Hard boolean functions handled by oracles

As mentioned in Section 6.1, generalising [17], it should be possible to show th a t there are

boolean functions such as the satisfiable pigeonhole formulas PHP™ which do not have polysize representations of bounded hardness even for the relative condition. One way to overcome this barrier is to generalise the theory started here via the use of oracles as in [104, 110] (recall Subsection 4.3.2), and then employing oracles which can handle pigeonhole formulas. The basic definitions are as follows.

D e fin itio n 8.2 . 1 A v a lid oracle fo r generalised unit-clause propagation is some U C U S A T

with {_L} G U which is stable under application of partial assignments. The oracle is s tr o n g if hio c U, where Uq := { F G CCS : _L G F }.

Consider k G N0. In [104] the reduction r^ : CCS —> CCS has been defined. A n equivalent definition (generalising Definition 3.1.1) is as follows fo r F G CCS:

' (F) : = if F e U otherwise

M ( F s .= J rfc+i((; r -^ !> * F ) i / 3 x G l i t ( F ) : r “ ( ( : r - > 0 ) * F ) = {.L}

f c + 1 | F otherwise

Note rfc = r^°. Generalising Definitions 3.3.1, 3.4.1:

D e fin itio n 8.2 .2 Consider a valid oracle U. The hardness h d u ( F ) G No ( “hardness with oracle U ”) of an unsatisfiable F G CCS is the minimal k G No such that r ( F ) = {_L}. And fo r general F G CCS we define h d ^ (T ) := 0, while fo r F / T let

hd u( F) := max{hd^(c£ * F) : ip G P A S S A ip * F £ U S A T } G No.

We have hd = hd^0, and if U is strong then for all F holds h d ^ (F ) < h d (F ). An interesting oracle U (with polytime membership decision) is given by the class of unsatisfiable clause-sets defined in [49] via semidefinite programming, for which we get hd£/(PHP™) = 0.

An im portant aspect of the theory to be developed m ust be the usefulness of th e representation (with oracles) in context, th a t is, as a “constraint” in a bigger problem: a boolean function / represented by a clause-set F is typically contained in F ' DF , where F ' is th e SAT problem to be solved (containing also other constraints). One approach is to require from the oracle also stability under addition of clauses, as we have it already for th e resolution-based reductions like r^, so th a t the (relativised) reductions r*f can always run on th e whole clause-set (an instantiation of F '). However for example for th e semidefinite programming oracle m entioned above, this would be prohibitively expensive. And for some oracles, like detection of m inim ally unsatisfiable clause-sets of a given deficiency, the problems would tu rn from polytim e to N P -hard in this way ([60, 32]). Furtherm ore, th a t we have some representation does not m ean th a t in other p arts of the problems also th a t oracle will be of help. So in m any cases it is b e tte r to restrict the application of the oracle U to the subset F C F ', where to achieve th e desired hardness the

oracle is required.

8.3

Exploring (t-,w -)hardness

In [104] the notions of hardness and w-hardness for unsatisfiable clause-sets were extensively explored, along w ith their relations to tree- and full-resolution complexity. However, there remain

b oth open questions regarding th e new generalisation to satisfiable clause-sets (hd from Definition 3.4.1) and also new questions raised for unsatisfiable clause-sets related to more recent results.

8.3.1

Exploring hardness

Regarding the underlying hardness notions discussed in C hapter 3, two directions of future research are further charactisations of hardness and th e placement of th e UCk hierarchy in the landscape of all CNF clause-sets. In particular, the following are two possibilities for future research:

1. In Lemma 3.3.2 we saw th a t for unsatisfiable clause-sets F G CCS th a t h d (F ) is the optim al value for the Pudlak-Im pagliazzo game from [133]. C an this characterisation be extended to give a gam e-theoretic characterisation of hardness (from Definition 3.4.1) for

all clause-sets?

2. In [62] a probabilistic argum ent is used to estim ate th e proportion of fc-CNF clause-sets th a t are in SCUTZ and other simple poly-tim e SAT classes. A next step in this direction would be to try to generalise these results to UCk (= SCUlZk) to estim ate how the proportion of CCS th a t UCk makes up grows in k (tending towards 1 as A; tends to oo - recall UfceN0 UCk =

CCS).

8.3.2

Exploring w-hardness

It is to be expected th a t w-hardness can behave very differently from hardness. For example, as expressed by Conjecture 8.1.1, already its second level should contain short clause-sets not

representable in any UCk• However yet we do not have tools at hand to handle w-hardness (we do not even have yet a conjectured example for such a separation). A first task is to investigate which of the results on hardness from this thesis and from [78] can be adapted to w-hardness.

Can the classes WCk go beyond m onotone circuits, which were shown in [17] to be strongly related to the expressive power of arc-consistent CN F representations (see Section 8 . 2 for some

further remarks)? Conjecture 6.1.7 would show the contrary, nam ely th a t in th e (unrestricted) presence of new variables also w-hardness boils down, modulo polytim e com putations, to VC\ (under the relative condition!). If this is true, then th e believable greater power of WCk over UCk would all take place inside arc-consistency; and by Conjecture 8.1.3 it would take place strictly inside arc-consistency.

8.4

A theory of “go o d ” SAT representations

The main future application, which brings the WC-perspective and the <S£ZV7?,-perspective to ­ gether, is in the area of “good SAT representations” . This thesis considers th e approach of representing a boolean function / via a clause-set F G UCk as th e first beginning of w hat is envisaged as a theory of good SAT representations. The main open questions here and future directions for research are now enum erated.

1. Full co n stra in t tran slation : T hroughout this thesis the focus has been on representing

boolean functions. However, in general, when translating to SAT one typically follows a more “full” process:

P ro b lem --- >• C o n stra in ts ---> B oolean functio n s > SAT.

M o d e ll in g E n c o d i n g T r a n s l a t i o n

' --- v --- '

As mentioned in Section 2 . 8 of C hapter 2, the current main m ethodology for deriving

“good” SAT translations is to attack the problem from the constraint perspective - trying to derive SAT translations which m aintain certain forms of consistency in the original constraint network via mechanisms such as unit-clause propagation in th e SAT solver. In light of the UCk,VCk and WCk hierarchies, another possibility is to consider the tra n s­ lation from the SAT perspective, relying on these “target-classes” for guarantees on solver performance, rather th an on consistency notions at the higher level. Interesting future directions are:

(a) E n c o d in g v s tran slation : The focus in this thesis on translation of boolean func­ tions to CNF is both conceptually simpler th a n the constraint to CNF translation, useful on its own in a variety of cases, and useful as the translation component of the above SAT translation process. In future, it would be interesting to look both purely at the encoding part of SAT translation, i.e., assuming an ideal translation and asking when different encodings (i.e., direct, log, order etc) can m aintain different types of constraint consistency, and also for fixed encodings, what are the best translations. (b) U n ions: Proven in [65], it is a well-known fact in the constraint community th a t

m aintaining arc-consistency on acyclic binary constraint networks enforces satisfia­ bility (i.e., if the network is inconsistent the m aintenance procedure will produce an em pty domain for some constraint). From th e SAT perspective, instead if one has clause-sets Fq, . . . , Fm e VCk (for k E No) w ith an acyclic variable interaction hyper­ graph (nodes are variables, hyperedges are var(Fj) for all i € { 1 , . . . , m}) then in the same way one should have U m 'efi m} £ V C k■ This would then provide another m ethod for upper-bounding the p-hardness of SAT instances and constructing (in a tree-like manner) p-soft representations. O f particular interest would be to consider additional constraints one could place on th e variable-interaction hypergraph to al­ low only a constant or bounded increase in (p-)hardness, for example bounds on the tree-width and restrictions on the type of constraint.