Variants of Quantified Linear Programming and Quantified Linear Implication

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1 Varants of Quantfed Lnear Programmng and Quantfed Lnear Implcaton Potr Wojcechowsk LDCSEE West Vrgna Unversty Morgantown, WV Pavlos Ernaks DMST Athens Unversty of Economcs and Busness Athens, Greece K. Subraman LDCSEE West Vrgna Unversty Morgantown, WV Abstract A Quantfed Lnear Program (QLP) conssts of a set of lnear nequaltes and a correspondng quantfer strng, n whch each unversally quantfed varable s bounded. By extendng the quantfcaton of varables to mplcatons of two lnear systems, we explore Quantfed Lnear Implcatons (QLIs). QLPs and QLIs offer a rch language that s deal for expressng specfcatons n real-tme schedulng and for modellng reactve systems. In ths paper, we show that the varants of QLP and QLI that arse when the unversally quantfed varables are partally bounded or unbounded can be decded n polynomal tme. Moreover, we show that for each class of the polynomal herarchy (PH), there exsts a form of QLI that s complete for that class, thus provdng a contnuous analogue of the way Quantfed Boolean Formulas cover the PH. We also prove that the generc QLI problem s PSPACE-complete and solve some open problems on QLIs wth one quantfer alternaton. 1 Introducton Quantfed lnear programmng s the problem of checkng whether a lnear system s satsfable wth respect to a gven quantfer strng that specfes whch varables are exstentally quantfed and whch are unversally quantfed. Hence, quantfed lnear programmng s a (non-trval) generalzaton of lnear programmng. Accordngly, a Quantfed Lnear Program (QLP) conssts of a set of lnear nequaltes and a correspondng quantfer strng, n whch a bounded regon of values s also specfed for each unversally quantfed varable (Subraman, 2007). A study on QLPs wth unbounded unversally quantfed varables appears n (Rugger et al., 2013). The problem that arses by extendng the quantfcaton of varables to mplcatons of Ths research was supported n part by the Natonal Scence Foundaton through Award CCF and Award CCF , and Ar Force Offce of Scentfc Research through Award FA Ths research has been co-fnanced by the European Unon (European Socal Fund ESF) and Greek natonal funds through the Operatonal Program Educaton and Lfelong Learnng of the Natonal Strategc Reference Framework (NSRF) - Research Fundng Program: Thales. Investng n knowledge socety through the European Socal Fund. two lnear systems s called Quantfed Lnear Implcaton (QLI) (Ernaks et al., 2012a; Ernaks et al., 2013). QLPs represent a rch language that s deal for expressng schedulablty specfcatons n real-tme schedulng (Gerber et al., 1995; Cho and Agrawala, 2000). In real-tme schedulng, however, t may be the case that the dspatcher has already obtaned a schedule (soluton) but then some constrants are slghtly altered. QLIs can be then utlzed to decde whether the dspatcher needs to recompute a soluton or can stll use the current one. Moreover, QLPs and QLIs can be used to model reactve systems (Koo et al., 1999; Pftzmann and Wadner, 2000; Kam et al., 2001; Hall, 2002), where the unversally quantfed varables represent the envronmental nput, whle the exstentally quantfed varables represent the system s response. In ths paper, we examne some new varants of QLPs and QLIs, and establsh ther computatonal complextes. We also settle several open questons n the correspondng lterature on QLIs. More specfcally, we prove that the varants of QLP and QLI that arse when the unversally quantfed varables are partally bounded or unbounded can be decded n polynomal tme. Moreover, wth respect to each class of the polynomal herarchy (PH), we show that there exsts some nstantaton to the QLI framework that s complete for ths class. Ths s nterestng due to the fact that QLIs cover the PH usng only contnuous (and not dscrete) varables, thus provdng a contnuous analogue to the results of Stockmeyer (1977), where the PH s generated usng Quantfed Boolean Formulas (QBFs). We also show that the generc QLI problem s PSPACE-complete. Fnally, we examne the computatonal complextes of some classes of QLIs wth one quantfer alternaton not dentfed by Ernaks et al. (2013). The rest of ths paper s organzed as follows: Secton 2 formally descrbes the problems that we consder n ths paper. Sectons 3 and 4 examne polynomally-solvable varants of QLPs. Secton 5 answers several open questons concernng QLIs, whle Secton 6 concludes the paper. 2 Statement of Problems A QLP s a conjunctve system of lnear constrants n whch each varable s ether exstentally or unversally quantfed accordng to a gven quantfer strng: x 1 y 1 [l 1, u 1 ]... x n y n [l n, u n ] A x + N y b

2 where x 1... x n s a partton of x wth, possbly, x 1 empty; y 1... y n s a partton of y wth, possbly, y n empty; and l, u are lower and upper bounds n R for y, = 1,..., n. Note that n a QLP each unversally quantfed varable s bounded from above and below. Let us ntroduce two varants of QLP both of whch change the nature of the bounds on the unversal varables. A Partally bounded Quantfed Lnear Program (PQLP) s a QLP n whch each unversally quantfed varable s only bounded on one sde. Wthout loss of generalty, we can assume ths sngle bound forces each such varable to be nonnegatve: x 1 y 1 [0, )... x n y n [0, ) A x + N y b An Unbounded Quantfed Lnear Program (UQLP), also studed by Rugger et al. (2013), s a QLP where there are no bounds on any unversal varable: x 1 y 1... x n y n A x + N y b QLIs extend the noton of ncluson of two lnear systems to arbtrary quantfers: x 1 y 1... x n y n [A x + N y b C x + M y d] We say that a QLI holds f t s true as a frst-order formula over the doman of the reals. The decson problem for a QLI conssts of checkng whether t holds or not. A nomenclature s ntroduced n (Ernaks et al., 2012a) to represent the classes of QLIs. Consder a trple A, Q, R. Let A denote the number of quantfer alternatons n the quantfer strng and Q the frst quantfer. Also, let R be an (A + 1)-character strng, whch specfes whether each quantfed set of varables n the quantfer strng appears on the Left, on the Rght, or on Both sdes of the mplcaton. For nstance, 1,, BL ndcates a problem descrbed by: y x [A x + N y b M y d] We extend the noton of partally bounded and unbounded unversally quantfed varables to QLIs. Note that due to the nature of mplcatons, only the constrants n the Left Hand Sde (LHS) restrct the values that unversally quantfed varables can take. Hence, a Partally bounded Quantfed Lnear Implcaton (PQLI) s a QLI n whch each unversally quantfed varable s only bounded by a sngle absolute constrant. Smlarly to PQLPs, we can assume (wthout loss of generalty) that ths sngle constrant s a nonnegatvty constrant. Thus, a PQLI has the followng form: x 1 y 1... x n y n [(A x b y 0) C x + M y d] Accordngly, an Unbounded Quantfed Lnear Implcaton (UQLI) s a QLI n whch each unversally quantfed varable does not appear n the LHS of the mplcaton at all. Thus, a UQLI has the followng form: x 1 y 1... x n y n [A x b C x + M y d] It s mportant to note that 2-person game semantcs have been proposed both for QLPs (Subraman, 2007) and QLIs (Ernaks et al., 2013). More specfcally, for QLPs such a game ncludes an exstental player X, who chooses values for the exstentally quantfed varables, and a unversal player Y, who chooses values for the unversally quantfed varables. X and Y make ther choces accordng to the order of the varables n the quantfer strng. If, at the end, the nstantated lnear system n the QLP s true, then X wns the game (and we say that X has a wnnng strategy). Otherwse, Y wns the game (and we say that Y has a wnnng strategy). In the case of QLIs, an exstental player X and a unversal player Y also choose ther moves accordng to the order of the varables n the quantfer strng. In any game of ths form, the goals of the players are the followng: X selects the values of the exstentally quantfed varables so as to volate the constrants n the LHS or to satsfy the constrants n the Rght-Hand Sde (RHS) of the mplcaton. On the other hand, Y selects the values of the unversally quantfed varables so as to satsfy the constrants of the LHS and at the same tme to volate the constrants of the RHS of the mplcaton. X wns the game (and has a wnnng strategy) f at the end of the game the nstantated mplcaton n the QLI s true. Otherwse, Y wns the game (and has a wnnng strategy). In both cases, the QLP or the QLI holds precsely when X has a wnnng strategy. It s apparent that these game semantcs can be readly appled to the aforementoned varants. 3 Partally bounded varants In ths secton, we utlze the noton of drecton of a convex set (Ignzo and Cavaler, 1993) to show that both PQLP and PQLI are n P. Frst, we obtan an ntermedate result for PQLPs. Theorem 3.1 PQLP wth a fxed number of quantfer alternatons s n P. Proof: Let L n denote the followng PQLP problem: x n y n [0, ) x n 1 y n 1 [0, )... x 1 n n y 1 [0, ) x 0 (A x ) + (B y ) c =0 For n = 0, L 0 s a Lnear Program (LP), hence n P (Khachyan, 1979). Moreover, assume that L k s n P and consder the problem L : x y [0, ) x k y k [0, )... x 1 y 1 [0, ) x 0 (A x ) + (B y ) c (1) =0 We examne whether t s possble to choose n P an x that makes (1) feasble. Such an x needs to handle the case where y = 0. Thus, the followng nstance of L k needs to be feasble: x y x k y k [0, )... x 1 y 1 [0, ) (A x ) + (B y ) c, y = 0 x 0 =0

3 Moreover, y needs to be unbounded. Let m denote the dmenson of y and each e j, j = 1... m, denote a drecton of the system after x s chosen. Note that the drectons of that system do not depend on the value of x. Ths s because the choce of x only changes the value of the constant term and hence does not affect whether y = e j s a drecton or not. So we only need to check whether the followng m nstances of L k are feasble: y x k y k [0, )... x 1 y 1 [0, ) x 0 k (A x ) + (B y ) 0, y = e j : j = 1... m =0 Thus, the feasblty of (1) can be decded based on the feasblty of m + 1 nstances of L k, whch have been assumed to be n P; so, L s also n P. The nductve proof of Theorem 3.1 mples a recursve procedure for decdng any PQLP wth a fnte number of quantfer alternatons n P. However, f the number of alternatons s unbounded, an exponental number of sub-problems s generated. However, many of these subproblems are superfluous and hence need not be calculated. Next, we show that even f the number of alternatons s unbounded, only polynomally-many LPs are generated. Theorem 3.2 PQLP s n P. Proof: Let L be the followng PQLP problem: x n y n [0, ) x n 1 y n 1 [0, )... x 1 y 1 [0, ) x 0 A x + B y c (2) Usng the recursve method mpled by the proof of Theorem 3.1, L would take exponental tme to solve snce 2 n sub-problems are generated. We show that many of these sub-problems are the same and so they can be precomputed. Each tme we check whether y = 1 s a drecton of the system, we always compute: y x 1 y 1 [0, )... x 1 y 1 [0, ) x 0 A x + B y 0 : y = 1 (3) Ths s because pror choces only change the value of the constant term and hence do not affect whether y = 1 s a drecton or not. Checkng whether y = 1 s a drecton wll nvolve subsequently checkng whether y 1 = 1 through y 1 = 1 are also drectons; ths can be handled by performng these computatons frst. Thus, we frst check whether y 1 = 1 s a drecton by solvng: y 1 x 0 A x + B y 0 : y 1 = 1 Then, when computng whether y 2 = 1 s a drecton, we need to solve: y 2 x 1 y 1 [0, ) x 0 A x + B y 0 : y 2 = 1 Snce we already know that y 1 = 1 s a drecton, we only need to solve: y 2 x 1 y 1 x 0 A x + B y 0 : y 2 = 1, y 1 = 0 Now assume that we know that y 1 = 1 through y 1 = 1 are drectons. Recall that n order to check whether y = 1 s a drecton, we need to solve (3). Snce y 1 = 1 s a drecton, we only need to solve: y x 1 y 1... x 1 y 1 [0, ) x 0 A x + B y 0 : y = 1, y 1 = 0 Applyng the same for y 2 = 1 through y 1 = 1, we get: y x 1 y 1... x 1 y 1 x 0 A x + B y 0 : y = 1, y 1,..., y 1 = 0 Thus, to show that each y = 1 s a drecton, we only need to solve n LPs, each of the form above. Let us now show how L can be solved n P. Snce y n = 1 s a drecton, we only need to check: x n y n x n 1 y n 1 [0, )... x 1 y 1 [0, ) x 0 A x + B y c : y n = 0 Snce y n 1 = 1 through y 1 = 1 are also drectons, the problem can be smplfed down to: x n y n x n 1 y n 1... x 1 y 1 x 0 A x + B y c : y = 0 (4) Thus, to solve L, we only need to solve n + 1 LPs. Furthermore, we can use these LPs to generate the functon whch determnes the value of each x. Let ˆx n contan the values of x that solve (4), whch s equvalent to ˆx n satsfyng: x n x n 1... x 1 x 0 A x c (5) Moreover, for each = 1,..., n, let ˆx 1 contan the values of x that solve: y x 1 y 1... x 1 y 1 x 0 A x + B y 0 : y = 1, y 1... y 1 = 0 whch s equvalent to ˆx 1 satsfyng: x 1... x 1 x 0 A x + B 0 (6) Now we expand the ˆx 1 s wth zeroes so that for each ˆx 1, = 1,..., n, we have ˆx 1 = x. Then, the followng functon can be used to determne the values of x: n x = ˆx n + y ˆx 1 (7) Note that the value that each x takes depends only on the y j s that precede t n the quantfer strng. But then: ( ) n n A x + B y = A ˆx n + y ˆx 1 + B y = A ˆx n + n y (A ˆx 1 + B ) c + n y 0 = c snce A ˆx n c by (5) and A ˆx 1 + B 0 for any = 1,..., n by (6). But ths means that the orgnal PQLP (2) can also be satsfed by the values gven to x by (7). Note that f nstead of varables y n (2), we have a vectors y, then when checkng for drectons, nstead of solvng one LP, we need to solve y LPs. Thus, n that stuaton the total number of LPs needed to be solved s y +1, whch can also be done n P. We wll utlze ths result to show that PQLI s also n P.

4 Corollary 3.1 PQLI s n P. Proof: Consder the followng PQLI problem: x 1 y 1... x n y n [A x b, y 0 C x + D y f] (8) If x A x b s nfeasble (.e., f A x b contans even one constrant), then (8) s trvally satsfed. Otherwse, (8) s equvalent to: x 1 y 1... x n y n [y 0 C x + D y f] whch can be rewrtten as x 1 y 1 [0, )... x n y n [0, ) C x + D y f The latter s a PQLP and hence n P by Theorem Unbounded varants In ths secton, we show that UQLP and UQLI are also n P. Theorem 4.1 UQLP s n P. Proof: Consder the followng UQLP problem: x n y n x n 1 y n 1... x 1 y 1 x 0 A x + B y c (9) To show that ths can be solved n polynomal tme, we wll reduce t to a PQLP. We start by creatng addtonal vectors y and y for each y. Then, for each y, we add constrants y = y y. By applyng unversal quantfcaton to y and y and exstental quantfcaton to y, we construct the followng PQLP: x n y n [0, ) y n [0, ) y n x n 1 y n 1 [0, ) y n 1 [0, ) y n 1... x 1 y 1 [0, ) y 1 [0, ) y 1 x 0 A x + B y c, y = y y (10) where y 1,..., y n s a partton of y and y 1,..., y n s a partton of y. Now consder the correspondng 2-person game for (10). The unversal player has full control over the values assumed by y, (whch can take any value n the range (, )), snce y = y y and both y and y are n the range [0, ). Thus, the quantfer sequence y [0, ) y [0, ) y s equvalent to y. Therefore, (10) s equvalent to (9), whch means that UQLP s n P. Observe that the above theorem has been proven ndependently and usng dfferent technques n (Rugger et al., 2013). We wll use ths result to show that UQLI s also solvable n polynomal tme. Corollary 4.1 UQLI s n P. Proof: Consder the followng UQLI problem: x 1 y 1... x n y n [A x b C x + D y f] (11) Smlarly to the proof of Corollary 3.1, f x A x b s nfeasble, then (11) s trvally satsfed. Otherwse, (11) s equvalent to: x 1 y 1... x n y n C x + D y f whch s a UQLP and hence n P by Theorem QLI and the polynomal herarchy In ths secton, we prove that for each class of the PH, there exsts a form of QLI that s complete for that class. Ths s nterestng, snce QLIs use contnuous varables and not dscrete. Hence, we provde a contnuous analogue to the results n Stockmeyer (1977), where the PH s generated usng QBFs. Moreover, we show the generc QLI problem tself s PSPACE-complete. Let B denote the strng B }. {{.. B }. The followng results were obtaned n (Ernaks et al., 2013, Theorems 14 and 15): 1. k,, B wth k odd s Σ P k -hard. 2. k,, B wth k even s Π P k -hard. To establsh the computatonal complextes of k,, B when k s even and k,, B when k s odd, we frst provde a reducton from Q3DNF (see Appendx A) to QLI. Theorem 5.1 Q3DNF can be reduced to QLI. Proof: We wll reduce the Q3DNF problem to a QLI. Consder a Q3DNF nstance Q(x, y) φ(x, y), where Q(x, y) represents the quantfer strng, x s the set of exstentally quantfed varables, y s the set of unversally quantfed varables, and φ s a dsjuncton of 3-lteral terms. We want to produce a correspondng QLI whch wll hold f and only f Q(x, y) φ(x, y) s satsfable. Let E represent the set of constrants on the LHS of the mplcaton and F the set of constrants on the RHS of the constructed mplcaton. For each exstentally quantfed varable x n the nstance of Q3DNF, we add an exstentally quantfed varable x and a unversally quantfed varable r. We also add the constrants r x and r 1 x to E and the constrants r 0 to F. Note that these constrants are equvalent to r mn(x, 1 x ) r 0. Moreover, we add 0 x 1 to F. For each unversally quantfed varable y n the nstance of Q3DNF, we add an exstentally quantfed varable s and a unversally quantfed varable y. We also add the constrants 0 y 1 to E and the constrants 0 s 1, 2y 1 s, and s 2y to F. Note that these are the only constrants that use y varables, snce the clause constrants wll only use x and s varables. For each clause φ j n the nstance of Q3DNF, we add the exstentally quantfed varable w j, and 3 correspondng constrants to F. These constrants ask for w j to be less than or equal to the exstental varables correspondng to the lterals of φ j. Note that these constrants contan only exstental varables. Even n the case of a unversal varable y n φ j, the constrant contans the exstental varable s of the QLI. Negated lterals are also treated approprately. For example, f φ j = (x, y k, x l ), we add w j x, w j s k, and w j 1 x l to F, whle f φ j = (x, ȳ k, x l ), we add w j x, w j 1 s k, and w j 1 x l to F. Furthermore, we add w 1 + w w m 1 to F. We create the quantfer strng of the QLI accordng to Q(x, y): For x n Q(x, y), we ntroduce x r ; for y n Q(x, y), we ntroduce y s ; fnally, we ntroduce w.

5 To complete the proof, we wll show that the constructed QLI holds f and only f Q(x, y) φ(x, y) s satsfable. We start by establshng that all varables that partcpate n the constrants correspondng to the clauses of the Q3DNF nstance are effectvely restrcted to values 0 or 1. To do so, we utlze the game semantcs ntroduced n Secton 2. Frst note that the exstental player X wll only choose x from the set {0, 1}. Ths s because f x [0, 1], then at least one of the constrants n F would be volated. But then the mplcaton would not hold (hence the unversal player Y would wn the game), snce X cannot cause any constrant n E to be volated. On the other hand, f x (0, 1), then Y could choose to set r = mn(x, 1 x ) > 0, whch would cause the mplcaton not to hold (causng the exstental player to lose the game). To sum up, any choce of x {0, 1} would cause X to lose the game. The unversal player Y wll also choose y from the set {0, 1}. We wll show why ths assumpton s not restrctve. Suppose that Y can wn by choosng y [0, 1]. Then at least one constrant of E s volated (.e., 0 y 1) and the mplcaton holds (.e., a contradcton). Suppose now that Y can wn by choosng y (0, 1). If y (0, 1 2 ], then we have that s [0, 2y ]. However, f nstead Y had chosen y = 0, then s {0} [0, 2y ], thus restrctng the possble responses of X. Snce y only appears n the constrants descrbed above, we have that ths s a strctly better move for Y. Smlarly, choosng y = 1 s strctly better for Y than choosng y [ 1 2, 1). To sum up, we can safely assume that the unversal player would only choose y {0, 1}. Snce y s n the set {0, 1}, we have that the exstental player s forced to set s = y. Any other choce of s would volate at least one constrant of F, causng (agan) the exstental player to lose the game. Hence, s varables are also restrcted n the set {0, 1}. Let us show that for the QLI obtaned from a Q3DNF nstance of the form Q(x, y) φ(x, y), the exstental player has a wnnng strategy for the QLI f and only f Q(x, y) φ(x, y) s satsfable. Only-f part. Assume that the exstental player X has a wnnng strategy U for the constructed QLI. Ths means that for every sequence of moves V made by the unversal player Y, the mplcaton holds. From constrant w 1 + w w m 1, we must have that at least one w j > 0. Ths w j corresponds to the clause φ j of the orgnal Q3DNF, and we can assume wlog that t has the form (x, y k, x l ). Snce the x s and s s are restrcted to the set {0, 1}, the constrants w j x, w j s k, and w j x l force each varable to be 1. Thus at least one term of the orgnal Q3DNF formula s satsfed, meanng that the entre formula s. If part. Assume that Q(x, y) φ(x, y) s satsfable. Then there exst values x for x such that x = [c 1, f 1 (y 1 ), f 2 (y 1, y 2 ),..., f n 1 (y 1, y 2,..., y n 1 ] T and for any values y = [y 1, y 2,..., y n ] T gven to y, the Q3SAT expresson s satsfed. Note that f () are Skolem functons and are used to represent that the values of the elements of x depend on the values of the correspondng elements of y. Snce the Q3DNF expresson s satsfed, at least one clause, say φ j, must be satsfed. Consder the constrants constructed from φ j, assumng wlog that t s of the form (x, y k, x l ). Snce φ j s satsfed, we must have that x, y k, x l are all true. Ths means that x = s k = x l = 1. Thus, X can set w j = 1 and the other clause varables to 0, satsfyng the constrant w 1 +w 2 + +w m 1. Snce the x and y varables can be restrcted to the set {0, 1} and snce s = y, we have that for each, r x and r 1 x mply that r 0 (thus satsfyng the correspondng constrant of F ). For the same reason, for any the constrants 0 x 1, 0 y 1, 0 s 1, 2y 1 s, and s 2y are all satsfed. Hence, E F s satsfed and so X has a wnnng strategy for the correspondng QLI. Snce the last quantfer of a Q3DNF formula s always, ths reducton ncreases the number of quantfer alternatons by one and the produced QLI always has as the fnal quantfer. Ths allows us to obtan the followng two results. Corollary 5.1 k,, B wth k even s Σ P k -hard. Proof: Consder the class of Q3DNF formulas wth k quantfers startng wth an exstental one,.e., wth a quantfer strng of the form.... Such a class s Σ k P- complete (the assumpton that k s even s essental). The prevous proof reduces such a class to a QLI wth a quantfer strng obtaned by addng an exstental quantfer at the end, namely to a k,, B formula. Hence, the result. Corollary 5.2 k,, B wth k odd s Π P k -hard. Proof: Consder the class of Q3DNF formulas wth k quantfers startng wth a unversal one,.e., wth the quantfer strng of the form.... Such a class s Π k P- complete (the assumpton that k s odd s essental). The prevous proof reduces such a class to a QLI wth a quantfer strng obtaned by addng an exstental quantfer at the end, namely to a k,, B formula. Hence, the result. Corollares 5.1 and 5.2 when pared wth (Ernaks et al., 2013, Theorems 14 and 15) show the mportance of the fnal quantfer n a QLI. If the fnal quantfer s, then the QLI corresponds to a Q3SAT nstance wth one fewer quantfer alternaton and so covers the correspondng class n the polynomal herarchy. Smlarly, f the fnal quantfer s, then the QLI corresponds to a Q3DNF nstance wth one fewer quantfer alternaton and so covers the correspondng class n the polynomal herarchy. However these results cover only the hardness of these forms of QLI; to show completeness, we need prove to that each of these problems s also contaned wthn ts level of the polynomal herarchy. Theorem 5.2 k,, B s n Σ P k, k,, B s n Π P k. Proof: Sontag (1985) showed that decdng an arbtrary boolean combnaton of lnear constrants under a gven quantfer strng s contaned wthn PSPACE. QLIs are a sub-problem of ths (snce t can be rewrtten as a quantfed conjuncton of the RHS constrants and the negaton of the LHS constrants) and thus also n PSPACE. Hence, QLIs can be decded even f each varable s restrcted to values that are polynomally-szed wth respect to the nput. Consder problem k,, B. After k rounds of each player choosng polynomaly-szed values, the remanng

6 QLI, ether 0,, B or 0,, B, can be solved n P (Ernaks et al., 2013, Theorems 1 and 6). Thus, k,, B s n Σ P k. Smlarly, k,, B s n Π P k. The varous forms of QLI cover the polynomal herarchy as shown n Fgure 1. Moreover, snce QLIs are n general PSPACE-hard (Ernaks et al., 2013, Theorem 5), the proof of Theorem 5.2 has the followng mplcaton. Corollary 5.3 QLI s PSPACE-complete. Π P 2 2,, BBB conp 1,, BB PH. P 0,, B 0,, B Σ P 2 2,, BBB NP 1,, BB Fgure 1: QLIs and the Polynomal Herarchy 5.1 Related Problems In ths secton, we examne the computatonal complextes of varous classes of QLIs wth one quantfer alternaton that were left open n (Ernaks et al., 2013). Corollary 5.4 1,, RB,.e., decdng whether y x [M x n A x + B y c] holds, s n P. Proof: Frst, we check whether x M x n s nfeasble (.e., whether M x n contans even one constrant). If t s, then the QLI trvally holds. If t s not, then y x [M x n A x+b y c] reduces to y x A x+b y c, whch s a UQLP and hence n P by Theorem 4.1. The decson problem for formula x y [A x + B y c M y n] s NP-complete (Ernaks et al., 2012a). However, by movng the exstentally quantfed varables x to the RHS, the problem becomes tractable. Corollary 5.5 1,, BL,.e., decdng whether x y [A x + B y c M x n] holds, s n P. Proof: Frst, we check whether x M x n holds, whch can be done n P (Khachyan, 1979). If x M x n holds, then x y [A x+b y c M x n] trvally holds. If x M x n does not hold, x y [A x + B y c M x n] reduces to x y A x + B y c, whch can be checked to hold n P by Theorem 4.1. Let us turn our attenton to the class 1,, BR and ts super-class 1,, BB, shown conp-hard n (Ernaks et al., 2013). Note that these classes are also n conp by Theorem 5.2. We provde an alternatve proof for the latter. Lemma 5.1 1,, BR and 1,, BB are n conp. Proof: Problems n 1,, BR are descrbed by: y x [N y b C x + M y d] (12) We consder two cases: N y b beng (a) bounded and (b) unbounded. (a) Let N y b be bounded. Then, f (12) does not hold, there wll exst some y for whch x [N y b C x + M y d] does not hold. Note that ths means that y satsfes N y b (otherwse the mplcaton would be trvally true). But then, snce N y b s bounded, there wll also exst some extreme pont y of N y b, such that x [N y b C x + M y d] does not hold. Our clam follows from the fact that y s an extreme pont of N y b and hence polynomally-szed. (b) Now let N y b be unbounded. For each soluton of N y b to satsfy C x + M y d, we need to have that the extreme ponts of N y b satsfy C x + M y d and that the extreme drectons of N y b are drectons of C x + M y d. For the latter, we need to check whether each soluton of N y 0 also satsfes C x + M y 0,.e., we need to decde y x [N y 0 C x + M y 0]. But snce we are searchng for extreme drectons, we can restrct the values of y to the nterval [ 1, 1]. Hence, consder a case where (12) does not hold. We would frst decde y x [N y 0, 1 y 1 C x + M y 0], whch s n conp, snce the LHS s bounded (case (a)). If ths QLI does not hold, we are done. Otherwse, there wll exst a (polynomally-szed) extreme pont y of N y b such that x [N y b C x + M y d] does not hold (case (a)). Let us now consder problem 1,, BB,.e., y x [A x + N y b C x + M y d]. If A 0, ths QLI s trvally true; for any choce of values for the unversally quantfed varables, there wll exst a choce of the exstentally quantfed varables such that the LHS of the QLI s false. Otherwse (f A = 0), problem 1,, BB reduces to 1,, BR, whch was shown above to be n conp. Corollares 5.4 and 5.5, and Lemma 5.1 solve some open problems on the computatonal complextes of QLIs wth one quantfer alternaton (Ernaks et al., 2013). A complete representaton of all classes wth one quantfer alternaton QLIs startng wth an exstental or wth a unversal quantfer s gven n Appendx B, n Fgure 2 and Fgure 3 respectvely. The symmetry s apparent. 6 Concluson In ths paper, we examned several varants of QLP and QLI that arse when the unversally quantfed varables are partally bounded or unbounded and showed that all these varants are n P. Furthermore, we proved that k,, B s Σ P k -complete and k,, B s Π P k -complete. Hence, we showed that any class of the PH can be represented by some nstantaton to the QLI framework (usng only contnuous varables), thus provdng a contnuous analogue to the way QBFs cover the PH. Moreover, we proved that the

7 generc QLI problem s PSPACE-complete. Fnally, we answered several open questons on the computatonal complextes of classes of QLIs wth one quantfer alternaton, thus completng the map of the computatonal complexty of all such classes of QLIs. References Cho, S. and Agrawala, A Dynamc dspatchng of cyclc real-tme tasks wth relatve tmng constrants. Real-Tme Systems 19(1): Gerber, R.; Pugh, W.; and Saksena, M Parametrc dspatchng of hard real-tme tasks. IEEE Transactons on Computng 44(3): Ernaks, P; Rugger, S; Subraman, K; and Wojcechowsk, P Computatonal complextes of ncluson queres over polyhedral sets. Internatonal Symposum on Artfcal Intellgence and Mathematcs, Fort Lauderdale, FL. Ernaks, P; Rugger, S; Subraman, K; and Wojcechowsk, P Quantfed Lnear Implcatons. Annals of Mathematcs and Artfcal Intellgence (n press) - DOI: /s Hall, R Specfcaton, valdaton, and synthess of emal agent controllers: A case study n functon rch reactve system desgn. Automated Software Engneerng 9(3): Ignzo, J.P. and Cavaler, T.P Lnear Programmng. Prentce Hall Kam, N.; Cohen, I.R.; and Harel, D The mmune system as a reactve system: modelng T cell actvaton wth statecharts. In Proceedngs of the 2001 IEEE Symposa on Human-Centrc Computng Languages and Envronments, Khachyan, L.G A polynomal algorthm n lnear programmng. Doklady Akadema Nauk SSSR 224: Koo, T; Snopol, B.; Sangovann-Vncentell, A.; and Sastry, S A formal approach to reactve system desgn: unmanned aeral vehcle flght management system desgn example. In Proceedngs of the 1999 IEEE Internatonal Symposum on Computer Aded Control System Desgn, Pftzmann, B. and Wadner, M Composton and ntegrty preservaton of secure reactve systems. In Proceedngs of the 2000 ACM Conference on Computer Communcatons Securty, Rugger, S; Ernaks, P; Subraman, K; and Wojcechowsk, P On the complexty of quantfed lnear systems. Theoretcal Computer Scence (n press) - DOI: /j.tcs Sontag, E.D Real Addton and the Polynomal Herarchy. Informaton Processng Letters 20(3): Stockmeyer, L.J The Polynomal-Tme Herarchy. Theoretcal Computer Scence 3: Subraman, K On a decson procedure for quantfed lnear programs. Annals of Mathematcs and Artfcal Intellgence 51(1): Appendx A A quantfed boolean expresson s an nstance of Q3DNF f t s a dsjuncton of terms each of whch s a conjuncton of three lterals. Thus, the followng problem s an nstance of Q3DNF. x 1 y 1 x 2 y 2 x 3 y 3 (x 1 y 3 x 2 ) (x 3 y 2 x 1 ) (x 2 y 1 x 3 ) We can defne the two player game semantcs for ths problem as follows. The exstental and unversal players make moves accordng to the quantfer strng. The exstental player wns a term f all the lterals n the term are set to True. Conversely the unversal player wns the term f at least on lteral s set to False. The exstental player wns the game f he wns at least one term n the dsjuncton, thus causng the expresson to evaluate to True. The unversal player wns f he wns every term n the dsjuncton, thus causng the expresson to evaluate to False. Ths can be consdered as a reversal of the players objectves n an nstance of Q3SAT, where the exstental player needs to wn all the clauses, and the unversal player only needs to wn a sngle clause. Ths occurs because the Q3DNF and Q3SAT problems are complements of each other. 1,, BL 1,, LL Appendx B 1,, BB (NP-complete) 1,, LB (NP-complete) 1,, LR 6 1,, BR 1,, RL 1,, RB 1,, RR Fgure 2: Complexty of classes of QLIs. Arrows denote nclusons.

8 1,, BL 1,, LL 1,, BB (conp-complete) 1,, LB 1,, LR 1,, RL 6 1,, BR 1,, RB (conp-complete) 1,, RR Fgure 3: Complexty of classes of QLIs. Arrows denote nclusons.