Lower bounds for algebraic machines, semantically
aa r X i v : . [ c s . CC ] F e b Lower bounds for algebraic machines, semantically
Luc Pellissier
LACL, Faculté des Sciences et Technologie61 avenue du Général de Gaulle94010 Créteil, FRANCEEmail: [email protected]
Thomas Seiller
CNRS, LIPN – UMR 7030Université Sorbonne Paris Nord99, Avenue Jean-Baptiste Clément93430, Villetaneuse, FRANCEEmail: [email protected]
Abstract —This paper presents a new semantic method for prov-ing lower bounds in computational complexity. We use it to provethat maxflow , a P
TIME complete problem, is not computablein polylogarithmic time on parallel random access machines(
PRAM s ) working with integers, showing that NC Z = P TIME ,where NC Z is the complexity class defined by such machines,and P TIME is the standard class of polynomial time computableproblems (on, say, a Turing machine). On top of showing thisnew separation result, we show our method captures previouslower bounds results from the literature: Steele and Yao’s lowerbounds for algebraic decision trees [1], Ben-Or’s lower boundsfor algebraic computation trees [2], Cucker’s proof that NC R is not equal to P TIME R [3], and Mulmuley’s lower bounds for“ PRAM s without bit operations” [4]. I. I
NTRODUCTION
Complexity theory has traditionally been concerned withproving separation results between complexity classes. Manyproblems remain open, such as the much advertised P
TIME vs NP
TIME question which concerns the difference betweenfeasible sequential computability in deterministic and non-deterministic models, or equivalently the difference betweenfeasible computation and feasible verification. This paperinvestigates questions related to the NC vs P
TIME question,which concerns the difference between efficient sequentialcomputation and (more than) efficient parallel computation.Proving that two classes B ⊂ A are not equal can bereduced to finding lower bounds for problems in A : by provingthat certain problems cannot be solved with less than certainresources on a specific model of computation, one can showthat two classes are not equal. Conversely, proving a separationresult B ( A provides a lower bound for the problems that are A -complete [5] – i.e. problems that are in some way universal for the class A .The proven lower bound results are however very rough,and many separation problems remain as generally acceptedconjectures. For instance, a proof that the class of non-deterministic exponential problems is not included in what isthought of as a very small class of circuits was not achieveduntil very recently [6].The failure of most techniques of proof has been studiedin itself, which lead to the proof of the existence of negativeresults that are commonly called barriers . Altogether, theseresults show that all proof methods we know are ineffectivewith respect to proving interesting lower bounds. Indeed, thereare three barriers: relativisation [7], natural proofs [8] and algebrization [9], and almost every known proof method hitsat least one of them: this shows the need for new methods .However, to this day, only one research program aimed atproving new separation results is commonly believed to havethe ability to bypass all barriers: Mulmuley and Sohoni’s Geo-metric Complexity Theory ( GCT ) program [10]. This researchprogram was inspired from an earlier lower bounds result byMulmuley [4] which we strengthen in this paper.
A. Mulmuley’s result
The NC vs P
TIME question is one of the foremost openquestions in computational complexity. In laymen’s terms, itasks whether a problem efficiently computable on a sequentialmachine can be computed substantially more efficiently on aparallel machine. It is well known that any problem in NC,i.e. that is computable in polylogarithmic time on a parallelmachine (with a polynomial number of processors), belongs toP
TIME , i.e. is computable in polynomial time on a sequentialmachine. The converse, however, is expected to be false.Indeed, although many problems in P
TIME can be shown tobe in NC, some of them seem to resist efficient parallelisation.In particular it is not known whether the maxflow problem,known to be P
TIME -complete [11], belongs to NC.As part of the investigations on the NC vs P
TIME question,a big step forward is due to K. Mulmuley. In 1999 [4], heshowed that a notion of machine introduced under the name“
PRAM s without bit operations” does not compute maxflow inpolylogarithmic time. This notion of machine, quite exotic atfirst sight, corresponds to an algebraic variant of
PRAM s, whereregisters contain integers and individual processors are allowedto perform sums, subtractions and products of integers. It isargued by Mulmuley that this notion of machine provides anexpressive model of computation, able to compute some nontrivial problems in NC such as Neff’s algorithm for computingapproximate roots of polynomials [12]. Although Mulmuley’sresult has represented a big step forward in the quest for aproof that P
TIME and NC are not equal, the result was notstrenghtened or reused in the last 20 years, and remained thestrongest known lower bound result. In the words of S. Aaronson and A. Wigderson [9], “We speculatethat going beyond this limit [algebrization] will require fundamentally newmethods.” . Contributions.
The main contribution of this work is a strengthening ofMulmuley’s lower bounds result. While the latter proves that maxflow is not computable in polylogarithmic time in themodel of “
PRAM s without bit operations”, we show here that maxflow is not computable in polylogarithmic time in themore expressive model of
PRAM s over integers, making anadditional step in the direction of a potential proof that NC isdifferent from P
TIME . Indeed, our result can be stated as
Theorem 1. NC Z = P TIME , where NC Z is the set of problems decidable in polylogarithmictime by a (not necessarily uniform) family of PRAM s over Z . The second contribution of the paper is the proof methoditself, which is based on dynamic semantics for programs bymeans of graphings , a notion introduced in ergodic theoryand recently used to define models of linear logic by Seiller[13], [14], [15], [16]. The dual nature of graphings, bothcontinuous and discrete, is essential in the present work, as itenables invariants from continuous mathematics, in particularthe notion of topological entropy for dynamical systems, whilethe finite representability of graphings is used in the keylemma (as the number of edges appears in the upper boundsof Lemma 5).In particular, we show how this proof method capturesknown lower bounds and separation results in algebraic modelsof computation, namely Steele and Yao’s lower bounds for al-gebraic decision trees [1], Ben-Or’s lower bounds on algebraiccomputation trees [2], Cucker’s proof that NC R is not equalto P TIME R (i.e. answering the NC vs P TIME problem forcomputation over the real numbers).
C. A more detailed view of the proof method
One of the key ingredients in the proof is the representationof programs as graphings, and quantitative soundness results.We refer to the next section for a formal statement, andwe only provide an intuitive explanation for the moment.Since a program P is represented as a graphing | [ P ] | , whichis in some way a dynamical system, the computation P ( a ) on a given input a is represented as a sequence of values | [ a ] | ) , | [ P ] | ( | [ a ] | ) , | [ P ] | ( | [ a ] | ) , . . . . Quantitative soundness statesthat not only | [ P ] | computes exactly as P , but it does so inthe same number of steps, i.e. if P ( a ) terminates on a value b in time k , then | [ P ] | k ( | [ a ] | ) = | [ b ] | .The second ingredient is the dual nature of graphings, bothcontinuous and discrete objects. Indeed, a graphing represen-tative is a graph-like structure whose edges are representedas continuous maps, i.e. a finite representation of a (partial)continuous dynamical system. Given a graphing, we define its k th cell decomposition , which separates the input space intocells such that two inputs in the same cell are indistinguishablein k steps, i.e. the graphing’s computational traces on bothinputs are equal. We can then use both the finiteness of thegraphing representatives and the topological entropy of the associated dynamical system to provide upper bounds on thesize of a further refinement of this geometric object, namelythe k -th entropic co-tree of a graphing – a kind of finalapproximation of the graphing by a computational tree As we deal with algebraic models of computation, thisimplies a bound on the representation of the k th cell de-composition as a semi-algebraic variety. In other words, the k th cell decomposition is defined by polynomial in ·equalitiesand we provide bounds on the number and degree of theinvolved polynomials. The corresponding statement is themain technical result of this paper (Lemma 5).This lemma can then be used to obtain lower boundsresults. Using the Milnor-Thom theorem to bound thenumber of connected components of the k th cell decom-position, we then recover the lower bounds of Steele andYao on algebraic decision trees, and the refined result ofBen-Or providing lower bounds for algebraic computationtrees. A different argument based on invariant polynomialsprovides a proof of Cucker’s result that NC R = Ptime R by showing that a given polynomial that belongs to Ptime R cannot be computed within NC R . Lastly, follow-ing Mulmuley’s geometric representation of the maxflow problem, we are able to strenghten his celebrated result toobtain lower bounds on the size (depth) of a PRAM over theintegers computing this problem. This proves the followingtheorem, which has Theorem 1 as a corollary.
Theorem 2.
Let c be a positive integer, M a PRAM over Z with O ((log N ) c ) processors, with N the length of the inputs.Then M does not decide maxflow in O ((log N ) c ) steps. II. P
ROGRAMS AS D YNAMICAL SYSTEMS
A. Abstract models of computation and graphings
We consider computations as dynamical processes,hence model them as a dynamical system with two maincomponents: a space X that abstracts the notion of con-figuration space and a monoid acting on this space thatrepresents the different operations allowed in the model ofcomputation. Although the notion of space considered canvary (one could consider e.g. topological spaces, measurespaces, topological vector spaces), we restrict ourselves totopological spaces in this work. Definition 1. An abstract model of computation ( AMC ) is amonoid action α : M y X , i.e. a monoid morphism from M to the group of endomorphisms of X . The monoid M is oftengiven by a set G of generators and a set of relations R . Wedenote such an AMC as α : h G, R i y X . Programs in an amc α : h G, R i y X is then definedas graphings , i.e. graphs whose vertices are subspaces ofthe space X (representing sets of configurations on whichthe program act in the same way) and edges are labelledby elements of M h G, R i , together with a global control Intuitively, the k -th entropic co-tree mimicks the behaviour of the graphingfor k steps of computation. tate. More precisely, we use here the notion of topologicalgraphings [14]. Definition 2. An α -graphing representative G w.r.t. a monoidaction α : M y X is defined as a set of edges E G togetherwith a map that assigns to each element e ∈ E G a pair ( S Ge , m Ge ) of a subspace S Ge of X – the source of e – andan element m Ge ∈ M – the realiser of e . While graphing representatives are convenient to ma-nipulate, they do provide too much information about theprograms. Indeed, if one is to study programs as dynamicalsystems, the focus should be on the dynamics , i.e. on howthe object acts on the underlying space. The followingnotion of refinement captures this idea that the samedynamics may have different graph-like representations.
Definition 3 (Refinement) . An α -graphing representative F isa refinement of an α -graphing representative G , noted F G ,if there exists a partition ( E Fe ) e ∈ E G of E F such that ∀ e ∈ E G : (cid:16) ∪ f ∈ E Fe S Ff (cid:17) △ S Ge = ∅ ; ∀ f = f ′ ∈ E Fe , S Ff △ S Ff ′ = ∅ ; ∀ f ∈ E Fe , m Ff = m Ge . This induces an equivalence relation defined as F ∼ ref G ⇔ ∃ H, H F ∧ H G. The notion of graphing is therefore obtained by consider-ing the quotient of the set of graphing representatives w.r.t. ∼ ref . Intuitively, this corresponds to identifying graphingswhose actions on the underlying space are equal . Definition 4. An α - graphing is an equivalence class of α -graphing representatives w.r.t. the equivalence relation ∼ ref . We can now define the notion of abstract program.These are defined as graphings
Definition 5.
Given an
AMC α : M y X , an α -program A is a ¯ α -graphing G A w.r.t. the monoid action ¯ α = α × S k y X × S A , where S A is a finite set of control states of cardinality k and S k is the group of permutations of k elements. Now, as a sanity check, we will show how the notion ofgraphing do capture the expected dynamics. For this, werestrict to deterministic graphings , and show the notionrelates to the usual notion of dynamical system.
Definition 6. An α -graphing representative G is deterministicif for all x ∈ X there is at most one e ∈ E G such that x ∈ S Ge . An α -graphing is deterministic if its representativesare deterministic. An abstract program is deterministic if itsunderlying graphing is deterministic. Lemma 1.
There is a one-to-one correspondence between theset of deterministic graphings w.r.t. the action M y X and the While “measured” graphings were already considered [14], the definitionadapts in a straightforward manner to allow for other notions such as graphingsover topological vector spaces – which would be objects akin to the notionof quiver used in representation theory. set of partial dynamical systems f : X ֒ → X whose graph iscontained in the preorder { ( x, y ) | ∃ m ∈ M, α ( m )( x ) = y } . Lastly, we define some restrictions of α -programs thatwill be important later. First, we will restrict the pos-sible subspaces considered as sources of the edges, asunrestricted α -programs could compute even undecidableproblems by, e.g. encoding it into a subspace used as thesource of an edge. Given an integer k ∈ ω , we define thefollowing subspaces of R ω , for ⋆ ∈ { >, > , = , = , , < } : R ωk⋆ = { ( x , . . . , x k , . . . ) ∈ R ω | x k ⋆ } . Definition 7 (Computational graphings) . Let α : h G, R i y X be an AMC . A computational α -graphing is an α -graphing T with distinguished states ⊤ , ⊥ ∈ S A which admits a finiterepresentative such that each edge e has its source equal to oneamong R ω , R ωk > , R ωk , R ωk> , R ωk< , R ωk =0 , and R ωk =0 . Definition 8 (treeings) . Let α : h G, R i y X be an AMC . An α -treeing is an acyclic and finite α -graphing, i.e. an α -graphing F for which there exists a finite α -graphing representative T whose set of control states S T = { , . . . , s } can be endowedwith an order < such that every edge of T is state-increasing,i.e. for each edge e of source S e , for all x ∈ S e , π S T ( α ( m e )( x )) > π S T ( x ) , where π S T denotes the projection onto the control states space.A computational α -treeing is an α -treeing T which is acomputational α -graphing with the distinguished states ⊤ , ⊥ being incomparable maximal elements of the state space. B. Quantitative Soundness
As mentioned in the introduction, we will use the prop-erty of quantitative soundness of the dynamic semanticsjust introduced. This result is essential, as it connectsthe time complexity of programs in the model considered(e.g.
PRAM s , algebraic computation trees) with the lengthof the orbits of the considered dynamical system. Wehere only state quantitative soundness for computationalgraphings , i.e. graphings that have distinguished states ⊤ and ⊥ representing acceptance and rejection respectively.In other words, we consider graphings which compute decision problems .Quantitative soundness is expressed with respect toa translation of machines as graphings, together with atranslation of inputs as points of the configuration space.In the following section, these operations are defined foreach model of computation considered in this paper. In allthese cases, the representation of inputs is straightforward. Definition 9.
Let α be an abstract model of computation, and M a model of computation. A translation of M w.r.t. α isa pair of maps | [ · ] | which associate to each machine M in When α is a group action acting by measure-preserving transformations,this is a Borel equivalence relation R , and the condition stated here boilsdown to requiring that f belongs to the full group of α . computing a decision problem a computational α -graphing | [ M ] | and to each input ι a point | [ ι ] | in X × S . Definition 10.
Let α be an abstract model of computation, M a model of computation. The AMC α is quantitatively sound for M w.r.t. a translation | [ · ] | if for all machine M computing adecision problem and input ι , M accepts ι (resp. rejects ι ) in k steps if and only if | [ M ] | k ( | [ ι ] | ) = ⊤ (resp. | [ M ] | k ( | [ ι ] | ) = ⊥ ). C. The algebraic
AMC s We now define the actions α full and α R full . Those willcapture all algebraic models of computation considered inthis paper, and the main theorem will be stated for thismonoid action.As we intend to consider PRAM s at some point, weconsider from the beginning the memory of our machinesto be separated in two infinite blocks Z ω , intended torepresent both shared and a private memory cells . Definition 11.
The underlying space of α full is X = Z Z ∼ = Z ω × Z ω . The set of generators is defined by their action onthe underlying space, writing k//n the floor ⌊ k/n ⌋ of k/n with the convention that k//n = 0 when n = 0 : • const i ( c ) initialises the register i with the constant c ∈ Z : α full ( const i ( c ))( ~x ) = ( ~x { x i := c } ) ; • ⋆ i ( j, k ) ( ⋆ ∈ { + , − , × , // } ) performs the algebraic oper-ation ⋆ on the values in registers j and k and store the re-sult in register i : α full ( ⋆ i ( j, k ))( ~x ) = ( ~x { x i := x j ⋆ x k } ) ; • ⋆ ci ( j ) ( ⋆ ∈ { + , − , × , // } ) performs the algebraic op-eration ⋆ on the value in register j and the constant c ∈ Z and store the result in register i : α full ( ⋆ ci ( j ))( ~x ) =( ~x { x i := c ⋆ x j } ) ; • copy ( i, j ) copies the value stored in register j in register i : α full ( copy ( i, j ))( ~x ) = ( ~x { x i := x j } ) ; • copy ( ♯i, j ) copies the value stored in register j in theregister whose index is the value stored in register i : α full ( copy ( ♯i, j ))( ~x ) = ( ~x { x x i := x j } ) ; • copy ( i, ♯j ) copies the value stored in the register whoseindex is the value stored in register j in register i : α full ( copy ( i, ♯j ))( ~x ) = ( ~x { x i := x x j } ) ; • n √ i ( j ) computes the floor of the n -th root of the valuestored in register j and store the result in register i : α full ( n √ i ( j ))( ~x ) = ( ~x { x i := n √ x j } ) . We also define the real-valued equivalent, which will beessential for the proof of lower bounds. The corresponding amc α R full is defined in the same way than the integer-valued one, but with underlying space X = R Z and withinstructions adapted accordingly: • the division and n -th root operations are the usualoperations on the reals; • the three copy operators are only effective on integers. Definition 12.
The underlying space of α R full is X = R Z ∼ = R ω × R ω . The set of generators is defined by their action on Obviously, this could be done without any explicit separation of theunderlying space, but this will ease the constructions of the next section. the underlying space, with the convention that k/n = 0 when n = 0 : • const i ( c ) initialises the register i with the constant c ∈ R : α R full ( const i ( c ))( ~x ) = ( ~x { x i := c } ) ; • ⋆ i ( j, k ) ( ⋆ ∈ { + , − , × , / } ) performs the algebraic op-eration ⋆ on the values in registers j and k and storethe result in register i : α R full ( ⋆ i ( j, k ))( ~x ) = ( ~x { x i := x j ⋆ x k } ) ; • ⋆ ci ( j ) ( ⋆ ∈ { + , − , × , / } ) performs the algebraic operation ⋆ on the value in register j and the constant c ∈ R andstore the result in register i : α R full ( ⋆ ci ( j ))( ~x ) = ( ~x { x i := c ⋆ x j } ) ; • copy ( i, j ) copies the value stored in register j in register i : α R full ( copy ( i, j ))( ~x ) = ( ~x { x i := x j } ) ; • copy ( ♯i, j ) copies the value stored in register j in theregister whose index is the floor of the value stored inregister i : α R full ( copy ( ♯i, j ))( ~x ) = ( ~x { x ⌊ x i ⌋ := x j } ) ; • copy ( i, ♯j ) copies the value stored in the register whoseindex is the floor of the value stored in register j inregister i : α R full ( copy ( i, ♯j ))( ~x ) = ( ~x { x i := x ⌊ x j ⌋ } ) ; • n √ i ( j ) computes the n -th real root of the valuestored in register j and store the result in register i : α R full ( n √ i ( j ))( ~x ) = ( ~x { x i := n √ x j } ) .III. A LGEBRAIC MODELS OF COMPUTATIONS AS AMCS
A. Algebraic computation trees
The first model considered here will be that of algebraiccomputation tree as defined by Ben-Or [2]. Let us notethis model refines the algebraic decision trees model ofSteele and Yao [1], a model of computation consistingin binary trees for which each branching performs a testw.r.t. a polynomial and each leaf is labelled
YES or NO .Algebraic computation trees only allow tests w.r.t. , whileadditional vertices corresponding to algebraic operationscan be used to construct polynomials. Definition 13 (algebraic computation trees, [2]) . An algebraiccomputation tree on R n is a binary tree T with a functionassigning: • to any vertex v with only one child (simple vertex) anoperational instruction of the form f v = f v i ⋆ f v j , f v = c ⋆ f v i , or f v = p f v i , where ⋆ ∈ { + , − , × , / } , v i , v j areancestors of v and c ∈ R is a constant; • to any vertex v with two children a test instruction of theform f v i ⋆ , where ⋆ ∈ { >, = , > } , and v i is an ancestorof v or f v i ∈ { x , . . . , x n } ; • to any leaf an output YES or NO . Let W ⊆ R n be any set and T be an algebraic com-putation tree. We say that T computes the membershipproblem for W if for all x ∈ R n , the traversal of T following x ends on a leaf labelled YES if and only if x ∈ W .As algebraic computation trees are trees , they will berepresented by treeings, i.e. α R full -programs whose setof control states can be ordered so that any edge inthe graphing is strictly increasing on its control statescomponent. efinition 14. Let T be a computational α R full -treeing. Theset of inputs In( T ) (resp. outputs Out( T ) ) is the set of integers k (resp. i ) such that there exists an edge e in T satisfying that: • either e is realised by one of + i ( j, k ) , + i ( k, j ) , − i ( j, k ) , − i ( k, j ) , × i ( j, k ) , × i ( k, j ) , / i ( j, k ) , / i ( k, j ) + ci ( k ) , − ci ( k ) , × ci ( k ) , / ci ( k ) , n √ i ( k ) ; • or the source of e is one among R ωk > , R ωk , R ωk> , R ωk< , R ωk =0 , and R ωk =0 .The effective input space In E ( T ) of an α act -treeing T isdefined as the set of indices k ∈ ω belonging to In( T ) but notto Out( T ) . The implicit input space In I ( T ) of an α act -treeing T is defined as the set of indices k ∈ ω such that k Out( T ) . Definition 15.
Let T be an α R full -treeing, and assume that , , . . . , n ∈ In I ( T ) . We say that T computes the membershipproblem for W ⊆ R n in k steps if k successive iterations of T restricted to { ( x i ) i ∈ ω ∈ R ω | ∀ i n, x i = y i } × { } reach state ⊤ if and only if ( y , y , . . . , y n ) ∈ W . Remark . Let ~x = ( x , x , . . . , x n ) be an elementof R n and consider two elements a, b in the subspace { ( y , . . . , y n , . . . ) ∈ R ω | ∀ > i > n, y i = x i } × { } .One easily checks that π S ( T k ( a )) = ⊤ if and only if π S ( T k ( b )) = ⊤ , where π S is the projection onto the statespace and T k ( a ) represents the k -th iteration of T on a . It istherefore possible to consider only a standard representative | [ ~x ] | of ~x ∈ R n , for instance ( x , . . . , x n , , , . . . ) ∈ R ω , todecide whether ~x is accepted by T . Definition 16.
Let T be an algebraic computation tree on R n ,and T ◦ be the associated directed acyclic graph, built from T by merging all the leaves tagged YES in one leaf ⊤ and all theleaves tagged NO in one leaf ⊥ . Suppose the internal verticesare numbered { n + 1 , . . . , n + ℓ } ; the numbers , . . . , n beingreserved for the input.We define | [ T ] | as the α act -graphing with control states { n +1 , . . . , n + ℓ, ⊤ , ⊥} and where each internal vertex i of T ◦ defines either: • a single edge of source R ω realized by: – ( ⋆ i ( j, k ) , i t ) ( ⋆ ∈ { + , − , ×} ) if i is associated to f v i = f v j ⋆ f v k and t is the child of i ; – ( ⋆ ci ( j ) , i t ) ( ⋆ ∈ { + , − , ×} ) if i is associated to f v i = c ⋆ f v k and t is the child of i ; • a single edge of source R ωk =0 realized by: – ( / i ( j, k ) , i t ) if i is associated to f v i = f v j /f v k and t is the child of i ; – ( / ci ( k ) , i t ) if i is associated to f v i = c/f v k and t is the child of i ; • a single edge of source R ωk > × { i } realized by ( √ i ( k ) , i t ) if i is associated to f v i = p f v k and t is the child of i ; • two edges if i is associated to f v i ⋆ (where ⋆ ranges in > , > ) and its two sons are j and k . Those are of respectivesources R ωk⋆ × { i } and R ωk ¯ ⋆ × { i } (where ¯ ⋆ = ′ ′ if ⋆ = ′ > ′ , ¯ ⋆ = ′ < ′ if ⋆ = ′ > ′ , and ¯ ⋆ = ′ = ′ if ⋆ = ′ = ′ .),respectively realized by ( Id , i j ) and ( Id , i k ) Proposition 1.
Any algebraic computation tree T of depth k isfaithfully and quantitatively interpreted as the α R full -program | [ T ] | . I.e. T computes the membership problem for W ⊆ R n if and only if | [ T ] | computes the membership problem for W in k steps – that is π S ( | [ T ] | k ( | [ ~x ] | )) = ⊤ . As a corollary of this proposition, we get quantitativesoundness.
Theorem 3.
The representation of
ACT s as α R full -programsis quantitatively sound.B. Algebraic circuits As we will recover Cucker’s proof that NC R = Ptime R ,we introduce the model of algebraic circuits and theirrepresentation as α R full -programs. Definition 17.
An algebraic circuit over the reals with inputsin R n is a finite directed graph whose vertices have labels in N × N , that satisfies the following conditions: • There are exactly n vertices v , , v , , . . . , v ,n with firstindex , and they have no incoming edges; • all the other vertices v i,j are of one of the following types:1) arithmetic vertex: they have an associated arithmeticoperation { + , − , × , / } and there exist natural numbers l, k, r, m with l, k < i such that their two incomingedges are of sources v l,r and v k,m ;2) constant vertex: they have an associated real number y and no incoming edges;3) sign vertex: they have a unique incoming edge ofsource v k,m with k < i .We call depth of the circuit the largest m such that there exist avertex v m,r , and size of the circuit the total number of vertices.A circuit of depth d is decisional if there is only one vertex v d,r at level d , and it is a sign vertex; we call v d,r the endvertex of the decisional circuit. To each vertex v one inductively associates a function f v of the input variables in the usual way, where a signnode with input x returns if x > and otherwise. Theaccepted set of a decisional circuit C is defined as the set S ⊆ R n of points whose image by the associated functionis , i.e. S = f − v ( { } ) where v is the end vertex of C .We represent algebraic circuit as computational α R full -treeings as follows. The first index in the pairs ( i, j ) ∈ N × N are represented as states, the second index is representedas an index in the infinite product R ω , and vertices arerepresented as edges. Definition 18.
Let C be an algebraic circuit, defined as afinite directed graph ( V, E, s, t, ℓ ) where V ⊂ N × N , and ℓ : V → { init , + , − , × , /, sgn } ∪ { const c | c ∈ R } is a vertexlabelling map. We suppose without loss of generality that foreach j ∈ N , there is at most one i ∈ N such that ( i, j ) ∈ V .We define N as max { j ∈ N | ∃ i ∈ N , ( i, j ) ∈ V } .We define the α R full -program | [ C ] | by choosing as set ofcontrol states { i ∈ N | ∃ j ∈ N , ( i, j ) ∈ V } and the collectionf edges { e ( i,j ) | i ∈ N ∗ , j ∈ N , ( i, j ) ∈ V } ∪ { e +( i,j ) | i ∈ N ∗ , j ∈ N , ( i, j ) ∈ V, ℓ ( v ) = sgn } realised as follows: • if ℓ ( v ) = const c , the edge e ( i,j ) is realised as (+ n v j ( c ) , i ) of source R ωn v =0 × { } ; • if ℓ ( v ) = ⋆ ( ⋆ ∈ { + , − , ×} ) of incoming edges ( k, l ) and ( k ′ , l ′ ) , the edge e ( i,j ) is of source R ω × { max( k, k ′ ) } and realised by ( ⋆ j ( l, l ′ ) , max( k, k ′ ) i ) ; • if ℓ ( v ) = / of incoming edges ( k, l ) and ( k ′ , l ′ ) , the edge e ( i,j ) is of source R ωl ′ =0 × { max( k, k ′ ) } and realised by ( / j ( l, l ′ ) , max( k, k ′ ) i ) ; • if ℓ ( v ) = sgn of incoming edge ( k, l ) , the edges e ( i,j ) and e +( i,j ) are of respective sources R ωn v =0 ∧ x l × { k } and R ωn v =0 ∧ x l > × { k } realised by ( Id , k i ) and (+ j ( n v , , k i ) respectively. As each step of computation in the algebraic circuit istranslated as going through a single edge in the correspond-ing α R full -program, the following result is straightforward. Theorem 4.
The representation of
ALGCIRC as α R full -programs is quantitatively sound.C. Algebraic RAM s In this paper, we will consider algebraic parallel randomaccess machines, that act not on strings of bits, but onintegers. In order to define those properly, we first definethe notion of (sequential) random access machine (
RAM )before considering their parallelisation.A
RAM command is a pair ( ℓ, I ) of a line ℓ ∈ N ⋆ andan instruction I among the following, where i, j ∈ N , ⋆ ∈{ + , − , × , / } , c ∈ Z is a constant and ℓ, ℓ ′ ∈ N ⋆ are lines: skip ; X i := c ; X i := X j ⋆ X k ; X i := X j ; X i := ♯ X j ; ♯ X i := X j ; if X i = goto ℓ else ℓ ′ . A RAM machine M is then a finite set of commands suchthat the set of lines is { , , . . . , | M |} , with | M | the length of M . We will denote the commands in M by ( i, Inst M ( i )) ,i.e. Inst M ( i ) denotes the line i instruction.Following Mulmuley [4], we will here make the assump-tion that the input in the RAM (and in the
PRAM modeldefined in the next section) is split into numeric and nonumeric data – e.g. in the maxflow problem the non-numeric data would specify the network and the numericdata would specify the edge-capacities – and that indirectreferences use pointers depending only on nonnumericdata . We refer the reader to Mulmuley’s article for moredetails.Machines in the RAM model can be represented asgraphings w.r.t. the action α full . Intuitively the encodingworks as follows. The notion of control state allows to Quoting Mulmuley: "We assume that the pointer involved in an indirectreference is not some numeric argument in the input or a quantity that dependson it. For example, in the max- flow problem the algorithm should not use anedge-capacity as a pointer—which is a reasonable condition. To enforce thisrestriction, one initially puts an invalid-pointer tag on every numeric argumentin the input. During the execution of an arithmetic instruction, the same tagis also propagated to the result if any operand has that tag. Trying to use amemory value with invalid-pointer tag results in error." [4, Page 1468]. represent the notion of line in the program. Then, theaction just defined allows for the representation of allcommands but the conditionals. The conditionals arerepresented as follows: depending on the value of X i onewants to jumps either to the line ℓ or to the line ℓ ′ ; this iseasily modelled by two different edges of respective sources H ( i ) = { ~x | x i = 0 } and H ( i ) c = { ~x | x i = 0 } . Definition 19.
Let M be a RAM machine. We define thetranslation | [ M ] | as the α ram -program with set of control states { , , . . . , L, L +1 } where each line ℓ defines (in the following, ⋆ ∈ { + , − , ×} and we write ℓ ++ the map ℓ ℓ + 1 ): • a single edge e of source X × { ℓ } and realised by: – ( Id , ℓ ++ ) if Inst M ( ℓ ) = skip ; – ( const i ( c ) , ℓ ++ ) if Inst M ( ℓ ) = X i := c ; – ( ⋆ i ( j, k ) , ℓ ++ ) if Inst M ( ℓ ) = X i := X j ⋆ X k ; – ( copy ( i, j ) , ℓ ++ ) if Inst M ( ℓ ) = X i := X j ; – ( copy ( i, ♯j ) , ℓ ++ ) if Inst M ( ℓ ) = X i := ♯ X j ; – ( copy ( ♯i, j ) , ℓ ++ ) if Inst M ( ℓ ) = ♯ X i := X j . • an edge e of source H ( k ) ×{ ℓ } realised by ( // i ( j, k ) , ℓ ++ ) if Inst M ( ℓ ) is X i := X j / X k ; • a pair of edges e, e c of respective sources H ( i ) × { ℓ } and H ( i ) c × { ℓ } and realised by respectively ( Id , ℓ ℓ ) and ( Id , ℓ ℓ ) , if the line is a conditional if X i = ℓ else ℓ .The translation | [ ι ] | of an input ι ∈ Z d is the point (¯ ι, where ¯ ι is the sequence ( ι , ι , . . . , ι k , , , . . . ) . Now, the main result for the representation of
RAM s is the following. The proof is straightforward, as eachinstruction corresponds to exactly one edge, except for theconditional case (but given a configuration, it lies in thesource of at most one of the two edges translating theconditional). Theorem 5.
The representation of
RAM s as α full -program isquantitatively sound w.r.t. the translation just defined.D. The Crew operation and PRAM s Based on the notion of
RAM , we are now able to considertheir parallelisation, namely
PRAM s . A PRAM M is givenas a finite sequence of RAM machines M , . . . , M p , where p is the number of processors of M . Each processor M i hasaccess to its own, private, set of registers ( X ik ) k > and a shared memory represented as a set of registers ( X k ) k > .One has to deal with conflicts when several processorstry to access the shared memory simultaneously. Wehere chose to work with the Concurrent Read, ExclusiveWrite ( crew ) discipline: at a given step at which severalprocessors try to write in the shared memory, only theprocessor with the smallest index will be allowed to do so.In order to model such parallel computations, we abstractthe crew at the level of monoids. For this, we supposethat we have two monoid actions M h G, R i y X × Y and M h H, Q i y X × Z , where X represents the sharedmemory. We then consider the subset ⊂ G × H of pairsof generators that potentially conflict with one another –the conflict relation. efinition 20 (Conflicted sum) . Let M h G, R i , M h G ′ , R ′ i be two monoids and ⊆ G × G ′ . The conflicted sum of M h G, R i and M h G ′ , R ′ i over , noted M h G, R i∗ M h G ′ , R ′ i ,is defined as the monoid with generators ( { }× G ) ∪ ( { }× G ′ ) and relations ( { } × R ) ∪ ( { } × R ′ ) ∪ { ( , e ) } ∪ { ( , e ′ ) }∪{ (cid:0) (1 , g )(2 , g ′ ) , (2 , g ′ )(1 , g ) (cid:1) | ( g, g ′ ) / ∈ } where , e , e ′ are the units of M h G, R i∗ M h G ′ , R ′ i , M h G, R i and M h G ′ , R ′ i respectively.In the particular case where G × H ′ ) ∪ ( H × G ′ ) ,with H, H ′ respectively subsets of G and G ′ , we will writethe sum M h G, R i ∗ H H ′ M h G ′ , R ′ i . Remark . When the conflict relation is empty, this definesthe usual direct product of monoids. This corresponds to thecase in which no conflicts can arise w.r.t. the shared memory.In other words, the direct product of monoids corresponds tothe parallelisation of processes without shared memory .Dually, when the conflict relation is full ( G × G ′ ), thisdefines the free product of the monoids. Definition 21.
Let α : M y X × Y be a monoid action. Wesay that an element m ∈ M is central relatively to α (or just central ) if the action of m commutes with the first projection π X : X × Y → X , i.e. α ( m ); π X = α ( m ) ; in other words m acts as the identity on X . Intuitively, central elements are those that will not affectthe shared memory. As such, only non-central elements require care when putting processes in parallel.
Definition 22.
Let M h G, R i y X × Y be an AMC . We note Z α the set of central elements and ¯ Z α ( G ) = { m ∈ G | n Z α } . Definition 23 (The
CREW of AMC s) . Let α : M h G, R i y X × Y and β : M h H, Q i y X × Z be AMC s. We define the
AMCCREW ( α, β ) : M h G, R i ∗ ¯ Z α ( G ) ¯ Z β ( G ′ ) M h G ′ , R ′ i y X × Y × Z by letting CREW ( α, β )( m, m ′ ) = α ( m ) ∗ β ( m ′ ) on elementsof G × G ′ , where: α ( m ) ∗ β ( m ′ ) = (cid:26) ∆; [ α ( m ); π Y , β ( m ′ )] if m ¯ Z α ( G ) , m ′ ∈ ¯ Z β ( G ′ ) , ∆; [ α ( m ) , β ( m ′ ); π Z ] otherwise,with ∆ : X × Y × Z → X × Y × X × Z ; ( x, y, z ) ( x, y, x, z ) . We can now define amc of PRAM s and thus the in-terpretations of PRAM s as abstract programs. For eachinteger p , we define the amc crew p ( α full ) . This allowsthe consideration of up to p parallel RAM s : the translationof such a RAM with p processors is defined by extendingthe translation of RAM s by considering a set of states equalto L × L × · · · × L p where for all i the set L i is the setof lines of the i -th processor.Now, to deal with arbitrary large PRAM s , i.e. witharbitrarily large number of processors, one considers thefollowing amc defined as a direct limit . Here and in the following, we denote by ; the sequential composition offunctions. I.e. f ; g denotes what is usually written g ◦ f . Definition 24 (The
AMC of PRAM s) . Let α : M y X × X be the AMC α full . The AMC of PRAM s is defined as α pram =lim −→ CREW k ( α ) , where CREW k − ( α ) is identified with a restric-tion of CREW k ( α ) through CREW k − ( α )( m , . . . , m k − ) CREW k ( α )( m , . . . , m k − , . Remark . We notice that the underlying space of the
PRAMAMC α pram is defined as the union ∪ n ∈ ω Z ω × ( Z ω ) n whichwe will write Z ω × ( Z ω ) ( ω ) . In practise a given α pram -program admitting a finite α pram representative will only useelements in CREW p ( α full ) , and can therefore be understood asa CREW p ( α ) -program. Theorem 6.
The representation of
PRAM s as α pram -programis quantitatively sound. This result, here stated for integer-valued
PRAM s , caneasily be obtained for real-valued PRAM s translated as α R full -programs. IV. E
NTROPY AND C ELLS
A. Topological Entropy
Topological Entropy is a standard invariant of dynam-ical system. It is a value representing the average ex-ponential growth rate of the number of orbit segmentsdistinguishable with a finite (but arbitrarily fine) precision.The definition is based on the notion of open covers.
Definition 25 (Open covers) . Given a topological space X , an open cover of X is a family U = ( U i ) i ∈ I of open subsets of X such that ∪ i ∈ I U i = X . A finite cover U is a cover whoseindexing set is finite. A subcover of a cover U = ( U i ) i ∈ I isa sub-family S = ( U j ) j ∈ J for J ⊆ I such that S is a cover,i.e. such that ∪ j ∈ J U j = X . We will denote by Cov( X ) (resp. FCov( X ) ) the set of all open covers (resp. all finite opencovers) of the space X . Definition 26.
An open cover U = ( U i ) i ∈ I , together with acontinuous function f : X → X , defines the inverse imageopen cover f − ( U ) = ( f − ( U i )) i ∈ I . Given two open covers U = ( U i ) i ∈ I and V = ( V j ) j ∈ J , we define their join U ∨ V asthe family ( U i ∩ V j ) ( i,j ) ∈ I × J . Remark . If U , V are finite, f − ( U ) and U ∨ V are finite.
Traditionally [17], entropy is defined for continuousmaps on a compact set. However, a generalisation ofentropy to non-compact sets can easily be defined byrestricting the usual definition to finite covers . This isthe definition we will use here. Definition 27.
Let X be a topological space, and U = ( U i ) i ∈ I be a finite cover of X . We define the quantity H X ( U ) =min { log (Card( J )) | J ⊆ I, ∪ j ∈ J U j = X } . In other words, if k is the cardinality of the smallestsubcover of U , H ( U ) = log ( k ) . This is discussed by Hofer [18] together with another generalisation basedon the Stone- ˇCech compactification of the underlying space. efinition 28.
Let X be a topological space and f : X → X be a continuous map. For any finite open cover U of X , define H k X ( f, U ) = k H X ( U ∨ f − ( U ) ∨ · · · ∨ f − ( k − ( U )) . One can show that the limit lim n →∞ H n X ( f, U ) existsand is finite; it will be noted h ( f, U ) . The topologicalentropy of f is then defined as the supremum of thesevalues, when U ranges over the set of all finite covers FCov( X ) . Definition 29.
Let X be a topological space and f : X → X be a continuous map. The topological entropy of f is definedas h ( f ) = sup U∈ FCov( X ) h ( f, U ) . B. Graphings and Entropy
We now need to define the entropy of deterministicgraphings . As mentioned briefly already, deterministicgraphings on a space X are in one-to-one correspondencewith partial dynamical systems on X . Thus, we only needto extend the notion of entropy to partial maps, and wecan then define the entropy of a graphing G as the entropyof its corresponding map [ G ] .Given a finite cover U , the only issue with partialcontinuous maps is that f − ( U ) is not in general a cover.Indeed, { f − ( U ) | U ∈ U} is a family of open sets bycontinuity of f but the union ∪ U ∈U f − ( U ) is a strictsubspace of X (namely, the domain of f ). It turns outthe solution to this problem is quite simple: we notice that f − ( U ) is a cover of f − ( X ) and now work with coversof subspaces of X . Indeed, U ∨ f − ( U ) is itself a coverof f − ( X ) and therefore the quantity H X ( f, U ) can bedefined as (1 / H f − ( X ) ( U ∨ f − ( U )) .We now generalise this definition to arbitrary iterationsof f by extending Definitions 28 and 29 to partial maps asfollows. Definition 30.
Let X be a topological space and f : X → X be a continuous partial map. For any finite open cover U of X , we define H k X ( f, U ) = k H f − k +1 ( X ) ( U ∨ f − ( U ) ∨ · · · ∨ f − ( k − ( U )) . The entropy of f is then defined as h ( f ) =sup U∈ FCov( X ) h ( f, U ) , where h ( f, U ) is again defined as thelimit lim n →∞ H n X ( f, U ) . We now consider the special case of a graphing G withset of control states S G . For an intuitive understanding,one can think of G as the representation of a PRAM machine. We focus on the specific open cover indexed bythe set of control states, i.e. S = ( X × { s } s ∈ S G ) , and call it the states cover . We will now show how the partial entropy H k ( G, S ) is related to the set of admissible sequence ofstates . Let us define those first. Definition 31.
Let G be a graphing, with set of controlstates S G . An admissible sequence of states is a sequence s = s s . . . s n of elements of S G such that for all i ∈{ , , . . . , n − } there exists a subset C of X – i.e. a set ofconfigurations – such that G contains an edge from C × { s i } to a subspace of X × { s i +1 } . Example . As an example, let us consider the very simplegraphing with four control states a, b, c, d and edges from X ×{ a } to X × { b } , from X × { b } to X × { c } , from X × { c } to X × { b } and from X × { c } to X × { d } . Then the sequences abcd and abcbcbc are admissible, but the sequences aba , abcdd ,and abcba are not. Lemma 2.
Let G be a graphing, and S its states cover. Thenfor all integer k , the set Adm k ( G ) of admissible sequences ofstates of length k > is of cardinality k.H k ( G, S ) . A tractable bound on the number of admissible se-quences of states can be obtained by noticing that thesequence H k ( G, S ) is sub-additive , i.e. H k + k ′ ( G, S ) H k ( G, S ) + H k ′ ( G, S ) . A consequence of this is that H k ( G, S ) kH ( G, S ) . Thus the number of admissiblesequences of states of length k is bounded by k H ( G, S ) .We now study how the cardinality of admissible sequencescan be related to the entropy of G . This is deduced fromLemma 2 and the following general result (which does notdepend on the chosen cover). Lemma 3.
For all ǫ > and all cover U , there exists a naturalnumber N such that ∀ k > N , H k ( G, U ) < h ([ G ]) + ǫ . The two previous lemmas combine to give the following.
Lemma 4.
Let G be a graphing. Then Card(Adm k ( G )) = O (2 k.h ([ G ]) ) as k → ∞ .C. Cells Decomposition Now, let G be a deterministic graphing with its statecover S . We fix k > and consider the partition ( C [ s ]) s ∈ Adm k ( G ) of the space [ G ] − k +1 ( X ) , where the sets C [ s ] = C [( s s . . . s k − , s k )] are defined inductively asfollow: • C [ s , s ] is the set { x ∈ X | [ G ]( x, s ) ∈ X × { s }} ; • C [( s s . . . s k − , s k )] is the set { x ∈ X | ∀ i ∈{ , . . . , k } , [ G ] i − ( x, s ) ∈ X × { s i }} .This decomposition splits the set of initial configurationsinto cells satisfying the following property: for any twoinitial configurations contained in the same cell C [ s ] , the k -th first iterations of G go through the same admissiblesequence of states s . Definition 32.
Let G be a deterministic graphing, with itsstate cover S . Given an integer k , we define the k -th celldecomposition of X along G as the partition { C [ s ] | s ∈ Adm k ( G ) } . Then Lemma 2 provides a bound on the cardinality ofthe k -th cell decomposition. Using the results in the previ-ous section, we can then obtain the following proposition. Proposition 2.
Let G be a deterministic graphing, withentropy h ( G ) . The cardinality of the k -th cell decomposition of X w.r.t. G , as a function c ( k ) of k , is asymptotically boundedby g ( k ) = 2 k.h ([ G ]) , i.e. c ( k ) = O ( g ( k )) . We also state another bound on the number of cellsof the k -th cell decomposition, based on the state coverntropy, i.e. the entropy with respect to the state coverrather than the usual entropy which takes the supremum ofcover entropies when the cover ranges over all finite coversof the space. This is a simple consequence of Lemma 2. Proposition 3.
Let G be a deterministic graphing. We considerthe state cover entropy h ([ G ]) = lim n →∞ H n X ([ G ] , S ) where S is the state cover. The cardinality of the k -th cell decompo-sition of X w.r.t. G , as a function c ( k ) of k , is asymptoticallybounded by g ( k ) = 2 k.h ([ G ]) , i.e. c ( k ) = O ( g ( k )) . V. E
NTROPIC C OTREES AND THE M AIN L EMMA
A. Lower Bounds through the Milnor-Thom theorem
The results stated in the last section can be used to provelower bounds in several models. These results rely on twoingredients: the above bounds on the cardinality of the k -th cell decomposition, and the Milnor-Thom theorem.The Milnor-Thom theorem, which was proven indepen-dently by Milnor [19] and Thom [20], states bounds on thesum of the Betti numbers (i.e. the rank of the homologygroups) of an algebraic variety. This theorem providesbounds on the number of connected components (i.e. the -th Betti number β ( V ) ) of a semi-algebraic variety V .We here use the statement of the Milnor-Thom theoremas given by Ben-Or [2, Theorem 2]. Theorem 7.
Let V ⊆ R n be a set defined by polynomialin ·equations ( n, m, h ∈ N ): { q i ( ~x ) = 0 | i m }∪ { p i ( ~x ) > | i s }∪ { p i ( ~x ) | s + 1 i h } . Then β ( V ) is at most d (2 d − n + h − , where d =max { , deg( q i ) , deg( p j ) } . The lower bounds proofs then proceed by the followingproof strategy:1) consider an algebraic model of computation, anddefine the corresponding amc ;2) show that the cells in the k -th cell decomposition aresemi-algebraic sets defined by systems of equations E with explicit upper bounds on the number of equationsand the degrees of the polynomials;3) bound the number of connected components of eachcell by the Milnor-Thom theorem;4) given an algebraic problem (e.g. a subset of R k ),deduce lower bounds on the length of the computa-tions deciding that problem based on its number ofconnected components.Among the lower bound proofs using this proof strategy,we point out Steele and Yao lower bounds on algebraicdecision trees [1], and Mulmuley’s proof of lower boundson “ PRAM s without bit operations” [4]. These results donot use the notion of entropy. Due to space constraints,we do not detail these in this paper.We will now explain how this method can be refinedfollowing Ben-Or’s proof of lower bounds for algebraic computational trees. Indeed, while Mulmuley’s [4] was notlater improved upon, Steele and Yao’s lower bounds wereextended by Ben-Or [2] to encompass algebraic computa-tional trees with sums, substractions, products, divisionsand square roots. The technique of Ben-Or improves onSteele and Yao in that it provides a method to dealwith divisions and square roots. We here abstract thismethod by considering k -th entropic co-trees which area refinement of the k -th cell decomposition. This allowsus to recover Ben-Or’s result, capture Cucker’s proof that NC R = Ptime R , and to strengthen Mulmuley’s resultby allowing the machines considered to use divisions andsquare roots. B. Entropic co-trees
The principle underlying the improvement of Ben-Or onSteele and Yao consists in adding additional variables toavoid using the square root or division, obtaining in thisway a system of polynomial equations instead of a singleequation for a given cell in the k -th cell decomposition.For instance, instead of writing the equation p/q < , onedefines a fresh variable r and considers the system { p = qr ; r < } .To adapt it to graphings, we consider the notion of entropic co-tree of a graphing that generalises the k -th celldecomposition to account for the instructions used at eachstep of the computation.As we explained in Remark 3, a given PRAM is inter-preted as a crew p ( α R full ) -program for a fixed integer p (the number of processors). It is therefore enough tostate the following definitions and results for the amccrew p ( α R full ) to apply them to the interpretations ofarbitrary PRAM s . Definition 33 ( k -th entropic co-tree) . Consider a deterministic
CREW p ( α R full ) -graphing representative T , and fix an element ⊤ of the set of control states. We can define the k -th entropicco-tree of T along ⊤ and the state cover inductively: • k = 0 , the co-tree CO T ( T ) is simply the root n ǫ = R n × {⊤} ; • k = 1 , one considers the preimage of n ǫ through T , i.e. T − ( R n × {⊤} ) the set of all non-empty sets α ( m e ) − ( R n × {⊤} ) and intersects it pairwise withthe state cover, leading to a finite family (of cardinalitybounded by the number of states multiplied by thenumber of edges fo T ) ( n ie ) i defined as n i = T − ( n ǫ ) ∩ R n ×{ i } . The first entropic co-tree CO T ( T ) of T is thenthe tree defined by linking each n ie to n ǫ with an edgelabelled by m e ; • k + 1 , suppose defined the k -th entropic co-tree of T ,defined as a family of elements n π e where π is a finitesequence of states of length at most k and e a sequenceof edges of T of the same length, and where n π e and n π ′ e ′ are linked by an edge labelled f if and only if π ′ = π.s and e ′ = f. e where s is a state and f an edge of T . Weconsider the subset of elements n π e ′ where π is exactlyf length k , and for each such element we define newvertices n π.se. e ′ defined as α ( m e ) − ( n π e ′ ) ∩ R n ×{ s } when itis non-empty. The k +1 -th entropic co-tree CO T k +1 ( T ) isdefined by extending the k -th entropic co-tree CO T k ( T ) ,adding the vertices n π.se. e ′ and linking them to n π e ′ with anedge labelled by e . We can easily obtain bounds on the size of the cotrees,refining the bounds on the k th cell decomposition. Proposition 4.
Let G be a deterministic CREW p ( α R full ) -graphing with a finite set of edges E , and Seq k ( E ) the set oflength k sequences of edges in G . We consider the state coverentropy h ([ G ]) = lim n →∞ H n X ([ G ] , S ) where S is the statecover. The cardinality of the length k vertices of the entropicco-tree of G , as a function c ( k ) of k , is asymptotically boundedby g ( k ) = Card(Seq k ( E )) . k.h ([ G ]) , which is itself boundedby Card( E ) . k.h ([ G ]) .C. The main lemma This definition formalises a notion that appears moreor less clearly in the work of Steele and Yao, and of Ben-Or, as well as in the proof by Mulmuley. The vertices forpaths of length k in the k -th co-tree corresponds to the k -th cell decomposition, and the corresponding path definesthe polynomials describing the semi-algebraic set decidedby a computational tree. While in Steele and Yao andMulmuley’s proofs, one obtain directly a polynomial foreach cell, we here need to construct a system of equationsfor each branch of the co-tree.Given a crew p ( α R full ) -graphing representative G wewill write ∂ √ G the maximal value of n for which aninstruction n √ i ( j ) appears in the realiser of an edge of G . Lemma 5.
Let G be a computational graphing represen-tative with edges realised only by generators of the AMCCREW p ( α R full ) , and Seq k ( E ) the set of length k sequences ofedges in G . Suppose G computes the membership problemfor W ⊆ R n in k steps, i.e. for each element of R n , π S ( G k ( x )) = ⊤ if and only if x ∈ W . Then W is a semi-algebraic set defined by at most Card(Seq k ( E )) . k.h ([ G ]) systems of pk equations of degree at most max(2 , ∂ √ G ) andinvolving at most pk + n variables. The proof of this theorem is long but simple to under-stand. We define, for each vertex of the k -th entropic co-tree, a system of algebraic equations (each of degree atmost 2). The system is defined by induction on k , and usesthe information of the specific instruction used to extendthe sequence indexing the vertex at each step. For instance,the case of division follows Ben-Or’s method, introducing afresh variable and writing down two equations as explainedin Section V-B. VI. R
ECOVERING RESULTS FROM THE LITERATURE
A. Ben-Or’s theorem
We now recover Ben-Or result by obtaining a boundon the number of connected components of the subsets W ⊆ R n whose membership problem is computed by agraphing in less than a given number of iterations. Thistheorem is obtained by applying the Milnor-Thom theoremon the obtained systems of equations to bound the numberof connected components of each cell. Notice that in thiscase p = 1 and ∂ √ G = 2 since the model of algebraiccomputation trees use only square roots. A mode generalresult holds for algebraic computation trees extended witharbitrary roots, but we here limit ourselves here to theoriginal model. Theorem 8.
Let G be a computational α R full -graphing repre-sentative translating an algebraic computational tree, Seq k ( E ) the set of length k sequences of edges in G . Suppose G computes the membership problem for W ⊆ R n in k steps.Then W has at most Card(Seq k ( E )) . k.h ([ G ])+1 k + n − connected components. Since a subset computed by a tree T of depth k iscomputed by | [ T ] | in k steps by Theorem 3, we get asa corollary the original theorem by Ben-Or relating thenumber of connected components of a set W and thedepth of the algebraic computational trees that computethe membership problem for W . Corollary 1 ([2, Theorem 5]) . Let W ⊆ R n be any set, andlet N be the maximum of the number of connected componentsof W and R n \ W . An algebraic computation tree computingthe membership problem for W has height Ω(log N ) .Remark . In the case of algebraic
PRAM s discussed in thenext sections, the k -th entropic co-tree CO T k ( T )[ M ] of a ma-chine M defines an algebraic computation tree which followsthe k -th first steps of computation of M . I.e. the algebraiccomputation tree CO T k ( T )[ M ] approximate the computationof M in such a way that M and CO T k ( T )[ M ] behave in theexact same manner in the first k steps. B. Cucker’s theorem
Cucker’s proof considers the problem defined as thefollowing algebraic set.
Definition 34.
Define F er to be the set: { x ∈ R ω | | x | = n ⇒ x n + x n = 1 } , where | x | = max { n ∈ ω | x n = 0 } . It can be shown to lie within
Ptime R , i.e. it is decided bya real Turing machine [21] – i.e. working with real numbersand real operations –, running in polynomial time. Theorem 9 (Cucker ([3], Proposition 3)) . The problem F er belongs to P TIME R . We now prove that F er is not computable by an alge-braic circuit of polylogarithmic depth. The proof followsCucker’s argument, but uses the lemma proved in theprevious section. Theorem 10 (Cucker ([3], Theorem 3.2)) . No algebraic circuitof depth k = log i n and size kp compute F er .roof. For this, we will use the lower bounds result obtained inthe previous section. Indeed, by Theorem 4 and Lemma 5, anyproblem decided by an algebraic circuit of depth k is a semi-algebraic set defined by at most Card(Seq k ( E )) . k.h ([ G ]) systems of k equations of degree at most max(2 , ∂ √ G ) = 2 (since only square roots are allowed in the model) and in-volving at most k + n variables. But the curve F R n defined as { x n + x n − | x , x ∈ R } is infinite. As a consequence,one of the systems of equations must describe a set containingan infinite number of points of F R n .This set S is characterized, up to some transformations onthe set of equations obtained from the entropic co-tree, by afinite system of in ·equalities of the form s ^ i =1 F i ( X , X ) = 0 ∧ t ^ j =1 G j ( X , X ) < , where t is bounded by kp and the degree of the polynomials F i and G i are bounded by k . Moreover, since F R n is acurve and no points in S must lie outside of it, we musthave s > .Finally, the polynomials F i vanish on that infinite subsetof the curve and thus in a 1-dimensional component of thecurve. Since the curve is an irreducible one, this impliesthat every F i must vanish on the whole curve. Using thefact that the ideal ( X n + X n − is prime (and thusradical), we conclude that all the F i are multiples of X n + X n − which is impossible if their degree is bounded by log i n as it is strictly smaller than n . VII. A
PROOF THAT NC Z = P TIME
In this section, we provide a new presentation of a resultof Mulmuley which is part of his lower bounds for “pramswithout bit operations”. The idea is to encode a specificdecision problem and the run of a
PRAM as two specificsubsets of the same space and show that no short run of themachine can define the set of all instances of the decisionproblem. More specifically, consider the problem maxflow :given a weighted graph, find the maximal flow from asource edge to a target edge. This is an optimization prob-lem. It can be turned into a decision problem by addinga new variable z —a threshold—and asking whether thereexists a solution greater than z . This decision problem isknown to be Ptime -complete [11].
A. Geometric Interpretation of Optimization Problems
Let P opt be an optimization problem on R d . Solving P opt on an instance t amounts to optimizing a function f t ( · ) over a space of parameters. We note Max P opt ( t ) thisoptimal value. An affine function Param : [ p ; q ] → R d iscalled a parametrization of P opt . Such a parametrizationdefines naturally a decision problem P dec : for all ( x, y, z ) ∈ Z , ( x, y, z ) ∈ P dec iff z > , x/z ∈ [ p ; q ] and y/z ≤ Max P opt ◦ Param ( x/z ) .In order to study the geometry of P dec in a way thatmakes its connection with P opt clear, we consider theambient space to be R , and we define the ray [ p ] of a point p as the half-line starting at the origin and containing p . The projection Π( p ) of a point p on a plane is theintersection of [ p ] and the affine plane A of equation z = 1 .For any point p ∈ A , and all p ∈ [ p ] , Π( p ) = p . It is clearthat for ( p, p ′ , q ) ∈ Z × N + , Π(( p, p ′ , q )) = ( p/q, p ′ /q, .The cone [ C ] of a curve C is the set of rays of pointsof the curve. The projection Π( C ) of a surface or a curve C is the set of projections of points in C . We note Front the frontier set
Front = { ( x, y, ∈ R | y = Max P opt ◦ Param ( x ) } . and we remark that [ Front ] = { ( x, y, z ) ∈ R × R + | y/z = Max P opt ◦ Param ( x/z ) } . A machine M decides the problem P dec in k steps if thepartition of accepting cells in Z induced by the machine– i.e. the k -th cell decomposition – is finer than the onedefined by the problem’s frontier [ Front ] (which is definedby the equation y/z ≤ Max P opt ◦ Param ( x/z ) ). Parametric Complexity.
We now further restrict the classof problems we are interested in: we will only consider P opt such that Front is simple enough.
Definition 35.
We say that
Param is an affine parametrization of P opt if Max P opt ◦ Param is convex, piecewise linearwith breakpoints λ < · · · < λ ρ , and such that all ( λ i ) i and (Max P opt ◦ Param ( λ i )) i are rational. The parametriccomplexity ρ ( Param ) is the number of breakpoints ρ . The bitsize of the parametrization is the maximum of the bitsizesof the numerators and denominators of the coordinates of thebreakpoints of Max P opt ◦ Param . An optimization problem admitting an affineparametrization of complexity ρ is thus representedby a quite simple surface [ Front ] : the cone of the graph ofa piecewise affine function, constituted of ρ segments. Wecall such a surface is a ρ -fan and define its bitsize as β if all its breakpoints are rational and the bitsize of theircoordinates is less than β .The restriction to such optimization problems seemsquite dramatic when understood geometrically. Nonethe-less, maxflow admits such a parametrization. Theorem 11 (Murty [22], Carstensen [23]) . There exists anaffine parametrization of bitsize O ( n ) and complexity Ω( n ) of the maxflow problem for directed and undirected networks,where n is the number of nodes in the network. Surfaces and fans.
An algebraic surface in R is a surfacedefined by an equation of the form p ( x, y, z ) = 0 where p is a polynomial. If S is a set of surfaces S i , each definedby a polynomial p i , the total degree of S is defined as thesum of the degrees of polynomials p i .Let K be a compact of R delimited by algebraicsurfaces and S be a finite set of algebraic surfaces of totaldegree δ . We can assume that K is delimited by two affineplanes of equation z = µ and z = 2 µ z and the cone of arectangle { ( x, y, | | x | , | y | µ x,y } , by taking any suchcompact containing K and adding the surfaces bounding K to S . S defines a partition of K by considering maximalompact subspaces of K whose boundaries are includedin surfaces of S . Such elements are called the cells of thedecomposition associated to S . Definition 36.
Let K be a compact of R . A finite set ofsurfaces S on K separates a ρ -fan Fan on K if the partitionon Z ∩ K induced by S is finer than the one induced by Fan . A major technical achievement of Mulmuley [4] – notexplicitly stated – was to prove the following theorem, ofpurely geometric nature. We refer to the long version ofthis work for a detailed proof of this result. Theorem 12 (Mulmuley) . Let S be a finite set of algebraicsurfaces of total degree δ . There exists a polynomial P suchthat, for all ρ > P ( δ ) , S does not separate ρ -fans.B. Strengthening Mulmuley’s result We will now prove our strengthening of Mulmuley’slower bounds for “
PRAM s without bit operations” [4]. Forthis, we will combine the results from previous sections toestablish the following result. Theorem 2.
Let G be a deterministic graphing interpreting a PRAM with O ((log N ) c ) processors, where N is the length ofthe inputs and c any positive integer.Then G does not decide maxflow in O ((log N ) c ) steps. So, let M be an integer-valued PRAM . We can associateto it a real-valued
PRAM ˜ M such that M and ˜ M accept thesame (integer) values, and the ratio between the runningtime of the two machines is a constant. Indeed: Proposition 5.
Euclidian division can be computed by aconstant time real-valued
PRAM .Proof of Theorem 2.
Suppose now that | [ M ] | has a finite setof edges E . Then | [ ˜ M ] | has too has a finite set of edgeof cardinality O (Card( E )) . Since the running time of theinitial PRAM over integers is equal, up to a constant, to thecomputation time of the
CREW p ( α R full ) -program | [ ˜ M ] | , wededuce that if M computes maxflow in k steps, then | [ ˜ M ] | computes maxflow in at most Ck steps where C is a fixedconstant.By Lemma 5, the problem decided by | [ ˜ M ] | in Ck stepsdefines a system of equations separating the integral inputsaccepted by M from the ones rejected. I.e. if M computes maxflow in Ck steps, then this system of equations definesa set of algebraic surfaces that separate the ρ -fan defined by maxflow . Moreover, this system of equation has a total degreebounded by Ck max(2 , ∂ √ G )2 p × O (Card( E )) × k.h ( | [ ˜ M ] | ) .By Theorem 11 and Theorem 12, there exists a polynomial P such that a finite set of algebraic surfaces of total degree δ cannot separate the Ω( n ) -fan defined by maxflow as longas Ω( n ) > P ( δ ) . But here the entropy of G is O ( p ) , as theentropy of a product f × g satisfies h ( f × g ) h ( f ) + h ( g ) For the purpose of double-blind reviews, we do not provide an explicitreference for the moment. [24]. Hence δ = O (2 p k ) , contradicting the hypotheses that p = 2 O ((log N ) c ) and k = 2 O ((log N ) c ) . This has Theorem 1 as a corollary, which shows that theclass NC Z does not contain maxflow , and hence is distinctfrom Ptime . The question of how this class relates to NC is open: indeed, while bit extractions cannot be performedin constant time by our machines (a consequence ofTheorem 5), they can be simulated in logarithmic time. R EFERENCES [1] J. M. Steele and A. Yao, “Lower bounds for algebraic decisiontrees,”
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Transactions of the American Mathematical Society A PPENDIX
Proof of Proposition 1.
A computation tree defines an α act -graphing [ T ] , and the natural α act -graphing representativeobtained from the inductive definition of [ T ] is clearly an α act -treeing because T is a tree. That this treeing representsfaithfully the computational tree T raises no difficulty.Let us now show that the membership problem of a subset W ⊆ R n that can be decided by a computational α act -treeing is also decided by an algebraic computation tree T .We prove the result by induction on the number of states ofthe computational α act -treeing. The initial case is when T theset of states is exactly { , ⊤ , ⊥} with the order defined by < ⊤ and < ⊥ and no other relations. This computational α act -treeing has at most 2 edges, since it is deterministic andthe source of each edge is a subset among R ω , R ωk > , R ωk , R ωk> , R ωk< , R ωk =0 , and R ωk =0 .We first treat the case when there is only one edge of source R n . An element ( x , . . . , x n ) ∈ R n is decided by T if themain representative (( x , . . . , x n , , . . . ) , is mapped to ⊤ .Since there is only one edge of source the whole space, eitherthis edge maps into the state ⊤ and the decided subset W isequal to R n , or it maps into ⊥ and the subset W is empty. Inboth cases, there exists an algebraic computation tree deciding W . For the purpose of the proof, we will however constructa specific algebraic computation tree, namely the one thatfirst computes the right expression and then accepts or rejects.I.e. if the only edge mapping into ⊤ (resp. ⊥ ) is realisedby an element m in the AMC of algebraic computation treeswhich can be written as a product of generators g , . . . , g k , weconstruct the tree of height k + 1 that performs (in that order)the operations corresponding to g , g , etc., and then answers"yes" (resp. "no").Now, the case where there is one edge of source a strictsubspace, e.g. R ωk > (all other cases are treated in a similarmanner) and mapping into ⊤ (the other case is treated by sym-metry). First, let us remark that if there is no other edge, onecould very well add an edge to T mapping into ⊥ and realisedby the identity with source the complementary subspace R ωk< .We build a tree as follows. First, we test whether the variable x k is greater or equal to zero; this node has two childrencorresponding to whether the answer to the test is "yes" or"no". We now construct the two subtrees corresponding tothese two children. The branch corresponding to "yes" isdescribed by the edge of source R ωk > : we construct the tree of height k + 1 performing the operations corresponding to thegenerators g , g , etc. whose product defined the realiser m of e , and then answers "yes" (resp. "no") if the edge e maps intothe state ⊤ (resp. ⊥ ). Similarly, the other subtree is describedby the realiser of the edge of source R ωk< .The result then follows by induction, plugging small sub-trees as described above in place of the leaves of smallersubtrees. Proof of Lemma 2.
We show that the set
Adm k ( G ) of admis-sible sequences of states of length k has the same cardinalityas the smallest subcover of S ∨ [ G ] − ( S ) ∨· · ·∨ [ G ] − ( k − ( S )) .Hence H k ( G, S ) = k log (Card(Adm k ( G ))) , which impliesthe result.The proof is done by induction. As a base case, we considerthe set of Adm ( G ) of length admissible sequences of statesand the cover V = S ∨ [ G ] − ( S ) of D = [ G ] − ( X ) . Anelement of V is an intersection X × { s } ∩ [ G ] − ( X × { s } ) ,and is therefore equal to C [ s , s ] ×{ s } where C [ s , s ] ⊂ X is the set { x ∈ X | [ G ]( x, s ) ∈ X ×{ s }} . This set is empty ifand only if the sequence s s belongs to Adm ( G ) . Moreover,given another sequence of states s ′ s ′ , the sets C [ s , s ] and C [ s , s ] are disjoint. Hence a set C [ s , s ] is removable fromthe cover V if and only if s s is not admissible. This provesthe case k = 2 .The step for the induction is similar. One considers the par-tition S k = W − ( k − i =0 [ G ] i ( S ) as S k − ∨ [ G ] − ( k − ( S ) . By thesame argument, one shows elements of S k − ∨ [ G ] − ( k − ( S ) are of the form C [ s = ( s s . . . s k − ) , s k ] × { s } where C [ s , s k ] is the set { x ∈ X | ∀ i = 2 , . . . , k, [ G ] i − ( x, s ) ∈ X × { s i }} . Again, these sets C [ s , s k ] are pairwise disjointand empty if and only if the sequence s s . . . s k − , s k is notadmissible. Proof of Lemma 3.
Let us fix some ǫ > . Notice that if welet H k ( G, U ) = H ( U ∨ [ G ] − ( U ) ∨ · · · ∨ [ G ] − ( k − ( U ))) ,the sequence H k ( U ) satisfies H k + l ( U ) H k ( U ) + H l ( U ) .By Fekete’s lemma on subadditive sequences, this impliesthat lim k →∞ H k /k exists and is equal to inf k H k /k . Thus h ([ G ] , U ) = inf k H k /k .Now, the entropy h ([ G ]) is defined as sup U lim k →∞ H k ( U ) /k . This then rewrites as sup U inf k H k ( U ) /k . We can conclude that h ([ G ]) > inf k H k ( U ) /k for all finite open cover U .Since inf k H k ( U ) /k is the limit of the sequence H k /k , thereexists an integer N such that for all k > N the followinginequality holds: | H k ( U ) /k − inf k H k ( U ) /k | < ǫ , whichrewrites as H k ( U ) /k − inf k H k ( U ) /k < ǫ . From this wededuce H k ( U ) /k < h ([ G ]) + ǫ , hence H k ( G, U ) < h ([ G ]) + ǫ since H k ( G, U ) = H k ( G, U ) . Proof of Proposition 4.
For a fixed sequence ~e , the numberof elements n π~e of length m in CO T k ( T ) is bounded bythe number of elements in the m -th cell decomposition of T , and is therefore bounded by g ( m ) = 2 m.h ([ T ]) by 3.The number of sequences ~e is bounded by Card(Seq k ( E )) nd therefore the size of CO T k ( T ) is thus bounded by Card(Seq k ( E )) . ( k +1) .h ([ T ]) . Proof of Lemma 5. If G computes the membership problemfor W in k steps, it means W can be described as the union ofthe subspaces corresponding to the nodes n π e with π of length k in CO T k ( T ) . Now, each such subspace is an algebraic set,as it can be described by a set of polynomials as follows.Finally let us note that, as in Mulmuley’s work [4], since inour model the memory pointers are allowed to depend only onthe nonnumeric parameters, indirect memory instructions canbe treated as standard – direct – memory instructions. In otherwords, whenever an instruction involving a memory pointer isencountered during the course of execution, the value of thepointer is completely determined by nonnumerical data, andthe index of the involved registers is completely determined,independently of the numerical inputs.We define a system of equations ( E e i ) i for each node n π e of the entropic co-tree CO T k ( T ) . We explicit the constructionfor the case p = 1 , i.e. for the AMC CREW ( α R full ) = α R full ;the case for arbitrary p is then dealt with by following theconstruction and introducing p equations at each step (onefor each of the p instructions in α R full corresponding to anelement of CREW p ( α R full ) ). This is done inductively on thesize of the path ~e , keeping track of the last modificationsof each register. I.e. we define both the system of equations ( E e i ) i and a function h ( e ) : R ω + ⊥ → ω (which is almosteverywhere null) . For an empty sequence, the system ofequations is empty, and the function h ( ǫ ) is constant, equalto .Suppose now that ~e ′ = ( e , . . . , e m , e m + 1) , with ~e =( e , . . . , e m ) , and that one already computed ( E e i ) i > m and thefunction h ( e ) . We now consider the edge e m +1 and let ( r, r ′ ) be its realizer. We extend the system of equations ( E e i ) i > m bya new equation E m +1 and define the function h ( e ′ ) as follows: • if r = + i ( j, k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = x h ( e ′ )( j ) j + x h ( e ′ )( k ) k ; • if r = − i ( j, k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = x h ( e ′ )( j ) j − x h ( e ′ )( k ) k ; • if r = × i ( j, k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = x h ( e ′ )( j ) j × x h ( e ′ )( k ) k ; • if r = / i ( j, k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = x h ( e ′ )( j ) j /x h ( e ′ )( k ) k ; • if r = + ci ( k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = c + x h ( e ′ )( k ) k ; The use of ⊥ is to allow for the creation of fresh variables not relatedto a register. • if r = − ci ( k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = c − x h ( e ′ )( k ) k ; • if r = × ci ( k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = c × x h ( e ′ )( k ) k ; • if r = / ci ( k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = c/x h ( e ′ )( k ) k ; • if r = n √ i ( k ) , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = i , and h ( e ′ )( x ) = h ( e )( x ) otherwise; then E m +1 is x h ( e ′ )( i ) i = n q x h ( e ′ )( k ) k ; • if r = Id, the source of the edge e q is of the form { ( x , . . . , x n + ℓ ) ∈ R n + ℓ | P ( x k ) } × { i } where P compares the variable x k with : – if P ( x k ) is x k = 0 , h ( e ′ )( x ) = h ( e )( x ) + 1 if x = ⊥ , and h ( e ′ )( x ) = h ( e )( x ) otherwise then E m +1 is x h ( e ′ )( ⊥ ) ⊥ x h ( e ′ )( k ) k − ; – otherwise we set h ( e ′ ) = h ( e ) and E m +1 equal to P .We now consider the system of equations ( E i ) ki =1 definedfrom the path e of length k corresponding to a node n π e of the k -th entropic co-tree of G . This system consists in k equations of degree at most max(2 , ∂ √ G ) and containingat most k + n variables, counting the variables x , . . . , x n corresponding to the initial registers, and adding at most k additional variables since an edge of ~e introduces at most onefresh variable. Since the number of vertices n π e is bounded by Card(Seq k ( E )) . k.h ([ G ]) by 4, we obtained the stated resultin the case p = 1 .The case for arbitrary p is then deduced by noticing thateach step in the induction would introduce at most p newequations and p new variables. The resulting system thuscontains at most pk equations of degree at most max(2 , ∂ √ G ) and containing at most pk + n variables. Proof of Theorem 8.
By Lemma 5 (using the fact that p =1 and ∂ √ G = 2 ), the problem W decided by G in k steps is described by at most Card(Seq k ( E )) . k.h ([ G ]) sys-tems of k equations of degree involving at most k + n variables. Applying Theorem 7, we deduce that each suchsystem of in ·equations (of k equations of degree in R k + n ) describes a semi-algebraic variety S such that β ( S ) < . ( n + k )+ k − . This begin true for each of the Card(Seq k ( E )) . k.h ([ G ]) cells, we have that β ( W ) < Card(Seq k ( E )) . k.h ([ G ])+1 k + n − . Proof of Corollary 1.
Let T be an algebraic computation treecomputing the membership problem for W , and consider thecomputational treeing [ T ] . Let d be the height of T ; bydefinition of [ T ] the membership problem for W is computedin exactly d steps. Thus, by the previous theorem, W has atmost Card(Seq k ( E )) . d.h ([ T ])+1 d + n − connected compo-nents. As the interpretation of an algebraic computational tree, h ([ T ]) is at most equal to , and Card(Seq k ( E )) is boundedby d . Hence N d . d +1 n − d , i.e. d = Ω(log N ) . roof of Proposition 5. To compute p//q , where p, q ∈ Z ,consider the real-valued machine such that the i th processorcomputes x = p/q − i and if < x , writes ii