We now extend linear systems to allow for quantified variables. We specifically look at Quantified Linear Programs (QLPs) and Quantified Linear Implications (QLIs)
Quantified Linear Programmingextends Linear Programming by allowing the variables
CHAPTER 2. CONSTRAINT SYSTEMS 21
interested in deciding the following query:
G : ∃x1∈ [a1, b1] ∀y1∈ [l1, u1] . . . ∃xn∈ [an, bn] ∀yn∈ [ln, un]
A · [x y]T≤ b, x ≥ 0 (2.3)
where
• A is an m × 2 · n matrix called the constraint matrix,
• x is a n−vector, representing the control variables (these are existentially quantified), • y is a n−vector, representing the variables that can assume values within a pre-
specified range; i.e., component yi has a lower bound of li and an upper bound of
ui(these are universally quantified),
• b is an m−vector,
• {ai, bi}, i = 1, 2, . . . , n are rational numbers bounding variable xi and {li, ui} are ra-
tional numbers bounding yi.
The pair (A, b) is called the Constraint System. Without loss of generality, we assume that the quantifiers are strictly alternating, since we can always add dummy variables (and constraints, if necessary) without affecting the correctness or complexity of the problem
[Pap94]. Let us say that an existentially quantified dummy variable xp is added to the
quantifier string. We can add the constraint xp= 0 to the constraint system. Note that
the value of xp is fixed and cannot depend on the values of other yi variables; further the
variable xpis not part of any constraint involving the original variables of the system.
The string ∃x1∈ [a1, b1] ∀y1∈ [l1, u1] ∃x2∈ [a2, b2] ∀y2∈ [l2, u2] . . . ∃xn∈ [an, bn] ∀yn∈
[ln, un] is called the quantifier string of the given QLP and is denoted by Q(x, y). The length
of the quantifier string, is denoted by |Q(x, y)| and it is equal to the dimension of A. Note that the range constraints on the existentially quantified variables can be included in the
CHAPTER 2. CONSTRAINT SYSTEMS 22
constraint matrix A (xi∈ [ai, bi] can be written as ai≤ xi, xi≤ bi) and thus the generic QLP
can be represented as:
G :∃x1∀y1∈ [l1, u1] ∃x2∀y2∈ [l2, u2] . . . ∃xn∀yn∈ [ln, un]
A · [x y]T≤ b (2.4)
However, the range constraints on the yi variables cannot be moved into the constraint
system.
It follows that the QLP problem can be thought of as checking whether a polyhedron de-
scribed by a system of linear inequalities (A · [x y]T≤ b) is non-empty vis-a-vis the spec-
ified quantifier string (say Q(x, y)). The pair < Q(x, y), (A, b) > is called a Parametric Polytope. In other words, Quantified Linear Programming is concerned with checking the non-emptiness of Parametric Polytopes, just as traditional linear programming is concerned with checking the non-emptiness of simple polytopes. For the rest of this paper, we shall assume that the generic QLP has the form described by System (2.4), so that the analysis is simplified. Accordingly, we observe that in a QLP, the dimension of the constraint matrix A and hence the length of the quantifier string is always even.
An example of a quantified linear program (QLP) is the following:
∃x1∈ [0, 1] ∀y1∈ [1, 4] ∃x2∈ [3, 9] x1+ y1+ x2≥ 4
3x1− 5y1+ 7x2≤ −5.
As with linear programs, the conjunction of inequalities is frequently written in matrix- vector form. Note that the feasibility version of a linear program is a QLP with all quanti- fiers existential.
Definition 2.2.18. A QLP is said to be feasible if it is true as a first-order sentence over the real numbers, using the standard semantics for bounded quantifiers.
CHAPTER 2. CONSTRAINT SYSTEMS 23
When we are interested in computational complexity, we restrict QLPs to be defined over the rational numbers.
There is an equivalent formula game definition of truth for QLPs. Player I (the ∃-player) chooses the values for the existentially quantified variables and Player II (the ∀-player) chooses the values for the universally quantified variables. The players must choose values consistent with the bounds on the variables. The ∃-player wins precisely when the values chosen constitute a feasible point for the matrix of the QLP. See [Sub07] for details.
In fact, we can place a further restriction on the values chosen be the ∀-player.
Lemma 2.2.1. [Sub07] Given a QLP φ , define the modified formula game of φ to be the
standard formula game with the added restriction that the∀-player is restricted to choosing
from the endpoints of the intervals over which the universally quantified variables can
range. For example, if the universally quantified variable eiis quantified over the interval
[2, 3], then the ∀-player may only choose 2 or 3 for the value of ei. Then the∀-player has
a winning strategy for the modified formula game if and only if the∀-player has a winning
strategy for the standard formula game.
Oftentimes we are interested in QLPs with a particular quantifier string.
Definition 2.2.19. Given a string of quantifiersS , a QLP whose quantifiers, when grouped
into blocks, are consistent withS is known as an S -QLP.
For example, an ordinary linear program (with arbitrarily many variables) is an ∃-QLP. Note that ∀∃-QLPs are known as F-QLPs in [Sub07].
Definition 2.2.20. A generalized quantified linear program (GQLP) is a QLP where uni-
versal variables are bounded not by constants but by systems of linear constraints. i.e. ∀ui
such that ui≤ a1v1+ b1u1+ · · · + ai−1vi−1+ bi−1ui−1where v1through vi−1and u1though
ui−1appear before uiin the quantifier string. If uiis bounded above by multiple constraints
it is bounded above by their minimum, similarly if it is bounded bellow by multiple con-
straints then it is bounded by their maximum. For example ∀v01 such that v01≤ 1, v01≥ 0,
CHAPTER 2. CONSTRAINT SYSTEMS 24
An example of a generalized quantified linear program (GQLP) is the following:
∃x1∈ [0, 1] ∀y1∈ [x1− 1, 3x1] ∃x2∈ [3, 9] x1+ y1+ x2≥ 4
3x1− 5y1+ 7x2≤ −5.
Note that Quantified Linear Programming is a generalization of the feasibility problem of linear programming; we have not mentioned an objective function to optimize. Multi-
level games are a generalization of Linear Programming which do incorporate objective
functions. Briefly, a multi-level game involves multiple players choosing values for real- valued variables. There is a common set of inequalities in these variables; each player must ensure that her choices do not violate these inequalities. Moreover, each player has her own objective function, which she seeks to optimize. It can be shown that games with (p + 1)
players can capture the Σpand Πplevels of PH [Jer85].
We also examine the problem of implication in quantified linear programs.
Consider now two linear systems P1: A · x ≤ b and P2: C · x ≤ d. We say that P1 is
includedin P2if every solution of P1is also a solution of P2. This holds if and only if the
logic formula ∀x [Ax ≤ b → C · x ≤ d] is true in the domain of the reals. We extend the notion of inclusion to arbitrary quantifiers by introducing Quantified Linear Implications of two linear systems:
∃x1∀y1 . . . ∃xn∀yn [A · x + N · y ≤ b → C · x + M · y ≤ d] (2.5)
where x1. . . xn and y1. . . yn are partitions of x and y respectively, and where x1and/or yn
may be empty. We say that a QLI holds if it is true as a first-order formula over the domain of the reals. The decision problem for a QLI consists of checking whether it holds or not.
Let Q(x, y) denote the quantifier string, namely ∃x1∀y1 . . . ∃xn∀yn in System (2.5).
We introduce a nomenclature to represent the different classes of QLIs that we will be examining. Consider a triple hA, Q, Ri. Let A denote the number of quantifier alternations
CHAPTER 2. CONSTRAINT SYSTEMS 25
in the quantifier string Q(x, y) and Q the first quantifier of Q(x, y). Also, let R be an (A + 1)-character string, specifying for each quantified set of variables in Q(x, y) whether they appear on the Left, on the Right, or on Both sides of the implication. For instance, h1, ∃, LBi indicates a problem described by:
∃x ∀y [A · x + N · y ≤ b → M · y ≤ d]
Example 20: Consider the following QLI of the class h2, ∀, LRBi:
∀s1∃r1∀x1∀x2 x1≥ 0 − 3r1 x1≤ 2 − 5r1 x2≥ 0 + 2r1 x2≤ 1 + 3r1 → x1+ x2≥ −1 + 3s1 x1+ x2≤ 1 + 4s1 x1− x2≥ −1 + 5s1 x1− 3x2≤ 3 + 7s1
Let P1denote the left-hand side and P2the right-hand side linear system of the implication.
Figure 2.4 presents P1 and P2 for specific values of s1 and r1, i.e., s1= r1= 0. Note that
for these values P1 is included in P2 (i.e., if both s1 and r1 were existentially quantified).
However, in order for the above QLI to hold, for all values of s1there must exist a value of
CHAPTER 2. CONSTRAINT SYSTEMS 26
x1 x2
P1for s1= r1= 0
P2for s1= 0
27
Chapter 3
Refutations
In this chapter, we discuss refutation systems and focus on the specific refutation sys- tems examined in this dissertation.
3.1
Refutations in Boolean Formulas
First we examine refutations for Boolean formulas.
We focus solely on resolution refutations. A resolution refutation consists of a series of resolution steps terminating in the empty clause t.
Definition 3.1.1. A resolution step derives a resolvent clause from two parent clauses. A
resolution step with parent clauses(α ∨ x) and (¬x ∨ β ) with resolvent (α ∨ β ), is denoted
as
(α ∨ x), (¬x ∨ β ) |Res1 (α ∨ β ).
The variable x is called the matching or resolution variable.
CHAPTER 3. REFUTATIONS 28
plying a resolution step with these clauses as parents results in:
(x1∨ ¬x2∨ x3), (¬x1∨ x3∨ ¬x4) |Res1 (¬x2∨ x3∨ ¬x4).
Note that despite x3 appearing in both parent clauses, it occurs only once in the resol-
vent.
Definition 3.1.2. A resolution refutation of a Boolean formula Φ is a sequence of resolu- tion steps such that
1. Each parent clause is either in Φ or is the resolvent of a previous resolution step.
2. The final resolvent is the empty clause,t.
A Boolean formula Φ is infeasible if and only if it has a resolution refutation. Such a
refutation is denoted as Φ |Res t.