30,753 research outputs found
The Complexity of Model Checking Higher-Order Fixpoint Logic
Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed
\lambda-calculus and the modal \lambda-calculus. This makes it a highly
expressive temporal logic that is capable of expressing various interesting
correctness properties of programs that are not expressible in the modal
\lambda-calculus.
This paper provides complexity results for its model checking problem. In
particular we consider those fragments of HFL built by using only types of
bounded order k and arity m. We establish k-fold exponential time completeness
for model checking each such fragment. For the upper bound we use fixpoint
elimination to obtain reachability games that are singly-exponential in the
size of the formula and k-fold exponential in the size of the underlying
transition system. These games can be solved in deterministic linear time. As a
simple consequence, we obtain an exponential time upper bound on the expression
complexity of each such fragment.
The lower bound is established by a reduction from the word problem for
alternating (k-1)-fold exponential space bounded Turing Machines. Since there
are fixed machines of that type whose word problems are already hard with
respect to k-fold exponential time, we obtain, as a corollary, k-fold
exponential time completeness for the data complexity of our fragments of HFL,
provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer
Scienc
Power of Quantum Computation with Few Clean Qubits
This paper investigates the power of polynomial-time quantum computation in
which only a very limited number of qubits are initially clean in the |0>
state, and all the remaining qubits are initially in the totally mixed state.
No initializations of qubits are allowed during the computation, nor
intermediate measurements. The main results of this paper are unexpectedly
strong error-reducible properties of such quantum computations. It is proved
that any problem solvable by a polynomial-time quantum computation with
one-sided bounded error that uses logarithmically many clean qubits can also be
solvable with exponentially small one-sided error using just two clean qubits,
and with polynomially small one-sided error using just one clean qubit. It is
further proved in the case of two-sided bounded error that any problem solvable
by such a computation with a constant gap between completeness and soundness
using logarithmically many clean qubits can also be solvable with exponentially
small two-sided error using just two clean qubits. If only one clean qubit is
available, the problem is again still solvable with exponentially small error
in one of the completeness and soundness and polynomially small error in the
other. As an immediate consequence of the above result for the two-sided-error
case, it follows that the TRACE ESTIMATION problem defined with fixed constant
threshold parameters is complete for the classes of problems solvable by
polynomial-time quantum computations with completeness 2/3 and soundness 1/3
using logarithmically many clean qubits and just one clean qubit. The
techniques used for proving the error-reduction results may be of independent
interest in themselves, and one of the technical tools can also be used to show
the hardness of weak classical simulations of one-clean-qubit computations
(i.e., DQC1 computations).Comment: 44 pages + cover page; the results in Section 8 are overlapping with
the main results in arXiv:1409.677
Delta-Complete Decision Procedures for Satisfiability over the Reals
We introduce the notion of "\delta-complete decision procedures" for solving
SMT problems over the real numbers, with the aim of handling a wide range of
nonlinear functions including transcendental functions and solutions of
Lipschitz-continuous ODEs. Given an SMT problem \varphi and a positive rational
number \delta, a \delta-complete decision procedure determines either that
\varphi is unsatisfiable, or that the "\delta-weakening" of \varphi is
satisfiable. Here, the \delta-weakening of \varphi is a variant of \varphi that
allows \delta-bounded numerical perturbations on \varphi. We prove the
existence of \delta-complete decision procedures for bounded SMT over reals
with functions mentioned above. For functions in Type 2 complexity class C,
under mild assumptions, the bounded \delta-SMT problem is in NP^C.
\delta-Complete decision procedures can exploit scalable numerical methods for
handling nonlinearity, and we propose to use this notion as an ideal
requirement for numerically-driven decision procedures. As a concrete example,
we formally analyze the DPLL framework, which integrates Interval
Constraint Propagation (ICP) in DPLL(T), and establish necessary and sufficient
conditions for its \delta-completeness. We discuss practical applications of
\delta-complete decision procedures for correctness-critical applications
including formal verification and theorem proving.Comment: A shorter version appears in IJCAR 201
The Complexity of Planning Revisited - A Parameterized Analysis
The early classifications of the computational complexity of planning under
various restrictions in STRIPS (Bylander) and SAS+ (Baeckstroem and Nebel) have
influenced following research in planning in many ways. We go back and
reanalyse their subclasses, but this time using the more modern tool of
parameterized complexity analysis. This provides new results that together with
the old results give a more detailed picture of the complexity landscape. We
demonstrate separation results not possible with standard complexity theory,
which contributes to explaining why certain cases of planning have seemed
simpler in practice than theory has predicted. In particular, we show that
certain restrictions of practical interest are tractable in the parameterized
sense of the term, and that a simple heuristic is sufficient to make a
well-known partial-order planner exploit this fact.Comment: (author's self-archived copy
Constraint LTL Satisfiability Checking without Automata
This paper introduces a novel technique to decide the satisfiability of
formulae written in the language of Linear Temporal Logic with Both future and
past operators and atomic formulae belonging to constraint system D (CLTLB(D)
for short). The technique is based on the concept of bounded satisfiability,
and hinges on an encoding of CLTLB(D) formulae into QF-EUD, the theory of
quantifier-free equality and uninterpreted functions combined with D. Similarly
to standard LTL, where bounded model-checking and SAT-solvers can be used as an
alternative to automata-theoretic approaches to model-checking, our approach
allows users to solve the satisfiability problem for CLTLB(D) formulae through
SMT-solving techniques, rather than by checking the emptiness of the language
of a suitable automaton A_{\phi}. The technique is effective, and it has been
implemented in our Zot formal verification tool.Comment: 39 page
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?
Many reasoning problems are based on the problem of satisfiability (SAT).
While SAT itself becomes easy when restricting the structure of the formulas in
a certain way, the situation is more opaque for more involved decision
problems. We consider here the CardMinSat problem which asks, given a
propositional formula and an atom , whether is true in some
cardinality-minimal model of . This problem is easy for the Horn
fragment, but, as we will show in this paper, remains -complete (and
thus -hard) for the Krom fragment (which is given by formulas in
CNF where clauses have at most two literals). We will make use of this fact to
study the complexity of reasoning tasks in belief revision and logic-based
abduction and show that, while in some cases the restriction to Krom formulas
leads to a decrease of complexity, in others it does not. We thus also consider
the CardMinSat problem with respect to additional restrictions to Krom formulas
towards a better understanding of the tractability frontier of such problems
Efficient Open World Reasoning for Planning
We consider the problem of reasoning and planning with incomplete knowledge
and deterministic actions. We introduce a knowledge representation scheme
called PSIPLAN that can effectively represent incompleteness of an agent's
knowledge while allowing for sound, complete and tractable entailment in
domains where the set of all objects is either unknown or infinite. We present
a procedure for state update resulting from taking an action in PSIPLAN that is
correct, complete and has only polynomial complexity. State update is performed
without considering the set of all possible worlds corresponding to the
knowledge state. As a result, planning with PSIPLAN is done without direct
manipulation of possible worlds. PSIPLAN representation underlies the PSIPOP
planning algorithm that handles quantified goals with or without exceptions
that no other domain independent planner has been shown to achieve. PSIPLAN has
been implemented in Common Lisp and used in an application on planning in a
collaborative interface.Comment: 39 pages, 13 figures. to appear in Logical Methods in Computer
Scienc
Model Checking CTL is Almost Always Inherently Sequential
The model checking problem for CTL is known to be P-complete (Clarke, Emerson, and Sistla (1986), see Schnoebelen (2002)). We consider fragments of CTL obtained by restricting the use of temporal modalities or the use of negations—restrictions already studied for LTL by Sistla and Clarke (1985) and Markey (2004). For all these fragments, except for the trivial case without any temporal operator, we systematically prove model checking to be either inherently sequential (P-complete) or very efficiently parallelizable (LOGCFL-complete). For most fragments, however, model checking for CTL is already P-complete. Hence our results indicate that in most applications, approaching CTL model checking by parallelism will not result in the desired speed up. We also completely determine the complexity of the model checking problem for all fragments of the extensions ECTL, CTL +, and ECTL +
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