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Symposium in Honour ofLauri Hella’s 60th birthday
Tampere, Finland, 4-6 July 2018
On Fragments of HigherOrder Logics that on Finite
Structures Collapse to aLower Order
Jose Marıa Turull-Torres
Universidad Nacional de LaMatanza, Argentina
In-progress joint work with FlavioFerrarotti and Senen Gonzalez.
1
Contents:
• Two Motivating Examples in ThirdOrder Logic: 3.
Hypercube graphs (two definitions)and Formula-Value query
• 0: Higher Order Logics: 17.
• 1: A General Schema of TO For-mulas: 38.
• 2: Downward polynomially boundedRelations; HOi,P: 60.
• 3: Valuating Relations of Poly-logarithmic Cardinality: 71.
• 4: Beyond Second Order; SATQBF:96.
• 5: Beyond Third Order; SATQBF(Σ2j):
118.
2
Two Motivating
Examples in
Third Order Logic
3
Example 1 in HO3:Hypercube graphs
An n-hypercube graph Qn, is anundirected graph whose vertices arebinary n-tuples. Two vertices ofQn are adjacent iff they differ inexactly one bit.
Note that we can build an (n+1)-cube Qn+1 starting with two iso-morphic copies of an n-cube Qn
and adding edges between correspond-ing vertices.
That is, multiplying an n-cubegraph by K2.
4
Using this fact, we can define inTO (HO3) the class of hypercubegraphs, by saying that:
• there is a sequence of graphs (i.e.,a third order linear digraph, whereevery TO node is an undirected(SO) graph)
• which starts with the graph K2,ends with a graph which is equalto the input graph, and such that
• every graph G2 in the sequenceresults from finding two total, in-jective functions f1, f2 from theprevious graph G1, so that
5
– f1 and f2 induce in G2 twoisomorphic copies of G1,
– the images of those functionsdefine a partition in the ver-tex set of G2, and
– there is an edge inG2 betweenthe images f1(x) and f2(x) ofevery node x in G1.
6
Actually, the expressive power ofHO3 is not required to character-ize hypercube graphs, since they canbe recognized in NP, and hence inESO.
7
Nevertheless, to define the classof hypercube graphs inESO seemsto be more challenging than to de-fine it in HO3.
(see the SO formula for the firststrategy considered for hypercubegraphs in [Ferrarotti, Ren, Turull-Torres, 2014], and Remark 4.1 there,indicating the way to translate it toan ESO formula).
8
A Second Definition ofHypercube graphs
Another definition of hypercubegraphs that yields a simple (TO)formula is the following.
9
We say that there is a proper nonempty subset V ′ of the vertex setV of the input graphG, and a (TO)bijective function F : V → P(V ′)(i.e., the power set of V ′), s. t.
for every pair of nodes x and y inG, there is an edge between them
iff
F(x) can be obtained from F(y)by adding or removing a single el-ement (note that V ′ is necessarilyof size log2 |V |).
Note that the corresponding SOformula is not so intuitive (see [Fer-rarotti, Ren, Turull-Torres, 2014]).
10
The SO formula that expressesthe second strategy is in the class∑1
2.
The existence of a formula in∑1
1that expresses this strategy is un-likely, since we must express thatevery subset S of V is identifiedwith some node in the graph.
11
Example 2 in HO3:Formula-Value Query
Given a propositional formula ϕin the constants {F, T}, representedas a word model, decide whether itis true.
• There is a sequence S of propo-sitional formulas represented asword models.
• S starts with ϕ and ends withthe formula “T”.
12
• Every formula ϕi in S (exceptthe first) results from the previ-ous formula ϕi−1 by either:
– Application to ϕi−1 of one of∨, ∧ and ¬ which is ready tobe evaluated.
∗ Like in “(T ∧ F )”.
– Or elimination of one pair ofredundant parenthesis in ϕi−1.
∗ Like in “((T ))”.
13
• Formula-Value query is in
DLOGSPACE [Beaudry, PierreMcKenzie, 1992].
•DLOGSPACE⊆ P⊆NP = ∃SO.
•Nevertheless, to define these queriesin ∃SO seems to be more chal-lenging than in TO (see [Ferrarotti,Ren, Turull-Torres, 2014]).
14
Note that in the two examples inHO3 the size of the valuating re-lations for the TO variables thatmake the formulas true, is polyno-mial (actually logarithmic and lin-ear, respectively) in the size of theinput structure.
15
On the other hand, if we considerthe query SATQBF (see below), wecan express it in EHO3, since theproblem is PSPACE complete, andit is known that EHO3 is power-ful enough as to characterize everyproblem in PSPACE.
Note that the existence of an SOformula that expresses SATQBF isvery unlikely, since SO = PH, andit is strongly conjectured that PH⊂ PSPACE.
16
0: Higher Order Logics
(HOi)
17
Higher Order VariablesTypes
•A first order variable type is τ1 =0,
• a second order variable type isτ2 ≥ 1, i.e., its arity,
• for i ≥ 3, an i-th order vari-able type is a sequence of typesof orders 1 ≤ j1, . . . , js ≤ i− 1,
τ i = (τj11 , . . . , τ
jss ), with s ≥ 1.
W.l.o.g., we assume that at least
one of the types τj11 , . . . , τ
jss is of
order i− 1.
18
In the alphabet of a Higher Or-der Logic of order i, HOi, for ev-ery order 2 ≤ j ≤ i, and for ev-ery variable type τ , we add to FOa countably infinite set of relation
variables X j,τ1 ,X j,τ2 , . . .
We use calligraphic letters like X iand Yi for variables of order i ≥ 3,upper case letters like X and Y forsecond order variables, and lowercase letters like x and y for firstorder variables.
19
Besides the atomic formulas in FOand SO, inHOi we can use atomicformulas like the following:
If X is a relation variable of or-der j, for some 3 ≤ j ≤ i, andof relation type τ , for some τ =(ρ1, . . . , ρs), with ρ1, . . . , ρs beingtypes of orders≤ j−1, andY1, . . . ,Ysare relation variables of orders andtypes according ρ1, . . . , ρs, respec-tively, thenX (Y1, . . . ,Ys) is an atomicformula.
20
Higher Order Relations
Let s ≥ 1. An SO relation ofarity s is a relation in the classi-cal sense, i.e., a set of s-tuples ofelements of the domain of a givenstructure.
For an arbitrary i ≥ 3, a rela-tion of order j of relation type τ =(ρ1, . . . , ρs), is a set of s-tuples ofrelations of orders and types ac-cording ρ1, . . . , ρs, respectively.
21
W.l.o.g., and for the sake of sim-plicity, we assume that the widthof a higher order relation is prop-agated downwards, i.e., the rela-tions of order i− 1 which form thes-tuples for a relation of order i,are themselves of width s, and soon, all the way down to the SO re-lations, which are also of arity s.
22
We define exp(0) = O(nO(1))),and for i ≥ 1
exp(i) = 2exp(i−1)
That is, exp(i) is a hyper expo-nential function, which we defineas a stack of i exponents 2, andthen O(nO(1))) as the topmost ex-ponent.
(*) actually the i exponents shouldbe O(1), but we write 2 for simplic-ity.
23
Maximum Cardinalities ofHO Relations
• SO relations: ≤ nO(1);
• TO relations: ≤ 2O(nO(1));
•HO4 relations: ≤ 2(2O(nO(1))) =exp(2);
•HO5 relations: ≤ 2(2(2O(nO(1)))) =exp(3);
• . . .
•HOi relations: ≤ exp(i− 2).
24
Σij
Let i, j ≥ 1, as it is usual in clas-sical Logic we denote by Σij the
class of formulas ϕ ∈ HOi+1 ofthe form
∃X11 . . . ∃X1s1∀X21 . . . ∀X2s2
∃X31
. . . ∃X3s3. . . QXj1 . . . QXjsj(ψ)
where ψ ∈ HOi, Q is either ∃ or∀, depending on whether j is oddor even, respectively.
25
That is, Σij is the class of HOi+1
formulas with j − 1 alternations ofquantifiers blocks of variables of or-der i+ 1, starting with an existen-tial quantifier.
Analogously, we define the classesof formulas Πij.
26
Expressibility of HigherOrder Logics
[Hella, Turull-Torres, 2006]
1. For every i ≥ 0, let
NEXPH0i =
NTIME(exp(i))
2. For every j ≥ 1, let
NEXPHji = NEXPH0
iΣpj−1
Recall that Σp1 = Σ1
1 = NP and
Σp0 = P.
27
[Hella, Turull-Torres, 2006]
• for i, j ≥ 1: Σij = NEXPHj−1i−1 .
That is, a stack of i−1 exponents2, and then O(nO(1)) as the top-most exponent, plus an oracle inΣpj−1.
• for i, j ≥ 1: Πij = co−NEXPHj−1i−1 .
28
Fragments of HOi
with Small ValuatingRelations
We have seen above sketches ofHO3 formulas for the queries Hy-percube graphs and Formula-Value.
As we pointed out then, the ex-pressive power of HO3 is not ac-tually required for any of them.
29
Could we...?
Could we take advantage of themuch higher expressibility and sim-plicity of HO3,
•and, still
be able to express a query in amore simple and intuitive way,though still formal (*),
• but
without having to pay the priceof a higher complexity to evaluatethe corresponding formulas?
(*) so we can still make use ofsemi-automatic theorem proving (seebelow).
30
Note that by the results given above
ESO =NTIME(nO(1)) ⊆ DTIME(2n
O(1)),
while
ETO =
NTIME(2nO(1)
) ⊆ DTIME(22nO(1)
).
31
What is good about HOi?
For all i ≥ 2, HOi+1 providestwo important features:
• exponentially bigger auxiliary re-lations than HOi;
• nesting of relations, like in (i +1)-th order graphs, where eachnode is actually an i-th order graph,
or
(i+1)-th order PERT networks,for large and complex projects,where a node may represent aPERT network itself, and the op-eration of zooming in or out al-lows navigation in depth.
32
But...
The complexity of the evaluationof anHOi+1 query is exponentiallyhigher than that of an HOi query(see above).
For instance, for Existential FourthOrder Logic queries (Σ3
1) the com-plexity is
=⋃c∈N
NTIME(22(nc))
While for Existential Third Or-der Logic queries (Σ2
1) is
=⋃c∈N
NTIME(2(nc))
33
What if...?
What happens if we bound thesize of the i-th order relations to bepolynomial in the size of the inputdbi?
We could still have nesting...
34
Besides being a requirement insome applications (like deep struc-tures where zoom operations arenecessary),
in many cases
• nesting provides a more pow-erful language which allows sim-pler and more intuitive expressionsfor a query.
This also happens when using pro-gramming languages with rich datastructures (like OOPL):
• it makes programs much sim-pler and less error-prone than us-ing the old Assembler languages ofthe sixties and seventies.
35
• This is convenient not only forapplications to Databases in the In-dustry, but also for Theoretical re-search.
• To prove that a query is in thepolynomial hierarchy (PH), in manycases using higher order construc-tions in HOi,P can be much simplerthan using SO (see below).
• To prove that a query is in thepoly-logarithmic hierarchy (PLH),in many cases using higher orderconstructions inHOi,plog(HO<i,plog) can be muchsimpler than using SOplog (see be-low).
36
• Is nesting still relevant as to ex-pressive power?
37
1: A General Schema of
TO Formulas
38
Let σ be a relational vocabulary,which may include constant sym-bols. We define T[σ] as the class ofTO formulas of the form:
∃C sOss(
TotalOrder(C,O) ∧
∀G[(
First(G)→ αFirst(G))
∧(Last(G)→ αLast(G)
)]∧
∀GpredGsucc
[Pred(Gpred , Gsucc)
→ ϕ(Gpred , Gsucc)])
where
39
• C ranges over TO relations of types = (i1, . . . , is).
• TotalOrder(C,O), First(G), Last(G)and Pred(Gpred , Gsucc) denote fixedSO formulas.
• αFirst(G) and αLast(G) denotearbitrary SO formulas.
• ϕ(Gpred , Gsucc) denotes an ar-bitrary SO formula.
40
This is a very usual, intuitive, andconvenient schema in the expres-sion of natural properties of finitemodels.
For a start, it can clearly be usedto express the hypercube and for-mula-value queries as described above.
41
Additional examples are providedby the different relationships be-tween pairs of undirected graphs(G,H) that can be defined as or-derings of special sorts (see [Downey,Fellows, 1999]).
Using the schema these relation-ships can be expressed by defininga set of possible operations thatcan be applied repeatedly toH , un-til a graph which is isomorphic toG is obtained.
42
In particular, the following rela-tionships fall into this category:
a) G ≤immersion H : G is an im-mersion in H ;
b) G ≤top H : G is topologicallyembedded or topologically containedin H ;
c) G ≤minor H : G is a minor ofH ;
d) G ≤induced−minor H : G is aninduced minor of H ;
Interestingly, in all these cases thelength of the sequence is at mostlinear.
43
The operations on graphs neededto define those orderings are:
(E) delete an edge,
(V) delete a vertex,
(C) contract an edge,
(T) degree 2 contraction, or sub-division removal,
(L) lift an edge.
44
In particular the set of allowableoperations for each of those order-ings are:
{E, V, L} for ≤immersion ,
{E, V, C} for ≤minor ,
{E, V, T} for ≤top,
{V,C} for ≤induced−minor .
45
[Ferrarotti, Gonzalez,Turull-Torres, 2017]
We have the following:
Theorem:Every TO formula Ψ of the above
schema T can be translated intoan equivalent SO formula Ψ′ when-ever the following conditions hold.
1. The sub formulas αFirst, αLastand ϕ of Ψ are SO formulas.
2. There is a d ≥ 0 such that for ev-ery valuation v with v(C) = R,if A, v |= ∃Ossψ(C,O), then|R| ≤ |dom(A)|d.
46
Planarity in Graphs
The classical Kuratowski defini-tion of planarity, provides yet an-other example of a property thatcan be defined using our schemaand also results in a linearly boundedsequence of structures.
By Wagner’s characterization (see[Bollobas, 2002]) a graph is planarif and only if it contains neither K5nor K3,3 as a minor.
47
Note that the more intuitive con-struction for planarity would be tosay that there is no transformationprocess of linear size that arrivesto a K5 or K3,3, starting from theinput graph and applying in eachtransition exactly one of the oper-ations in {E, V, C} above.
48
If we have the negation of a for-mula in the schema T, we can usethe same translation to SO, andthen add a negation in front of theSO formula.
Then we have the following:
Corollary:The negation ¬Ψ of a formula Ψ
of the above schema T can also betranslated into an equivalent SOformula ¬Ψ′ whenever the two con-ditions of the previous theorem hold.
49
[Ferrarotti, Gonzalez, Schewe,Turull-Torres, 2018]
By using the normal form for (SO+TC2) ([Imm,1999]) the following re-sult is straightforward:
Theorem:The class of TO formulas of the
above schema T is equivalent tothe logic (SO + TC2).
And, hence, equal to PSPACE.
Corollary:The class of TO formulas of the
schema T is closed under negation.
50
Translation to Non DetParallel ASM
[Ferrarotti, Gonzalez, Schewe,Turull-Torres, 2018]
By using the non deterministic,parallel Abstract State Machine model([Boerger, 2003]), it is not difficultto prove the following:
Theorem:Every formula Ψ of the above schemaT can be systematically translatedto an equivalent non determinis-tic, parallel ASM which doesn’t usehigher order formulas.
51
Note that for the sake of easilycomprehensible high-level specifi-cations it is advisable to extendrigorous methods to support alsohigher-order logic and to investi-gate strategies for refinement tofirst-order.
52
Theorem provers and Nondet Parallel ASM
It is well known that for manycases of ASM’s, there are theoremprovers which allow semi-automatictheorem proving support for manycases of ASM rules.
In particular, for non determin-istic parallel ASM’s there are veryinteresting results.
53
[Schellhorn,Ernst,Pfhler,Bodenmller,Reif , 2018]
• It is possible to compute an FOformula for each rule that im-plies clash-freedom (*) when prov-able (it is provable for many ASMsthat are used in practice).
(*) for each state S a rule r yieldsan update set ∆(S), i.e. a (fi-nite) set of (finite) sets of up-dates. There is a clash if thereare two updates (l, v1), (l, v2) in∆(S) with v1 6= v2.
(i.e., pairs location (i.e., n-ary function symbol and ann-tuple of values), and value)
54
• They give axioms that describethe transition relation for clash-free ASM rules as SO formu-las that can be used to verifypre/post-condition assertions, andto derive properties of ASM’s,using automated theorem provers.
• They provide a Calculus for clash-free ASM rules based on sym-bolic execution for deduction, whichcan be used for interactive theo-rem provers, like their tool KIV.
55
[Ferrarotti, Gonzalez, Schewe,Turull-Torres, 2018]
By using higher order logics HOi,P
(see below) the following result isstraightforward:
Theorem:For every ASM extended with HOi,P
formulas in its rules, we have anautomatic refinement of the HOi,P
extended ASM to an SO extendedASM.
56
Once we got the SO extended ASM,we can apply to it the naıve refine-ment strategy consisting on non-deterministically guessing the quan-tified relation variables.
As naıve refinements in a stan-dard way are always possible, webelieve that semi-automatic proofscould be conducted on such, thoughnot optimal refinements.
57
QBF Solvers
Alternatively, the use of QBF solversis worth exploring.
from “QBF Gallery 2014 (Com-petition)”, in the “QBF Solver Eval-uation Portal”,
www.qbflib.org/index eval.php
58
“Many problems from applicationdomains such as model checking,formal verification or synthesis arePSPACE-complete, and hence couldbe encoded in QBF”.
“Considerable progress has beenmade in QBF solving throughoutthe past years. However, in con-trast to SAT, QBF is not yet widelyapplied to practical problems inindustrial settings”.
59
Once we got an SO formula φ(see below):
• for every model A, there is atranslation fφ(A) to a QBF for-mula (see [Hella, Turull-Torres,
2006a] for a translation),
• we can then use a QBF solver.
60
2: Downward polynomially
bounded Relations
HOi,P
61
An i-th order relation R of typeτ in a structure A is downwardpolynomially bounded (dpb) by dif |R| ≤ |dom(A)|d,
and
for all 2 ≤ j ≤ i− 1, all the j-thorder relations that form the tuplesof (j + 1)-th order relations, are inturn dpb by d.
62
For i ≥ 3 we define HOi,P as theextension of HOi−1,P , where the i-th order quantifiers restrict the car-dinality to be bounded by a poly-nomial that depends on the quan-tifier.
In the alphabet of HOi,P , for ev-ery pair of positive integers d, andj, with i ≥ j ≥ 3, we have:
a j-th order quantifier ∃j,P,d
and
for every j-th order type τ , wehave countably many j-th order vari-able symbols X j,d,τ .
63
A valuation in a structure A as-signs to each i-th order relation vari-able X j,d,τ a dpb i-th order rela-tion R of type τ in A, such that|R| ≤ |dom(A)|d.
For any 2 < j ≤ i, the HOi,P
quantifier ∃j,P,d has the followingsemantics:
A |= ∃j,P,dX j,d,τϕ(X )
iff
there is a j-th order relation R oftype τ , such that A |= ϕ(X )[R]and R is dpb by d in A.
64
[Ferrarotti, Gonzalez,Turull-Torres, 2017]
We have the following:
Theorem:For all i ≥ 3, HOi,P collapses toSO. Moreover, every formula inHOi,P can be algorithmically trans-lated to an equivalent SO formula.
65
Strategy:
Basically, the strategy of the trans-lation is to use a relational databasewith referential integrity to encodeeach relation variable of order ≥2.
Let i ≥ k ≥ j ≥ 2. For eachvariable of order k, the db that rep-resents it consists of 2(k − 1) rela-tions.
For each j-th order variable wehave one relation with id’s for tu-ples of relations of order (j − 1),and one relation for id’s of rela-tions of order (j − 1).
66
Empty Relations
We must also have in mind thatthe tuples of relations of any order,can have empty relations in someof its components.
Then, the (SO) “relation” thatwe use to store the set of tuple iden-tifiers for a relation of type widths, is actually a set of 2s (SO) rela-tions, one for each possible combi-nation of empty relations in sucha tuple.
67
Then, for a given query, we canproceed as follows:
1. • Use an HOi,P formula, withan arbitrary order i, to expressthe query,
2. • translate algorithmically the
HOi,P formula into an SO for-mula,
3. • evaluate the SO formula.
Note that we have still (determin-istic) single exponential time com-plexity, (NP complete queries arestill there!) in the third step.
But we don’t have to deal withhyper exponential complexity.
68
A Note on the DifferentTranslations
The first translation (schema T ofTO) yields a more clear and in-tuitive SO formula, and the max-imum arity of the quantified SOrelation variables in general seemsto be much smaller.
For the case of hypercube graphsthe maximum arity obtained bythe schema translation is 4, whilefor the SO formulas obtained bythe HOi,P translation is 8.
69
And for the case of the Formula-Value query the maximum arityobtained by the schema translationis also 4, while for the SO formulasobtained by the HOi,P translationis 22.
Note that the arity of a relationsymbol in an SO formula is rele-vant for the complexity of its eval-uation (see among others [Hella,Turull-Torres,2006]).
70
Hence, and not surprisingly it makessense to study specific schemasof TO formulas that have equiva-lent SO formulae, aiming to findmore efficient translations thanthe general strategy used forHOi,P
formulas (which had the purposeof proving equivalence, rather thanlooking for efficiency in the trans-lation).
71
3: Valuating Relations of
Poly-logarithmic
Cardinality
72
A Query in TOplog
Graph Factoring[Ferrarotti, Gonzalez, Schewe,
Turull-Torres, 2018]
Roughly, let TOplog denote thefragment of TO where only valu-ations which assign TO relationsof poly-logarithmic cardinality, toTO variables are considered.
The SO sub-formulas in TOplog
are standard SO formulas.
For that matter we use typed TO
variables of the formX τ,logk, mean-ing that valuations can only assignto them relations of type τ and car-dinality ≤ (dlog ne)k.
73
The input structure is A of sig-nature σF = 〈VI , EI ,FI〉, where(V AI , EA
I ) is a connected and loop-less undirected graph (cu-graph),and FA
I is a TO relation which inturn consists of a set of pairs ofgraphs (V A
FI , EAFI), and (V A
K , EAK).
The first graph of each pair is a cu-graph, and the second graph is aclique.
74
We define graph factoring as adecision problem. A σF -structureA is in the class GraphFactoringiff the third-order relation FA
I is a
factoring of the graph (V AI , EA
I ),where the first graph of each pair inFAI is a cu-graph that is a factor of
the graph (V AI , EA
I ), and the sizeof the corresponding clique is theexponent.
75
A straightforward consequence ofthe definition of graph product isthat the size of any factoring cir-cuit C for a structure A is at most2 · dlog(|V A
I |)e, and the size of the
TO relationFIA on any given A ∈GraphFactoring is at most dlog(|V A
I |)e.
76
ϕGF ≡ ∃VCEC(
“FactoringCircuitForGI(VC, EC)
∧NodesCUgraphs(VC, EC)
∧RootsPrimeGraphsC∧RootsInFIC
∧SingleOutputGIC”)
where (VC, EC), is a TO graph ofsize at most 2 · dlog(|V A
I |)e, whosenodes are cu-graphs, and whose edgesare pairs of cu-graphs.
77
FactoringCircuitForGI(VC, EC) ≡(“Digraph(VC, EC) ∧
Acyclic(VC, EC) ∧
Connected(VC, EC) ∧
InDegree2C ∧
ProductOfParentsC ∧
LinearNonRootsC
∧NonIsomorphicRootsC”)
78
“InDegree2C” says that every nodein the circuit has either 1 or 2 inputnodes.
“ProductOfParentsC” says that ev-ery node in VC is a cu-graph thatis either the product of its two par-ents, or the square of its single par-ent.
79
Product(V1, E1, V2, E2, V3, E3) ≡
∃V×E×
([∀v1w1v2w2(
(V×(v1, w1)↔ (V1(v1)∧ V2(w1)))∧
[E×(v1, w1, v2, w2)↔((v1 = v2 ∧ E2(w1, w2)) ∨
(w1 = w2 ∧ E1(v1, v2)))])]
∧
“Isomorphic(V×, E×, V3, E3)”
)
80
LinearNonRootsC ≡ ∃VClECl(
“EqualTO(VCl, {int. nodes in C}
)∧
EqualTO(ECl, EC � {int. nodes in C}
)∧LinearDigraph(VCl, ECl)′′
)where EC � {int. nodes in C} is therestriction of the TO binary rela-tion EC to the subsetof internal nodes of the set VC.
81
NumbOfProductsC(V0, E0, VK0) ≡
∃H(“H : VK0 7→ ChildrenC(V0, E0)
quasi injective”)
The quasi injectivity of the func-tion in the formula above is dueto the fact that we avoid allowingmultiple edges between two givennodes in the circuit C, to make theformula simpler.
82
Note that the only possible casewhere one single edge means thata (factor) graph is actually beingused twice in the same product isat the (unique) node at depth onein the circuit.
An example for this situation isthe factoring circuit for an hyper-cube of order n, where the samefactor graph (K2) is used n times.
83
Note:As the sizes of the valuating TO
relations that make the formula ϕGFtrue are poly-logarithmic, then itseems straightforward to apply thesame encoding strategy as in HOi,P
and translate it to an SO formula.
Hence, we have the following:
Corollary:TOplog = SO.
84
Though the query graph factoringcan certainly be expressed in SO(for instance with a signature
σF = 〈V 1I , E
2I , V
2F , E
3F , V
2K, E
3K〉),
it doesn’t seem to be easy.
85
Roughly, let SOplog denote the frag-ment of SO where only valuationswhich assign SO relations of poly-logarithmic cardinality, to SO vari-ables are considered.
For that matter we use typed SO
variables of the formXr,logk, mean-ing that valuations can only assignto them relations of arity r and car-dinality ≤ (dlog ne)k.
And let TOplog(SOplog) denote thefragment of TOplog where only val-uations which assign SO relationsof poly-logarithmic cardinality, toSO variables are considered.
86
Expected result:
With the same strategy, we be-lieve that we can also prove:
• TOplog(SOplog) = SOplog.
87
[Ferrarotti, Gonzalez,Schewe, Turull-Torres,
2018a]
On the other hand, we proved thefollowing result:
•∑1,plog
1 (b∀) = NPolyLogTime.
• SOplog = PLH. (*)
[(*) Barrington gave a characteri-zation of the class of DCL-uniformfamilies of Boolean circuits of un-bounded fan-in, and quasi polyno-
mial size (i.e., 2(log n)O(1)) and con-
stant depth with an equivalent logic([Barrington, 1992]). From that re-sult the second result above follows.]
88
Where∑1,plog
1 (b∀) is the existen-
tial fragment of SOplog where theFO ∀ is bounded to poly-logarithmicsub-domains. And PLH denotesthe non deterministic Polylog-TimeHierarchy.
89
Expected result:
Then, we would have also that:
• TOplog(SOplog) = PLH.
This would mean that we can usea higher level language like TOplog
(SOplog) to prove that a given queryis in PLH.
That would make easier both theconstruction of the formulas andthe corresponding proofs.
90
Examples in TOplog(SOplog):
• There is an induced subgraph (V ′,E′) of size between dlog ne and(dlog ne)c, and there is a setF of
size at least (dlog ne)1/2, of dis-joint induced subgraphs (V ′i , E
′i),
s. t. the subgraphs in F are aset of prime factors of the sub-graph (V ′, E′).
• There are between dlog ne and(dlog ne)c disjoint induced sub-graphs that are cliques of sizesbetween dlog ne and (dlog ne)d.
Note that the first query, doesn’tseem to have an easy SOplog for-mula.
91
To express it in TOplog(SOplog)we can follow a similar strategy asfor Graph-Factoring above.
92
We believe that the following queriescan be also expressed in TOplog(SOplog):
•All the induced subgraphs of sizebetween dlog ne and (dlog ne)care prime.
• There are polylog disjoint inducedsubgraphs of polylog size s.t. foreach of them, all its prime fac-tors are disjoint induced sub-graphs of size polylog.
• For every polylog size set of dis-joint induced subgraphs of poly-log size in G1 there is a set ofthe same size of disjoint inducedsubgraphs of polylog size in G2,s.t. there is a bijectionF : V1→V2 so that the two graphs in ev-
93
ery pair in F are isomorphic.
94
So, proving that result, we wouldbe able to use TOplog(SOplog) logicto write probably many queries ina much simpler way than usingSOplog.
And still, in that way proving thatthe queries are in PLH.
But we believe that we can do bet-ter...
95
Expected result:
Finally, we also believe that withthe same strategy, we can prove:
•HOi,plog(HO<i,plog)
= SOplog = PLH.
96
4: Beyond Second Order
SATQBF
97
SATQBFk and SATQBF
QBFk denotes the set of quanti-fied propositional formulas of theform
φ ≡ ∃x1∀x2 . . . Qxk(ϕ),
where ϕ is a propositional formulaover X = {xij}1≤i≤k,1≤j≤li, n ≥0, and where for 1 ≤ i ≤ k, xi =(xi1, . . . , xili) is a tuple of li differ-ent variables from X .
98
Note that Q is “∃” if k is oddand “∀” if k is even, and the setsX1, . . . , Xk of variables in x1, . . . , xk,respectively, form a partition of X .
Let QBF =⋃k>0 QBFk.
The semantics of the quantifiersis as follows: ∃x(α(x)) ≡ α(0/x)∨α(1/x), and ∀x(α(x)) ≡ α(0/x)∧α(1/x).
99
Note that, in view of the seman-tics of the quantifiers, every quanti-fied propositional formula is equiv-alent to a propositional formula,which is longer (roughly, exponen-tially longer in the number of quan-tifiers).
φ is satisfiable if there is a par-tial valuation v1 : X1 → {T, F},s. t. for every partial valuationv2 : X2 → {T, F}, there is a par-tial valuation v3 : X3 → {T, F},s. t. . . . s. t. the valuation v =v1∪v2∪v3∪ . . .∪vk makes ϕ true.
100
We now define Boolean queries :
• For k > 0, SATQBFk is the setof quantified propositional formu-las in QBFk, represented as wordmodels in the signature
〈 ≤2, I1x, I
1∃, I
1∀, I
1∨, I
1∧, I
1¬,
I1(, I1
), I1| 〉
that are true.
• SATQBF is the set of quantifiedpropositional formulas in QBF thatare true.
Note that as formulas in QBF haveno free variables, such a formula issatisfiable iff it is true.
101
SATQBFk
• In Σ1k.
• It doesn’t look like there is a sim-ple SO formula to express SATQBFkon word models (see formula in[Ferrarotti, Ren, Turull-Torres, 2014]).
• [Pap,94] Complete for ΣPk under
PTIME reductions.
102
SATQBF
• In Σ21.
• It doesn’t look like there is a sim-ple TO formula to express SATQ
BF on word models (see formulain [Ferrarotti, Ren, Turull-Torres,2014]).
• [Imm,99,P.10.2]
PSPACE complete via (FO+ ≤+BIT ) reductions.
103
SATQBF in HO3
(known to be in HO3)
At the first level of abstraction:
“There is a third order alternating
valuation Tv applicable to ϕ,
which satisfies ϕ”.
104
At the second level of abstractionwe express the following:
“∃ av ∆3 = (V3∆, E
3∆,B
3∆) ”;
“∃ linear digraph Gq = (Vq, Eq)”;([“B3
∆ : V3∆→ {0, 1}”
]∧[
“Gq = (Vq, Eq) represents the
sequence of quantified variables
in ϕ”]∧[
“(V3∆, E
3∆,B
3∆) is an av applica-
ble to ϕ, i.e.”:
105
• “(V3∆, E
3∆) is a TO binary tree
with all its leaves at the samedepth, which is in turn equal tothe length of (Vq, Eq)”;
• “all the nodes in (V3∆, E
3∆) whose
depth correspond to a univer-sally quantified variable in theprefix of quantifiers of ϕ, haveexactly one sibling, and its valueunder B3
∆ is different than thatof the given node”;
• “all the nodes whose depth cor-respond to an existentially quan-tified variable in the prefix of quan-tifiers of ϕ, are either the root or
have no siblings”]
∧106
[“(V3
∆, E3∆,B
3∆) |=av ϕ”
] )with |=av we denote that every
leaf valuation of the av satisfiesthe quantifier-free sub formula of ϕ.
107
What about using HO4?
Next we use an HO4 formula in-stead.
We don’t need to say that the avis applicable to ϕ; we just describehow to build it, which we believeis more intuitive and simpler.
Note that for the valuation of thefourth order variables it is enoughif we consider only relations withcardinality exp(1).
108
SATQBF as a Sequence ofav’s
in HO4,exp(1) = HO3
(known to be in HO3)
At the second level of abstractionwe express the following:
“∃ sequence (of linear size) (T 4, E4)
of av’s ∆3 = (V3∆, E
3∆,B
3∆) (each
av of size exp(1))”;
“ ∃ bijection (of linear size) F4T ,ϕ :
T 4→ {x : (I∀(x) ∨ I∃(x))} that
preserves E4 and ≤ϕ”;
109
[“∀ av’s V3
∆, E3∆,B
3∆, V
3∆′, E
3∆′,B
3∆′ ”([
“B3∆ : V3
∆→ {0, 1}”]
∧[“FirstE(V3
∆, E3∆,B
3∆)”→
“(V3∆, E
3∆,B
3∆) is an av with just
one node”]
∧[“LastE(V3
∆, E3∆,B
3∆)”→
“(V3∆, E
3∆,B
3∆) |=av ϕ”
]∧
110
[“SuccE(V3
∆′, E3∆′,B
3∆′,V
3∆, E
3∆,B
3∆)”→
“av ∆ is an extension of av ∆′
by one level in depth, so that”:
111
[“I∃(F4T ,ϕ(V3
∆, E3∆,B
3∆))” →
“each leaf of av (V3∆′, E
3∆′,B
3∆′)
has
exactly 1 child in its image in
(V3∆, E
3∆,B
3∆), with an arbitrary
value in B3∆”]
∧[“I∀(F4T ,ϕ(V3
∆, E3∆,B
3∆))” →
“each leaf of av (V3∆′, E
3∆′,B
3∆′)
has exactly 2 children in its image
in (V3∆, E
3∆,B
3∆), with different
values in B3∆”] ] )]
112
where
“av ∆ = (V3∆, E
3∆,B
3∆) is an
extension of
av ∆′ = (V3∆′, E
3∆′,B
3∆′)”
is roughly expressed as follows:
“∃ a total injection (of size exp(1))
H3 : V3∆′ → V
3∆”(
“H3 preserves E3∆,B
3∆, E
3∆′,B
3∆′”
)
113
Note that in the formula abovefor the valuation of the 4-th ordervariables it is enough if we con-sider only relations of cardinalityexp(1).
We could then encode the HO4
relations in TO relations, using tu-ples of (SO) sets as identifiers fortuples of TO relations in the 4-thorder relations.
114
Expected result:
For every i ≥ k ≥ j ≥ 4 letHOi,exp(j−2) denote the fragmentsof HOi where the cardinality of thevaluating k-th order relations forthe k-th order variables are restrictedto be O(exp(j − 2)) w.r.t. the sizeof the model.
Then, we believe that, by usingbasically the same encoding strat-egy as for HOi,P , we can prove thefollowing:
• For every i ≥ k ≥ j ≥ 4:
HOi,exp(j−2) collapses to HOj.
115
In the encoding of relations of or-der k as above, the difference w.r.t.HOi,P is that we need more differ-ent identifiers to encode tuples ofHOk−1 relations.
Note that, as the cardinality of anHOk relation is at most exp(k −2), the number of different HOk
relations is at most exp(k − 1).
116
Then, to encode HO relations, ofwhichever order k, whose maximumcardinality is O(exp(j − 2)),
we need O(exp(j − 2)) differentidentifiers, and hence a tuple of re-lations of order (j − 1) is enough.
117
So that in the db that encodes arelation of order k:
• all the relations will use tuplesof relations of order (j − 1) asidentifiers for tuples of relationsof order k − 1,
• and hence, relations of order jsuffice to represent the db.
118
5: Beyond Third Order
SATQBF(Σik)
119
We will next see an example ofa query known to be expressible inHO4.
It doesn’t seem easy to expressit in HO4.
We will use HO6 and HO7 to ex-press it instead.
And then we will see that we (be-lieve that) we can translate bothformulas to HO5.
120
A more complexproblem:
High Order SATQBFk[Hella, Turull-Torres, 2006]
We want to build a variant of theproblem SATQBFk of a higher com-plexity, that is, a higher order vari-ant, considering the logics Σij for alli, k ≥ 1.
But we must remain as close aspossible to propositional logic.
121
With that in mind, we considerone single structure, that we callthe Boolean model,
B = 〈{a, b}, 0B, 1B〉
a two-element model where bothelements are interpretations of theconstant symbols 0 and 1.
122
Then, deciding whether a givenΣij sentence is “satisfiable” (in the
propositional logical sense), turnsinto deciding whether a Σij sentencein the vocabulary of the Booleanmodel, is true in the Boolean model.
That is, it means deciding the Σijtheory of the Boolean model:
Σij-Th(B).
123
The problem SATQBF(Σik)
For i, k ≥ 1 let SATQBF(Σik) de-note the Boolean query:
“given a Σik sentence φ in the vo-cabulary of the Boolean model, isφ ∈ Σik-Th(B)?”.
124
Descriptive Complexity ofSATQBF(Σik)
[Hella, Turull-Torres, 2006]
Then we have the following:
• For i, k ≥ 1, SATQBF(Σik) on
word models is complete for Σi+1k
under P reductions.
Note that each Σik sentence is rep-resented as a string in the alphabetof predicate logic of order i.
125
Note that the notion of complete-ness of the result above is w.r.t.a logic, not to a (computational)complexity class, i.e., it is a notionin the setting of descriptive com-plexity.
This means that for every Σi+1k
sentence ψ of an arbitrary vocab-ulary τ , and every τ -structure A,we build (in polynomial time) a Σiksentence fψ(A) on the Boolean mod-el, s. t.
B |= fψ(A) iff A |= ψ.
126
Computational Complexityof
SATQBF(Σik)[Hella, Turull-Torres, 2006]
Considering the expressibility ofΣi+1k given above, we also get:
• For i ≥ 1 and k ≥ 1, SATQBF(Σik)on word models is complete forNEXPHk−1
i under P reductions.
127
Note that these problems beingcomplete for NEXPHk−1
i , impliesthat they are provably intractable,that is, we know that for each i ≥1 and k ≥ 1, there is no algorithmin P that can decide SATQBF(Σik).
This is because there are provablyintractable problems in NTIME(2n
c),
and hence all the classes that in-clude it contain intractable prob-lems too [Garey,Johnson,1979].
The problems SATQBF(Σik) arethe first known family of completeproblems for all the levels of theNon deterministic Hyper-exponentialTime Hierarchy NEXPHk−1
i .
128
SATQBF(Σ2j)
In the word model for the inputformula ϕ ∈ Σ2
j, the variables andtheir types are encoded as follows(where Q ∈ {∃,∀}, and i, ri, ti ≥1):
• 1st order variable xi:
Qx|i
• 2nd order variable Ri of arity ri:
QR|i ∗ |ri
• 3nd order variableRi of type τi =(r1, . . . , rti):
QR|i ∗ (|r1, . . . , |rti)
129
The signature of the word modelis the following:
〈 ≤2, I1R, I
1R, I
1x, I
1∃, I
1∀, I
1∨, I
1∧, I
1¬,
I1(, I1
), I1, , I
1| , I
1∗ 〉
We assume that the quantifierblocks are arranged in the order〈3rd, 2nd, 1st〉 order quantifiers, andare then followed by a quantifierfree formula.
The first quantifier is always a3rd order existential quantifier.
130
Representation of HORelations
SO variables
An r-ary SO variable S2,r:
as a TO relation S3,τ2, with τ2 =
(1, 2, 2), i.e., a set of linear digraphsof size r with a Boolean assignment.
So that each such digraph repre-sents an r-tuple in the SO relationthat valuates S2,r.
131
Representation of HORelations
TO variables
A TO variable R3,τ3of type τ3 =
(r1, . . . , rs):
as an HO5 relation R5,τ5.
132
In the TO relation that valuatesR3,τ3
:
• each tuple of SO relations has scomponents which are SO rela-tions of arities r1, . . . , rs, respec-tively;
• hence, each such tuple is rep-
resented in R5,τ5as a sequence
of linear digraphs with Booleanassignments,
133
• that is, it is an HO4 linear di-graph of size s where each nodeis a TO set of linear (SO) di-graphs of sizes r1, . . . , rs, respec-tively;
then, a TO relation, i.e., a set oftuples of SO relations, is repre-
sented in R5,τ5as a set of HO4
linear digraphs of size s, hence asan HO5 relation.
τ5 =(((1, 2, 2)
),((1, 2, 2), (1, 2, 2)
))
134
SATQBF(Σ2j)
in HO6,exp(3) = HO5
(known to be in HO4)
“∃ av ∆6 = (V6∆, E
6∆) (of size exp(3))”;
“∃ linear digraph Gq = (Vq, Eq)”
that represents de sequence of
quantified variables in ϕ, ordered
as 〈3rd, 2nd, 1st〉 order variables;
“∃ Fq,ϕ :
Vq → {z : (Ix(z)∨IR(z)∨IR(z))}total bijection (of linear size) that
preserves Eq and ≤ϕ”;
135
“∃ F6∆,q : V6
∆→ Vq total surjective
function (of size exp(3)) that maps
every node in av ∆6 to its corre-
sponding quantified variable in ϕ”;
136
(1
[“(V6
∆, E6∆) is an out-tree with
all leaves at depth |Vq|”]∧[
“V6∆ is a set of tuples (I5, x1,S3,R5)”
]∧ ∀z
[2
Vq(z)→
(3
[4(
5
“IR(Fq,ϕ(z)
)∧
I∃(Pred≤ϕ(Fq,ϕ(z))
)”
)5
→
137
(5
[6
“FirstEq(z) ∧ ∃ I51 , x
11,S
31 ,R
51”
(“Root∆(I5
1 , x11,S
31 ,R
51) ∧
S31 = ∅ ∧ x1
1 = 3 ∧
’R51 is well formed as a
representation of a 3rd order
relation’ ”)]
6
∨
138
[6¬FirstEq(z)∧
∀ I51 , x
11,S
31 ,R
51, ∃ I
52 , x
12,S
32 ,R
52(
7
[“F∆,q(I5
1 , x11,S
31 ,R
51) =PredEq(z)”
]→[“(I5
2 , x12,S
32 ,R
52) is the unique child
of (I51 , x
11,S
31 ,R
51) in av ∆6” ∧
S32 = ∅ ∧ x1
2 = 3 ∧
’R52 is well formed...’ ”
])7
]6
)5
]4∧
139
[4“IR
(Fq,ϕ(z)
)∧I∀(Pred≤ϕ(Fq,ϕ(z))
)”
→ . . .]
4
∧[4“IR(Fq,ϕ(z)
)∧I∃(Pred≤ϕ(Fq,ϕ(z))
)”
→ . . .]
4
∧[4“IR(Fq,ϕ(z)
)∧I∀(Pred≤ϕ(Fq,ϕ(z))
)”
→ . . .]
4
∧[4“Ix(Fq,ϕ(z)
)∧I∃(Pred≤ϕ(Fq,ϕ(z))
)”
→ . . .]
4
∧140
[4“Ix(Fq,ϕ(z)
)∧I∀(Pred≤ϕ(Fq,ϕ(z))
)”
→ . . .]
4)3
]2
∧
141
∀ I51 , x
11,S
31 ,R
51[
2“Leaf∆(I5
1 , x11,S
31 ,R
51)→(
3“the valuation in the path from
the root of av ∆ to the leaf
(I51 , x
11,S
31 ,R
51) satisfies the q-free
sub-formula of ϕ”)
3
]2
)1
142
Note that with each leaf valua-tion we can build a propositionalformula in {F, T} from the q-freesub-formula of ϕ.
Each TO atomic formula in ϕR3,τ (S1, . . . , S|τ |) is replaced withthe truth value of the fact that thetuple of SO relations assigned tothe SO variables S1, . . . , S|τ |, be-longs to the TO relation assignedto R3,τ .
And we proceed similarly for theSO atomic formulas.
Then, to evaluate the resultingformula, we can use the TO for-mula in the fragment T for the For-mula-Value query mentioned above.
143
SATQBF(Σ2j) as a Sequenceof av’s
in HO7,exp(3) = HO5
(known to be in HO4)
“∃ sequence (of linear size) (V7S, E
7S)
of av’s ∆6 = (V6∆, E
6∆) out-trees
of size exp(3), and depth growing
from 1 to |Vq|”;
“∃ linear digraph Gq = (Vq, Eq)”
that represents de sequence of
quantified variables in ϕ, ordered
as 〈3rd, 2nd, 1st〉 order variables;
144
“∃ bijection (of linear size) F7VS,ϕ :
V7S → {x : (Ix(z)∨IR(z)∨IR(z))}
that preserves E7S and ≤ϕ, and
maps every av ∆6 to its corre-
sponding quantifier in ϕ”;(1
[“V6
∆ is a set of tuples (I5, x1,S3,R5)”]
∧ “∀ av’s V6∆, E
6∆, V
6∆′, E
6∆′ ”
(2
145
[3
“FirstE7S(V6
∆, E6∆)”→
(4“(V6
∆, E6∆) is an av with just
one node (I5, x1,S3,R5)”
∧
“S3 = ∅ ∧ x1 = 3 ∧
’R51 is well formed as a
representation of a 3rd order
relation’ ”])
4
]3
∧
146
[3
“SuccE7S(V6
∆′, E6∆′, V
6∆, E
6∆)”→
“av ∆ is an extension of av ∆′
by one level in depth, so that”:
147
(4
[5
(6“I∃(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
IR(V6
∆, E6∆
)”)
6→
∀ I51 , x
11,S
31 ,R
51, ∃ I
52 , x
12,S
32 ,R
52(
6“Leaf∆′(I5
1 , x11,S
31 ,R
51)” →[
“(I52 , x
12,S
32 ,R
52) is the unique child
of the image of (I51 , x
11,S
31 ,R
51)
in av ∆6” ∧ “S32 = ∅ ∧ x1
2 = 3 ∧
’R52 is well formed...’ ”
])6
]5∧
148
[5“I∀(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
IR(V6
∆, E6∆
)”)
6→ . . .
]5∧[
5“I∃(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
IR(V6
∆, E6∆
)”)
6→ . . .
]5∧[
5“I∀(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
IR(V6
∆, E6∆
)”)
6→ . . .
]5∧[
5“I∃(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
Ix(V6
∆, E6∆
)”)
6→ . . .
]5∧
149
[5“I∀(Pred≤ϕ(F7
VS,ϕ(V6∆, E
6∆))∧
Ix(V6
∆, E6∆
)”)
6→ . . .
]5
)4
]3∧
150
[3
“LastE7S(V6
∆, E6∆)”→
∀ I51 , x
11,S
31 ,R
51(
4
“Leaf∆(I51 , x
11,S
31 ,R
51)→
[5“the valuation in the path from
the root of av ∆ to the leaf
(I51 , x
11,S
31 ,R
51) satisfies the q-free
sub-formula of ϕ”]
5
)4
]3
)2
)1
151
References
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[Bollobas, 2002] B. Bollobas, “Modern Graph The-ory”, Springer, Graduate Texts in Mathematics, 184,2002.
[Downey, Fellows, 1999] R. G. Downey, M. R. Fellows,“Parameterized Complexity”, Springer, Monographs inComputer Science, 1999.
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[Ferrarotti, Gonzalez, Schewe, Turull-Torres,2018] F.Ferrarotti, S. Gonzalez, K.-D. Schewe, J. M. Turull-Torres,“Systematic Refinement of Abstract State Machines withHigher-Order Logic”, 6th International ABZ ConferenceASM, Alloy, B, TLA, VDM, Z, June 5th-8th, 2018, Southamp-ton, UK.
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