23,168 research outputs found
A novel EGs-based framework for systematic propositional-formula simplification
Funding: Bowles is partially supported by Austrian FWF Meitner Fellowship M-3338 N.This paper presents a novel simplification calculus for propositional logic derived from Peirce’s Existential Graphs’ rules of inference and implication graphs. Our rules can be applied to arbitrary propositional logic formulae (not only in CNF), are equivalence-preserving, guarantee a monotonically decreasing number of clauses and literals, and maximise the preservation of structural problem information. Our techniques can also be seen as higher-level SAT preprocessing, and we show how one of our rules (TWSR) generalises and streamlines most of the known equivalence-preserving SAT preprocessing methods. We further show how this rule can be extended with a novel n-ary implication graph to capture all known equivalence-preserving preprocessing procedures. Finally, we discuss the complexity and implementation of our framework as a solver-agnostic algorithm to simplify Boolean satisfiability problems and arbitrary propositional formula.Postprin
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Cook and Krajíček [9] have obtained the following Karp-Lipton result in bounded arithmetic: if the theory proves , then collapses to , and this collapse is provable in . Here we show the converse implication, thus answering an open question from [9]. We obtain this result by formalizing in a hard/easy argument of Buhrman, Chang, and Fortnow [3]. In addition, we continue the investigation of propositional proof systems using advice, initiated by Cook and Krajíček [9]. In particular, we obtain several optimal and even p-optimal proof systems using advice. We further show that these p-optimal systems are equivalent to natural extensions of Frege systems
The Complexity of Reasoning for Fragments of Default Logic
Default logic was introduced by Reiter in 1980. In 1992, Gottlob classified
the complexity of the extension existence problem for propositional default
logic as \SigmaPtwo-complete, and the complexity of the credulous and
skeptical reasoning problem as SigmaP2-complete, resp. PiP2-complete.
Additionally, he investigated restrictions on the default rules, i.e.,
semi-normal default rules. Selman made in 1992 a similar approach with
disjunction-free and unary default rules. In this paper we systematically
restrict the set of allowed propositional connectives. We give a complete
complexity classification for all sets of Boolean functions in the meaning of
Post's lattice for all three common decision problems for propositional default
logic. We show that the complexity is a hexachotomy (SigmaP2-, DeltaP2-, NP-,
P-, NL-complete, trivial) for the extension existence problem, while for the
credulous and skeptical reasoning problem we obtain similar classifications
without trivial cases.Comment: Corrected versio
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
Generic Modal Cut Elimination Applied to Conditional Logics
We develop a general criterion for cut elimination in sequent calculi for
propositional modal logics, which rests on absorption of cut, contraction,
weakening and inversion by the purely modal part of the rule system. Our
criterion applies also to a wide variety of logics outside the realm of normal
modal logic. We give extensive example instantiations of our framework to
various conditional logics. For these, we obtain fully internalised calculi
which are substantially simpler than those known in the literature, along with
leaner proofs of cut elimination and complexity. In one case, conditional logic
with modus ponens and conditional excluded middle, cut elimination and
complexity were explicitly stated as open in the literature
On Sub-Propositional Fragments of Modal Logic
In this paper, we consider the well-known modal logics ,
, , and , and we study some of their
sub-propositional fragments, namely the classical Horn fragment, the Krom
fragment, the so-called core fragment, defined as the intersection of the Horn
and the Krom fragments, plus their sub-fragments obtained by limiting the use
of boxes and diamonds in clauses. We focus, first, on the relative expressive
power of such languages: we introduce a suitable measure of expressive power,
and we obtain a complex hierarchy that encompasses all fragments of the
considered logics. Then, after observing the low expressive power, in
particular, of the Horn fragments without diamonds, we study the computational
complexity of their satisfiability problem, proving that, in general, it
becomes polynomial
Complexity of Non-Monotonic Logics
Over the past few decades, non-monotonic reasoning has developed to be one of
the most important topics in computational logic and artificial intelligence.
Different ways to introduce non-monotonic aspects to classical logic have been
considered, e.g., extension with default rules, extension with modal belief
operators, or modification of the semantics. In this survey we consider a
logical formalism from each of the above possibilities, namely Reiter's default
logic, Moore's autoepistemic logic and McCarthy's circumscription.
Additionally, we consider abduction, where one is not interested in inferences
from a given knowledge base but in computing possible explanations for an
observation with respect to a given knowledge base.
Complexity results for different reasoning tasks for propositional variants
of these logics have been studied already in the nineties. In recent years,
however, a renewed interest in complexity issues can be observed. One current
focal approach is to consider parameterized problems and identify reasonable
parameters that allow for FPT algorithms. In another approach, the emphasis
lies on identifying fragments, i.e., restriction of the logical language, that
allow more efficient algorithms for the most important reasoning tasks. In this
survey we focus on this second aspect. We describe complexity results for
fragments of logical languages obtained by either restricting the allowed set
of operators (e.g., forbidding negations one might consider only monotone
formulae) or by considering only formulae in conjunctive normal form but with
generalized clause types.
The algorithmic problems we consider are suitable variants of satisfiability
and implication in each of the logics, but also counting problems, where one is
not only interested in the existence of certain objects (e.g., models of a
formula) but asks for their number.Comment: To appear in Bulletin of the EATC
The Complexity of Reasoning for Fragments of Autoepistemic Logic
Autoepistemic logic extends propositional logic by the modal operator L. A
formula that is preceded by an L is said to be "believed". The logic was
introduced by Moore 1985 for modeling an ideally rational agent's behavior and
reasoning about his own beliefs. In this paper we analyze all Boolean fragments
of autoepistemic logic with respect to the computational complexity of the
three most common decision problems expansion existence, brave reasoning and
cautious reasoning. As a second contribution we classify the computational
complexity of counting the number of stable expansions of a given knowledge
base. To the best of our knowledge this is the first paper analyzing the
counting problem for autoepistemic logic
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