -languag e s ar e captured by FO[+]. Hence, F O[BC(CFL) N ] ⊆ F O[+]. Now Le mma 4.1 yields the following cor ollary . Corollary 4 .8. FO[DCFL N ] = F O[CFL N ] = F O[BC(CFL) N ] = F O[+] . W e no te that in particular , for any k ∈ N , the inclusion ( T k CFL) N ( FO [+] holds, wher e T k CFL denotes the languages definable as the intersection of ≤ k context-free languages : this is deduced from em b edding numerical pr e dicates derived from the infinite hierar c hy of context-free langua ges b y Liu and W einer int o CFL N [18]. Hence, CFL N ( · · · ( ( T k CFL) N ( ( T k +1 CFL) N ( · · · ( ( T CFL) N ⊆ FO[+] . Unfortunately , we could neither prov e nor refute FO [+] ⊆ BC(CFL) N . The difficult y in co mpa ring FO[+ ] and B C (CFL) N comes to so me extent fr o m the restriction on the syntactic representation of tuples in CFL; viz., c on text- free languages may only compare distances b etw een v ar iables, whereas the tuples defined by unary V n -languag e s count p ositions from the b eginning o f a word. This difference matters only for language classes that a r e sub ject to similar restrictions as the context-free language s (e. g., the regular languages are not capable of counting, the context-sensitiv e language s have the ability to co n vert betw een these tw o representations). T o account fo r this sp ecial b eha vior, we will render precisely CFL N in Theorem 4.9. But there is more to b e taken into acco un t. Consider, e . g., the relatio n R = { ( x, x, x ) : x ∈ N } . R is clea r ly definable in CFL N , yet the set E ( L ) of the defining languag e L , L N = R , is not stratified semilinea r. Sp ecifically , duplicate v ariables and p ermutations of the v ariables do not increas e the complexity of a unary V n -languag e L but affect L N . Let t b e an order type of ~ x = ( x 1 , . . . , x n ) and say that a r elation R ⊆ N n has o rder type t if, for all ~ x ∈ R , ~ x has order type t . F o r ~ x of order type t , let ~ x ′ = ( x ′ 1 , . . . , x ′ m ), m ≤ n , denote the v a r iables in ~ x with mutually distinct v alues a nd let π t denote a per m utatio n such tha t x ′ π t ( i ) < x ′ π t ( i +1) , 1 ≤ i < m . W e define functions sort : P ( N n ) → P ( N m ) a nd diff : P ( N n ) → P ( N n 0 ) a s sort ( R ) =  π t ( ~ x ′ ) : ~ x ∈ R ha s or der type t  , diff ( R ) = n ( x i ) 1 ≤ i ≤ n :  i X j =1 x j  1 ≤ i ≤ n ∈ R o . The function sort rearr anges the co mponents of R in an ascending order and eliminates duplicates, whereas diff transforms a tuple ( x 1 , . . . , x n ) with x 1 < x 2 < · · · < x n int o ( x 1 , x 2 − x 1 , x 3 − x 2 − x 1 , . . . , x n − P n − 1 i =1 x i ), a represe ntation more “suitable” to CFL (cf. E ( L ) in Theor em 4.3). Theorem 4.9 . L et R ⊆ N n . R ∈ CFL N if and only if ther e exists a p artition R = R 1 ∪ · · · ∪ R k such t hat e ach diff  sort ( R i )  , 1 ≤ i ≤ k , is a str atifie d semiline ar set. 14 Pr o of. F or the direc tio n from left to r ig h t, let L N ∈ CFL N , L N ⊆ N n and let t 1 , . . . , t p exhaust the po ssible o rder types of ~ x = ( x 1 , . . . , x n ). As CFL is c lo sed under int ersection with regular languages , L ca n be partitioned in to context-free languages L = L t 1 ∪ · · · ∪ L t p such that L N t i has o rder type t i , 1 ≤ i ≤ p . Fix any L t i . By desig n of sort , it holds that x 1 < x 2 < · · · < x m for all ( x 1 , . . . , x m ) ∈ sort ( L N t i ), hence diff  sort ( L N t i )  ∈ N m 0 is defined. Since ~ x has order type t i for all ~ x ∈ L t i , sort ( L N t i ) = ( ϕ ( L t i )) N for a homomorphism ϕ substituting the characters (0 , V ), V 6 = ∅ , with appropr iate (0 , V ′ ). W e thus obtain that sort ( L N t i ) ∈ CFL N . Say A N = sort ( L N t i ) for the context-free unar y V n - language A , then Theo rem 4.3 implies that the set E ( A ) is a stratified s emilinear set. Consider the finite sta te transducer T in Figure 1 . T defines the rational (0 , { x 1 } ) / a 1 (0 , { x 2 } ) / a 2 (0 , { x m − 1 } ) / a m − 1 (0 , { x m } ) / a m (0 , ∅ ) / a 1 (0 , ∅ ) / a 2 (0 , ∅ ) / a m (0 , ∅ ) / e Figure 1. T ransducer T transduction ψ : Γ ⋆ n → { a 1 , . . . , a m , e } ⋆ , ψ ( w ) = v 1 · · · v s , wher e w = w 1 · · · w s , | w | = s , is the given unary V n -structure and, for 1 ≤ i ≤ s , 1 ≤ j ≤ m , v i = ( a j , if w i · · · w s = (0 , ∅ ) l (0 , { x j } ) u for s ome l ∈ N 0 , u ∈ Γ ⋆ n , e , if w 1 · · · w i (0 , ∅ ) l = w for s ome l ∈ N 0 . W e cla im tha t diff  sort ( L N t i )  = E  ψ ( A )  . The claim co ncludes the direction from left to rig h t, since ψ ( A ) ∈ CFL due to the closure of CFL under r ational transductions. Claim. Let A ∈ CFL such tha t A N = sort ( L N t i ), then diff  sort ( L N t i )  = E  ψ ( A )  . T o prov e the claim, let V n = { x 1 , . . . , x m } and let A ∈ CFL b e a una ry V n -languag e such that A N = sort ( L N t i ). Fix an arbitrar y ~ c = ( c 1 , . . . , c m ) ∈ A N and cho o se w ∈ A s uch that w N = ~ c . Then c 1 < c 2 < · · · < c m and w = (0 , ∅ ) c 1 − 1 (0 , { x 1 } )(0 , ∅ ) c 2 − c 1 − 1 (0 , { x 2 } ) · · · (0 , ∅ ) c m − P m − 1 i =1 c i − 1 (0 , { x m } )(0 , ∅ ) d , where d = | w | − c m . Hence ψ ( w ) = a c 1 1 a c 2 − c 1 2 · · · a c m − P m − 1 i =1 c i m e d and E  ψ ( { w } )  = n c 1 , c 2 − c 1 , . . . , c m − m − 1 X i =1 c i o . On the other hand, diff ( { ~ c } ) = { ( c 1 , c 2 − c 1 , . . . , c m − P m − 1 i =1 c i ) } . Thus, for every ~ c ∈ sort ( L N t i ), E  ψ ( { ~ c } )  = diff ( { ~ c } ) and di ff  sort ( L N t i )  = E  ψ ( A )  . This implies the cla im and concludes the direction from left to right. 15 F or the direc tio n from rig h t to left, it suffices to show tha t R i ∈ CFL N for each 1 ≤ i ≤ k . By assumption, diff  sort ( R i )  ⊆ N m 0 is a stratified semilin- ear set. Thence there exists a b ounded la nguage A ∈ CFL such that E ( A ) = diff  sort ( R i )  . L et A w. l. o. g. b e bo unded by a 1 , . . . , a m ∈ Σ , i. e., A ⊆ a ⋆ 1 · · · a ⋆ m . Define the ratio nal transductio n χ : { a 1 , . . . , a m } ⋆ → Γ ⋆ n as χ ( w ) = v 1 · · · v s , where w = w 1 · · · w s , | w | = s , a nd, for 1 ≤ i ≤ s , 1 ≤ j ≤ m , v i = ( (0 , { x j } ) , if i = s and w i = a j or i < s and w i = a j 6 = w i +1 , (0 , ∅ ) if i < s and w i = w i +1 . Note that ( ψ ◦ χ )( w ) = w for all w ∈ A . An arg umen t analo gous to the ab ov e claim thus yields E ( A ) = E  ψ ( χ ( A ))  = diff  ( χ ( A )) N  and sort ( R i ) = ( χ ( A )) N . In particular, ( χ ( A )) N ∈ C FL N . Moreov er, c 1 < c 2 < · · · < c m for a ll ~ c = ( c 1 , . . . , c m ) ∈ ( χ ( A )) N , thus there exists a function π : { 1 , . . . , n } → { 1 , . . . , m } , n ≥ m , such tha t  ( x π (1) , . . . , x π ( n ) ) : ( x 1 , . . . , x m ) ∈ ( χ ( A )) N  = R i . Let the homomo rphism φ : Γ ⋆ n → Γ ⋆ n mimic the ab ov e tr ansformation by re plac- ing (0 , { x i } ) with (0 , V i ), where V i = { x j : 1 ≤ j ≤ n , π ( j ) = i } , 1 ≤ i ≤ m . Then R i =  ( φ ◦ χ )( A )  N ∈ C FL N . 5 The Uniformit y Dualit y and Context-F ree Languages Due to the previous s ection, w e may ex press the Uniformity Dualit y Prop erty for context-free languag es using Corolla r y 4 .8 in the following mor e intuitiv e way: let Q = {∃} and L b e such that FO[ L N ] = F O[ <, +] (e. g., DCFL ⊆ L ⊆ BC(CFL)), then the Unifor mit y Duality Pro perty for ( {∃} , L ) is equiv a len t to F O[ arb ] ∩ L = FO[ <, + ] ∩ L . (2) W e will hence examine whether (2) holds, and s ee that this is not the case. F or a binar y word u = u n − 1 u n − 2 · · · u 0 ∈ { 0 , 1 } ⋆ , we write b u for the in teger u n − 1 2 n − 1 + · · · + 2 u 1 + u 0 . Recall the Immerman la nguage L I ⊆ { 0 , 1 , a } ⋆ , that is, the languag e co ns isting of all words o f the form x 1 a x 2 a · · · a x 2 n , where x i ∈ { 0 , 1 } n , b x i + 1 = b x i +1 , 1 ≤ i < 2 n , and x 1 = 0 n , x 2 n = 1 n . F or e x am- ple, 00 a 01a10a1 1 ∈ L I and 000a 001a010 a 011a100a101a110a111 ∈ L I . W e prov e that despite its definition inv olving arithmetic, L I is simply the complement o f a co n text-fr e e language. Lemma 5 .1. The c omplement L I of the Immerman language is c ontext- fr e e. Pr o of. Let Σ = { 0 , 1 , a } . Thr oughout this pr oof, u a nd v stand for binary words. Claim. Let u = u n − 1 u n − 2 · · · u 0 , v = v n − 1 v n − 2 · · · v 0 and u 0 6 = v 0 . Then b u + 1 = b v (mo d 2 n ) (3) iff no ne of the words u 1 u 0 v 1 v 0 , u 2 u 1 v 2 v 1 , . . . , u n − 1 u n − 2 v n − 1 v n − 2 belo ngs to { 0010 , 0011 , 0100 , 0111 , 1001 , 1000 , 1110 , 1101 } . (4) 16 The claim implies that the fo llowing lang ua ge is context-free: A = { xu a v y : x ∈ Σ ⋆ , y ∈ Σ ⋆ , | u | = | v | , b u + 1 6 = b v (mo d 2 | u | ) } . Note that { xu a vy : x ∈ ( Σ ⋆ a) ⋆ , y ∈ (a Σ ⋆ ) ⋆ , | u | = | v | , b u + 1 6 = b v (mo d 2 | u | ) } ⊂ A ⊂ L I . Hence A do es not catc h all the w ords in L I , but it catches a ll the words of the corre c t “form” which vio la te the successor condition (mo dulo 2 | u | ). F or example, A catches 00 a01a11a 11a, but it do es not catc h a nor 0a1a0a1 nor 01 a10a11a 00 no r 00a 01a10a11 a00a01 nor 00a 01001a1 0a11. W e complete the pro of by expressing L I as fo llo ws : L I = A ∪ a ⋆ ∪ Σ ⋆ a0 ⋆ a Σ ⋆ ∪ Σ ⋆ a1 ⋆ a Σ ⋆ ∪ { 0 , 1 } ⋆ 1 Σ ⋆ ∪ Σ ⋆ 0 { 0 , 1 } ⋆ ∪ S | u |6 = | v | ( Σ ⋆ a) ⋆ u a v (a Σ ⋆ ) ⋆ . (5) Now we prov e the claim. The direction from left to r igh t is ea s y: if any word u i u i − 1 v i v i − 1 belo ngs to the forbidden set (4) then (3) trivially fails. F or the conv er se, we must pr o ve that if no forbidden word o ccurs then (3) holds. W e prov e this by induction on the common length n of the words u and v . When n = 1, u 0 and v 0 6 = u 0 are successo r s mo dulo 2. So supp ose that n ≥ 1 and that none o f the forbidden words o ccurs in the pa ir ( u n u, v n v ), wher e u n , v n ∈ { 0 , 1 } , u = u n − 1 · · · u 1 u 0 and v = v n − 1 · · · v 1 v 0 and u 0 6 = v 0 . Then by induction, b u + 1 = b v (mo d 2 n ). T her e are four cases to be tr eated: Case 1: u n − 1 = v n − 1 = 0. Then no overflo w into u n o ccurs when 1 is a dded to b u to obtain b v . Since the for bidden words leave only u n = v n as p ossibilities, d u n u + 1 = d v n v . Case 2: u n − 1 = v n − 1 = 1. Analogous. Case 3: 0 = u n − 1 6 = v n − 1 = 1. Analogous. Case 4: 1 = u n − 1 6 = v n − 1 = 0. This is an interesting cas e. The fa c t that b u + 1 = b v (mo d 2 n ) implies that u = 1 n and v = 0 n . Now the forbidden words imply u n 6 = v n . This means that either u n u = 01 n and v n v = 1 0 n , or u n u = 1 n +1 and v n v = 0 n +1 . In the former c a se, d u n u + 1 = d v n v , and in the latter case, d u n u + 1 = d v n v (mo d 2 n +1 ). F or a la ng uage L ⊆ Σ ⋆ , let Neutra l( L ) deno te L supplemented with a neutral letter e / ∈ Σ , i.e., Neutra l( L ) c o nsists of all words in L with p ossibly a rbitrary rep eated insertions of the neutra l letter. Theorem 5.2 . FO[ arb ] ∩ BC(CFL) ) F O[ <, +] ∩ BC(CFL) . Pr o of. F ro m the Cra ne Beach Conjecture by Bar rington et al. [4 , Lemma 5.4 ], we know that Neutra l( L I ) ∈ F O[ arb ] \ FO[ <, +] . So w e a re done if we can show Neutr a l( L I ) ∈ B C(CFL). W e proved in Lemma 5.1 that L I = { 0 , 1 , a } ⋆ \ L I is context-free. Therefore Neutral( L I ) is context-free. Now, for a n y w e ∈ { 0 , 1 , a , e } ⋆ , let w ∈ { 0 , 1 , a } ⋆ be the word obtained by deleting all o ccurrences of the neutral letter e fr om w e . Then for any w e , w e ∈ Neutr a l( L I ) ⇐ ⇒ w ∈ L I ⇐ ⇒ w / ∈ L I ⇐ ⇒ w e / ∈ Neutral( L I ) . In other words, Neutral( L I ) = { 0 , 1 , a , e } ⋆ \ Neutral( L I ) = Neutral( L I ) and thus Neutral( L I ) ∈ BC(CFL) . 17 Theorem 5.2 implies tha t the Uniformity Duality Prop erty fa ils for Q = {∃} and L = B C(CFL), since FO[ <, BC(CFL) N ] = FO[ <, +]. Y et, it even provides a witness for the failure o f the duality pr operty in the cas e of L = CFL, as the context-free langua ge Neutral( L I ) lies in FO[ a rb ] \ FO[ <, +]. W e will s tate this result as a corolla ry further b elow. F or now, consider the mo dified Immer man language R I defined as L I except that the successive binary words are r ev er sed in alternance, i. e., R I = { . . . , 0 00a(001) R a010a (011) R a100a (101) R a110a (111) R , . . . } . R I is the intersection of t w o deterministic cont ext-free la nguages. E v en more, the argument in Lemma 5.1 can actua lly b e extended to prov e that the complement of R I is a linear CFL. Hence, Theorem 5.3 . 1. FO [ arb ] ∩ BC(DCFL) ) FO[ <, +] ∩ BC(DCFL) . 2. F O [ arb ] ∩ BC(LinCFL) ) FO[ <, +] ∩ BC(LinCFL) . Pr o of. F or the first claim, o bserv e that Neutra l( R I ) / ∈ FO[ <, +], b ecause FO[ <, + ] has the Crane Beach Prop erty and R I / ∈ RE G . O n the other hand, Neutral( R I ) ∈ F O[ arb ]; and since R I ∈ BC(DCFL), Neutral( R I ) ∈ BC(DCFL). F or the second cla im, R I can b e expre s sed analogous ly to L I by substituting A with an a ppr opriate set A ′ in (5). F urther, fo r u a v ∈ Σ ⋆ with binar y words u and v , the co nditions c u R + 1 6 = b v (mod 2 | u | ) and b u + 1 6 = c v R (mo d 2 | u | ) ca n be chec ked by a linear CFL. Since the linear c o n text-fr ee la nguages are c lo sed under finite union, the cla im follows. The ro le o f neutral le tter s in the ab o v e theorems sug gests taking a closer lo ok at Neutra l (CFL). As the Uniformity Dualit y P roper t y for ( {∃} , Neutral(CFL)) would have it, a ll neutral-letter context-free languag es in AC 0 would b e regular and ap erio dic. T his is, howev er, not the case as witness ed b y Neutral( L I ). Hence, Corollary 5 .4. In the c ase of Q = {∃} , the Uniformity Duality Pr op erty fails in al l of the fol lowing c ases. 1. L = CFL , 2. L = BC(CFL) , 3. L = BC(DCFL ) , 4. L = BC(LinCFL) , 5. L = Neutral(CFL) . R emark 5.5. The class VP L of visibly pushdown langua ges [2] has g ained pro mi- nence recently b ecause it shar es with REG ma n y useful prop erties. But despite having acc ess to a stack, the VPL-numerical predicates coinc ide with RE G N , for each w ord ma y only contain constantly many characters different from (0 , ∅ ). It follows that the Uniformity Duality Prop erty fails for VPL and first-order quantifiers: consider, e. g., L = { a n b n : n > 0 } ∈ FO[ arb ] ∩ (VPL \ REG) then L ∈ F O[ arb ] ∩ VPL but L / ∈ FO[ <, VPL N ] ∩ VPL. 18 6 The Dualit y in Higher Classes W e hav e seen tha t the context-free lang ua ges do not e xhibit our conjectured Uniformity Duality . In this section we will show that the Uniformit y Duality Prop erty ho lds if the extensional uniformity condition imp osed by int ersecting with L is quite lo ose, in other words, if the la ng uage class L is p ow er ful. Recall the no tion of non-uniformity intro duced by Kar p and Lipton [1 3]. Definition 6 .1. F or a c omplexity class L , denote by L / poly the class L with p olynomial advic e. That is, L / p oly is the class of al l languages L such that, for e ach L , ther e is a function f : N → { 0 , 1 } ⋆ with 1. | f ( x ) | ≤ p ( | x | ) , for al l x , and 2. L f = { h x, f ( | x | ) i : x ∈ L } ∈ L , wher e p is a p olynomial dep ending on L . Without loss of gener ality, we wil l assume | f ( x ) | = | x | k for some k ∈ N . Note that, using the above notation, DLOGTIME-uniform AC 0 / p oly = AC 0 . As we further need to ma k e the advice string s acce ssible in a logic, we define the following predicates. F ollowing [5], we say that a Lindstr¨ om quantifier Q L is gr oup oidal , if L ∈ CFL. Definition 6 .2. L et Q b e any s et of gr oup oidal quantifiers. F urther, let L ∈ DLOGTIME -uniform AC 0 [ Q ] / p oly and let f b e the funct io n for which L f ∈ DLOGTIME -uniform AC 0 [ Q ] . L et r = 2 kl + 1 , wher e k and l ar e chosen su ch that the cir cuit family r e c o gnizing L in DLOGTIME -u n if orm AC 0 has size n l and | f ( x ) | = | x | k . We define A d vice f L, Q ∈ FO + Q [ arb ] to b e the ternary r elation Advice f L, Q = { ( i, n, n r ) : bit i of f ( n ) e qu als 1 } , and denote the set of al l r elations Advice f L, Q , for L ∈ L , by Advice L , Q . The inten tion of Advice f L, Q is to e ncode the advice string as a numerical relation. A p oint in this definition that will b ecome c le ar later is the thir d argu- men t of the Advice f L, Q -predicate; it will pa d words in the corr esponding unar y V n -languag e to the leng th of the advice string . T his padding will b e r equired for Theorem 6.4. Theorem 6.3 . L et L b e a language class and Q b e a set of gr oup oidal quantifiers. Then t he Un if ormity D uality Pr op erty for ( {∃} ∪ Q , L ) holds if Bit ∈ L N and Advice L , Q ∈ L N . Pr o of. Let L b e a language class that satisfies the requir emen ts of the claim. W e hav e to show that F O+ Q [ arb ] ∩ L = F O+ Q [ <, L N ] ∩ L . The inclusion from right to left is tr ivial. F or the other dir ection, let L ∈ F O+ Q [ arb ] ∩ L . Without loss of generality , we assume L ⊆ { 0 , 1 } ⋆ . In [5], 19 Barring ton et al. sta te that FO+ Q [ arb ] = F O+ Q [ Bit ] / p oly = DLOGTIME- uniform AC 0 [ Q ] / p oly for arbitra ry sets Q o f mono ida l q uan tifier s, but what is needed in fact is only the existence o f a neutra l element, which is given in our case. There hence exists a poly nomial p ( n ) = n k , k ∈ N , a function f with | f ( x ) | = p ( | x | ), and a DLOGTIME-uniform circuit family { C m } m> 0 that recog- nizes L f . F rom { C m } m> 0 , we construct a fo rm ula ϕ ∈ FO + Q [ <, L N ], ess en tially replacing the a dvice input gates with the relation Advice f L, Q . Let x 1 , . . . , x n , y 1 , . . . , y n k denote the input gates of c ircuit C m , m > 0, wher e x 1 · · · x n = x a nd y 1 · · · y n k = f ( | x | ). First cano nically tr ansform { C m } m> 0 int o a formula ϕ ′ ov er a n e xtended vocabular y σ that satisfies L ( ϕ ′ ) = L (cf. [5 , The- orem 9.1], [23, Theo rem IX.2.1 ] or [24, Theorem 4.73] for the construction of ϕ ′ ). Let l ∈ N be such that n l is a s ize bound on { C m } m> 0 . Then the transforma tion enco des gates as l -tuples of v a riables o v er { 1 , . . . , n } and ensures the co r rect structure using additional predica te symbols Pred , Input0 , Input1 , Output , And , Or , N ot and predicates Quant Q for the subset o f or acle ga tes from Q used in C m . F or example, the relatio n Pred holds on a tuple ( z 1 , . . . , z 2 l ) iff the gate enco ded by ( z 1 , . . . , z l ) is a predeces sor o f ( z l +1 , . . . , z 2 l ); and each of remaining predicates holds on a tuple ( z 1 , . . . , z l ) iff ( z 1 , . . . , z l ) enco des a gate of the corres ponding type. Note that each relation in σ is definable in F O[ <, L N ], as DLO GTIME ⊆ FO[ Bit ] ⊆ FO[ <, L N ]. Next, replace the rela tio ns Input0 and Input1 corres p onding to the input gates y i , 1 ≤ i ≤ n k , with the r espective advice predica tes from Definition 6 .2, ¬ Advice f L, Q ( i, n, n r ) and Advice f L, Q ( i, n, n r ); bo th of which are definable in L N by a ssumption. In the r esulting formula, r eplace the remaining pre dica tes besides < and Bit b y their defining F O + Q [ <, L N ]- formulae and e ventually obtain a for m ula defining L ∈ F O+ Q [ <, L N ]. W e ca n now give a lower b ound b eyond which the Uniformity Dua lity Pr op- erty holds. Let NTIME( n ) L denote the class of la nguages decidable in linea r time by nondeter ministic T uring machines with or acles from L . Theorem 6.4 . L et Q b e any set of gr oup oidal quantifiers and s upp ose L = NTIME( n ) L . Then t he Uniformity Duality Pr op erty for ( {∃} ∪ Q , L ) holds. Pr o of. Cho ose any L ∈ AC 0 [ Q ] ∩L and let f b e an advice function for which L f ∈ DLOGTIME-uniform AC 0 [ Q ]. W e hav e to show that the relation A d vice f L, Q is definable in L N . Let k ∈ N such that | f ( x ) | = | x | k and denote by { C m } m> 0 the DLOGTIME-uniform cir cuit family of s iz e ≤ n l that reco gnizes L f using o racle gates fro m Q . Let N b e a nondeterministic linear-time T ur ing machine deciding L using so me oracle L ′ ∈ L . W e will define a nondeterministic T uring machine M that decides x ∈ L Adv , where L Adv is s uc h that L N Adv = ADVICE f L, Q . Given input x , M pro ceeds as follows: 1 if x is not o f the for m x = [ x 1 = i, x 2 = n, x 3 = n r ] for s ome n > 0 a nd 1 ≤ i ≤ n 20 2 then reject; 3 for all str ings a o f length n k in lexicogra phic or dering do 4 t ← true ; 5 for all inputs y of length n do 6 if the output of C n + n k on y with a dvice a co n tra dicts the result of N on y 7 then t ← false ; 8 if t = true and bit i of a is 1 then accept; 9 el se r eject; That is, M guesses the advice s tr ing a using a na ¨ ıve tria l-and-error appr oac h; once the co rrect advice string a has b een found, it accepts x iff the i th bit in the advice string a is o n. Thu s M dec ides L Adv . As for the time required by M , note that in line 6 the circuit C n + n k can be ev aluated in time O (( n + n k ) 2 l ) = O ( n r ) = O ( | x | ). F urthermore, L = NTIME( n ) L implies that L is clo s ed under complemen t. Thus the ab o ve al- gorithm solves the pr oblem in NTIME( | x | ) coNTIME( | x | ) L = NTIME( | x | ) L = L and A D V ICE f L, Q ∈ L N . The claim no w follows from Theorem 6.3, b ecause L = NTIME( n ) L moreov er implies Bit ∈ L N . Corollary 6 .5. L et Q b e any set of gr oup oidal quantifiers. The Un iformity Dual- ity Pr op ert y holds for ( {∃} ∪ Q , L ) if L e quals t he deterministic c ontext- sensitive languages DSP ACE( n ) , the c ontext-sensitive languages CSL , the ru dimentary languages (i. e., t he line ar time hier ar chy [26]), PH , PSP A CE , or the r e curs ively enumer able languages. Pr o of. All of the ab ov e cla sses satisfy L = NTIME( n ) L . 7 Conclusion F or a set Q of quantifiers and a class L of la nguages, we have suggested tha t Q [ arb ] ∩ L defines an (extensionally) unifor m complexity class. After defining the notion of L -numerical pr edicates, we hav e pr opose d comparing Q [ arb ] ∩ L with its sub class Q [ <, L N ] ∩ L , a clas s equiv alently defined as the (intensionally) uniform circuit class FO [ <, L N ]-uniform AC 0 [ Q ] ∩ L . W e hav e no ted that the duality prop erty , defined to hold w hen both classes ab o v e ar e equal, encompass e s Straubing’s co njecture (1) as well a s so me po sitiv e and so me negative instances of the Crane Beach Conjecture. W e hav e then inv estig a ted the duality prop erty in sp ecific cases with Q = {∃} . W e hav e seen that the pr operty fails for several c la sses L inv olv ing the context-free langua ges. Exhibiting these failures ha s require d new insights, such as characterizatio ns of the context-free n umerical predicates and a pro of that the complement of the Immerman la nguage is c o n text- fr ee, but these failures have preven ted successfully tac kling complexity classes such as AC 0 ∩ CFL . Restricting the class of allowed relations o n the left ha nd side of the uniformity dua lit y prop ert y from arb to a sub class migh t lead to further insight and provide po s itiv e 21 examples of this mo dified duality pr operty (and address, e.g., the class of cont ext- free langua ges in different uniform versions of AC 0 ). Metho ds from embedded finite mo del theory should find applications here. More generally , the duality pr operty widens our p e r spective o n the relatio n- ship b et w een uniform cir cuits and descriptive co mplexit y b eyond the level of NC 1 . W e have noted for example that the prop erty holds for any se t of group oidal quan- tifiers Q ⊇ {∃} and co mplexit y clas ses L that ar e closed under nondeterministic linear-time T uring reductions. A p oint often made is that a satisfactory unifor mit y definition should ap- ply compar able res ource b ounds to a circuit family and to its co ns tructor. F or instance, although P -uniform NC 1 has merit [1], the classes AC 0 -uniform NC 1 and NC 1 -uniform NC 1 [5] seem more fundamental, provided that one ca n ma k e sense of the apparent cir c ularit y . As a by-product o f our work, we migh t suggest F O[ <, L N ] ∩ L as the minimal “unifor m sub class of L ” and thus as a meaning- ful (albeit restr ictiv e) definition of L -uniform L . Our c hoice o f FO[ < ] as the “b ottom cla ss of interest” is implicit in this definition and r e s ults in the co n tain- men t of L -uniform L in (non-uniform) AC 0 for a n y L . P rogressively less uniform sub c lasses of L would be the cla sses Q [ <, L N ] ∩ L for Q ⊇ {∃} . Restating ha rd ques tio ns such a s conjecture (1) in ter ms of a unifying prop- erty do es no t make these questions g o aw ay . But the duality proper t y raises further questions . As an example, can the duality pr operty fo r v arious ( Q , L ) b e shown to ho ld or to fail when Q includes the ma jority quantifier? This could help develop incis ive results concerning the clas s TC 0 . T o b e more precise, let us consider Q = {∃ , MAJ } . The ma jor it y quantifier is a particular gro upoidal (or , c ontext- fr e e ) quantifier [16], hence it seems natur al to co nsider the Uniformity Dualit y Prop erty for ( {∃ , MAJ } , CFL): F O+MAJ[ arb ] ∩ CFL = FO+MAJ[ <, +] ∩ CFL . (6) It is not hard to see that the Immer ma n langua ge in fact is in FO+MAJ[ <, +], hence our Theorem 5 .2 that refutes (2), the Uniformity Duality P roper t y for (F O , BC(CFL)), do es not sp eak to whether (6) holds. (Another prominent ex- ample that r efutes (2) is the “W otschk e lang uage” W = { (a n b) n : n ≥ 0 } , a gain a co- c o n text- fr ee language [25]. Similar to the case of the Immerman lang uage we o bserv e that W ∈ FO +MAJ[ <, + ], hence W do es not refute (6) either.) Observe that FO+MAJ[ arb ] = TC 0 [5] and that, on the other hand, F O+MAJ[ <, + ] = MAJ[ < ] = FO[+]-uniform linear fan-in TC 0 [15,6]. Let us call this latter class sTC 0 (for smal l TC 0 or strict TC 0 ). It is known that sTC 0 ( TC 0 [16]. Hence we co nc lude that if (6) holds, then in fact TC 0 ∩ CFL = sTC 0 ∩ CFL. Thus, if we can s how that so me la nguage in the B o olean clo s ure of the context-free lang uages is not in sTC 0 , we have a new TC 0 low er b ound. Thu s, to separate TC 0 from a sup erclass it suffices to se parate s TC 0 from a sup e rclass, a pos sibly less demanding g oal. This may b e a nother reas on to lo ok for a ppropriate uniform clas ses L such that F O+MAJ[ arb ] ∩ L = FO+MAJ[ <, +] ∩ L . 22 Ac kno wledgemen ts W e would like to thank K laus-J¨ orn Lange (p e rsonal comm unication) fo r sug- gesting Lemma 5.1. W e also acknowledge helpful discussions on v a rious topics of this pap er with Christoph Behle, Andrea s Krebs, Klaus -J¨ orn Lange and Thoma s Sch wen tick. W e also a c knowledge helpful comments from the a no n ymo us refer- ees. References 1. E. Allender. 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