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Incremental closure for systems of two variables per inequality
Subclasses of linear inequalities where each inequality has at most two vari- ables are popular in abstract interpretation and model checking, because they strike a balance between what can be described and what can be efficiently computed. This paper focuses on the TVPI class of inequalities, for which each coefficient of each two variable inequality is unrestricted. An implied TVPI in- equality can be generated from a pair of TVPI inequalities by eliminating a given common variable (echoing resolution on clauses). This operation, called result , can be applied to derive TVPI inequalities which are entailed (implied) by a given TVPI system. The key operation on TVPI is calculating closure: satisfiability can be observed from a closed system and a closed system also simplifies the calculation of other operations. A closed system can be derived by repeatedly applying the result operator. The process of adding a single TVPI inequality to an already closed input TVPI system and then finding the closure of this augmented system is called incremental closure. This too can be calcu- lated by the repeated application of the result operator. This paper studies the calculus defined by result , the structure of result derivations, and how deriva- tions can be combined and controlled. A series of lemmata on derivations are presented that, collectively, provide a pathway for synthesising an algorithm for incremental closure. The complexity of the incremental closure algorithm is analysed and found to be O (( n 2 + m 2 )lg( m )), where n is the number of variables and m the number of inequalities of the input TVPI system
Logahedra: A new weakly relational domain
Weakly relational numeric domains express restricted classes of linear inequalities that strike a balance between what can be described and what can be efficiently computed. Popular weakly relational domains such as bounded differences and octagons have found application in model checking and abstract interpretation. This paper introduces logahedra, which are more expressiveness than octagons, but less expressive than arbitrary systems of two variable per inequality constraints. Logahedra allow coefficients of inequalities to be powers of two whilst retaining many of the desirable algorithmic properties of octagons
An Improved Tight Closure Algorithm for Integer Octagonal Constraints
Integer octagonal constraints (a.k.a. ``Unit Two Variables Per Inequality''
or ``UTVPI integer constraints'') constitute an interesting class of
constraints for the representation and solution of integer problems in the
fields of constraint programming and formal analysis and verification of
software and hardware systems, since they couple algorithms having polynomial
complexity with a relatively good expressive power. The main algorithms
required for the manipulation of such constraints are the satisfiability check
and the computation of the inferential closure of a set of constraints. The
latter is called `tight' closure to mark the difference with the (incomplete)
closure algorithm that does not exploit the integrality of the variables. In
this paper we present and fully justify an O(n^3) algorithm to compute the
tight closure of a set of UTVPI integer constraints.Comment: 15 pages, 2 figure
Transfer Function Synthesis without Quantifier Elimination
Traditionally, transfer functions have been designed manually for each
operation in a program, instruction by instruction. In such a setting, a
transfer function describes the semantics of a single instruction, detailing
how a given abstract input state is mapped to an abstract output state. The net
effect of a sequence of instructions, a basic block, can then be calculated by
composing the transfer functions of the constituent instructions. However,
precision can be improved by applying a single transfer function that captures
the semantics of the block as a whole. Since blocks are program-dependent, this
approach necessitates automation. There has thus been growing interest in
computing transfer functions automatically, most notably using techniques based
on quantifier elimination. Although conceptually elegant, quantifier
elimination inevitably induces a computational bottleneck, which limits the
applicability of these methods to small blocks. This paper contributes a method
for calculating transfer functions that finesses quantifier elimination
altogether, and can thus be seen as a response to this problem. The
practicality of the method is demonstrated by generating transfer functions for
input and output states that are described by linear template constraints,
which include intervals and octagons.Comment: 37 pages, extended version of ESOP 2011 pape
Joint Scheduling and ARQ for MU-MIMO Downlink in the Presence of Inter-Cell Interference
User scheduling and multiuser multi-antenna (MU-MIMO) transmission are at the
core of high rate data-oriented downlink schemes of the next-generation of
cellular systems (e.g., LTE-Advanced). Scheduling selects groups of users
according to their channels vector directions and SINR levels. However, when
scheduling is applied independently in each cell, the inter-cell interference
(ICI) power at each user receiver is not known in advance since it changes at
each new scheduling slot depending on the scheduling decisions of all
interfering base stations. In order to cope with this uncertainty, we consider
the joint operation of scheduling, MU-MIMO beamforming and Automatic Repeat
reQuest (ARQ). We develop a game-theoretic framework for this problem and build
on stochastic optimization techniques in order to find optimal scheduling and
ARQ schemes. Particularizing our framework to the case of "outage service
rates", we obtain a scheme based on adaptive variable-rate coding at the
physical layer, combined with ARQ at the Logical Link Control (ARQ-LLC). Then,
we present a novel scheme based on incremental redundancy Hybrid ARQ (HARQ)
that is able to achieve a throughput performance arbitrarily close to the
"genie-aided service rates", with no need for a genie that provides
non-causally the ICI power levels. The novel HARQ scheme is both easier to
implement and superior in performance with respect to the conventional
combination of adaptive variable-rate coding and ARQ-LLC.Comment: Submitted to IEEE Transactions on Communications, v2: small
correction
Size-Change Termination, Monotonicity Constraints and Ranking Functions
Size-Change Termination (SCT) is a method of proving program termination
based on the impossibility of infinite descent. To this end we may use a
program abstraction in which transitions are described by monotonicity
constraints over (abstract) variables. When only constraints of the form x>y'
and x>=y' are allowed, we have size-change graphs. Both theory and practice are
now more evolved in this restricted framework then in the general framework of
monotonicity constraints. This paper shows that it is possible to extend and
adapt some theory from the domain of size-change graphs to the general case,
thus complementing previous work on monotonicity constraints. In particular, we
present precise decision procedures for termination; and we provide a procedure
to construct explicit global ranking functions from monotonicity constraints in
singly-exponential time, which is better than what has been published so far
even for size-change graphs.Comment: revised version of September 2
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Closure Algorithms for Domains with Two Variables Per Inequality
Weakly relational numeric domains express restricted classes of linear inequalities that strike a balance between what can be described and what can be efficiently computed. Such domains often restrict their attention of TVPI constraints which are systems of constraints where each constraint involves, at most, two variables. This technical report addresses the problem of deriving an incremental version of the closure operation. In this operation, a new constraint is added to a system that is already closed, and the computational problem is how to efficiently close the augmented system
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