3,173 research outputs found
A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report
Nested logic programs have recently been introduced in order to allow for
arbitrarily nested formulas in the heads and the bodies of logic program rules
under the answer sets semantics. Nested expressions can be formed using
conjunction, disjunction, as well as the negation as failure operator in an
unrestricted fashion. This provides a very flexible and compact framework for
knowledge representation and reasoning. Previous results show that nested logic
programs can be transformed into standard (unnested) disjunctive logic programs
in an elementary way, applying the negation as failure operator to body
literals only. This is of great practical relevance since it allows us to
evaluate nested logic programs by means of off-the-shelf disjunctive logic
programming systems, like DLV. However, it turns out that this straightforward
transformation results in an exponential blow-up in the worst-case, despite the
fact that complexity results indicate that there is a polynomial translation
among both formalisms. In this paper, we take up this challenge and provide a
polynomial translation of logic programs with nested expressions into
disjunctive logic programs. Moreover, we show that this translation is modular
and (strongly) faithful. We have implemented both the straightforward as well
as our advanced transformation; the resulting compiler serves as a front-end to
DLV and is publicly available on the Web.Comment: 10 pages; published in Proceedings of the 9th International Workshop
on Non-Monotonic Reasonin
On QBF Proofs and Preprocessing
QBFs (quantified boolean formulas), which are a superset of propositional
formulas, provide a canonical representation for PSPACE problems. To overcome
the inherent complexity of QBF, significant effort has been invested in
developing QBF solvers as well as the underlying proof systems. At the same
time, formula preprocessing is crucial for the application of QBF solvers. This
paper focuses on a missing link in currently-available technology: How to
obtain a certificate (e.g. proof) for a formula that had been preprocessed
before it was given to a solver? The paper targets a suite of commonly-used
preprocessing techniques and shows how to reconstruct certificates for them. On
the negative side, the paper discusses certain limitations of the
currently-used proof systems in the light of preprocessing. The presented
techniques were implemented and evaluated in the state-of-the-art QBF
preprocessor bloqqer.Comment: LPAR 201
Quantified Boolean Formula Games and Their Complexities
Consider QBF, the Quantified Boolean Formula problem, as a combinatorial game
ruleset. The problem is rephrased as determining the winner of the game where
two opposing players take turns assigning values to boolean variables. In this
paper, three common variations of games are applied to create seven new games:
whether each player is restricted to where they may play, which values they may
set variables to, or the condition they are shooting for at the end of the
game. The complexity for determining which player can win is analyzed for all
games. Of the seven, two are trivially in P and the other five are
PSPACE-complete. These varying properties are common for combinatorial games;
reductions from these five hard games can simplify the process for showing the
PSPACE-hardness of other games.Comment: 14 pages, 0 figures, for Integers 2013 Conference proceeding
On Generalizing Decidable Standard Prefix Classes of First-Order Logic
Recently, the separated fragment (SF) of first-order logic has been
introduced. Its defining principle is that universally and existentially
quantified variables may not occur together in atoms. SF properly generalizes
both the Bernays-Sch\"onfinkel-Ramsey (BSR) fragment and the relational monadic
fragment. In this paper the restrictions on variable occurrences in SF
sentences are relaxed such that universally and existentially quantified
variables may occur together in the same atom under certain conditions. Still,
satisfiability can be decided. This result is established in two ways: firstly,
by an effective equivalence-preserving translation into the BSR fragment, and,
secondly, by a model-theoretic argument.
Slight modifications to the described concepts facilitate the definition of
other decidable classes of first-order sentences. The paper presents a second
fragment which is novel, has a decidable satisfiability problem, and properly
contains the Ackermann fragment and---once more---the relational monadic
fragment. The definition is again characterized by restrictions on the
occurrences of variables in atoms. More precisely, after certain
transformations, Skolemization yields only unary functions and constants, and
every atom contains at most one universally quantified variable. An effective
satisfiability-preserving translation into the monadic fragment is devised and
employed to prove decidability of the associated satisfiability problem.Comment: 34 page
Invariant Synthesis for Incomplete Verification Engines
We propose a framework for synthesizing inductive invariants for incomplete
verification engines, which soundly reduce logical problems in undecidable
theories to decidable theories. Our framework is based on the counter-example
guided inductive synthesis principle (CEGIS) and allows verification engines to
communicate non-provability information to guide invariant synthesis. We show
precisely how the verification engine can compute such non-provability
information and how to build effective learning algorithms when invariants are
expressed as Boolean combinations of a fixed set of predicates. Moreover, we
evaluate our framework in two verification settings, one in which verification
engines need to handle quantified formulas and one in which verification
engines have to reason about heap properties expressed in an expressive but
undecidable separation logic. Our experiments show that our invariant synthesis
framework based on non-provability information can both effectively synthesize
inductive invariants and adequately strengthen contracts across a large suite
of programs
Decomposable Theories
We present in this paper a general algorithm for solving first-order formulas
in particular theories called "decomposable theories". First of all, using
special quantifiers, we give a formal characterization of decomposable theories
and show some of their properties. Then, we present a general algorithm for
solving first-order formulas in any decomposable theory "T". The algorithm is
given in the form of five rewriting rules. It transforms a first-order formula
"P", which can possibly contain free variables, into a conjunction "Q" of
solved formulas easily transformable into a Boolean combination of
existentially quantified conjunctions of atomic formulas. In particular, if "P"
has no free variables then "Q" is either the formula "true" or "false". The
correctness of our algorithm proves the completeness of the decomposable
theories.
Finally, we show that the theory "Tr" of finite or infinite trees is a
decomposable theory and give some benchmarks realized by an implementation of
our algorithm, solving formulas on two-partner games in "Tr" with more than 160
nested alternated quantifiers
Circuit Based Quantification: Back to State Set Manipulation within Unbounded Model Checking
In this paper a non-canonical circuit-based state set representation is used to efficiently perform quantifier elimination. The novelty of this approach lies in adapting equivalence checking and logic synthesis techniques, to the goal of compacting circuit based state set representations resulting from existential quantification. The method can be efficiently combined with other verification approaches such as inductive and SAT-based pre-image verifications
Conformant Planning as a Case Study of Incremental QBF Solving
We consider planning with uncertainty in the initial state as a case study of
incremental quantified Boolean formula (QBF) solving. We report on experiments
with a workflow to incrementally encode a planning instance into a sequence of
QBFs. To solve this sequence of incrementally constructed QBFs, we use our
general-purpose incremental QBF solver DepQBF. Since the generated QBFs have
many clauses and variables in common, our approach avoids redundancy both in
the encoding phase and in the solving phase. Experimental results show that
incremental QBF solving outperforms non-incremental QBF solving. Our results
are the first empirical study of incremental QBF solving in the context of
planning and motivate its use in other application domains.Comment: added reference to extended journal article; revision (camera-ready,
to appear in the proceedings of AISC 2014, volume 8884 of LNAI, Springer
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