1,179 research outputs found
FO(FD): Extending classical logic with rule-based fixpoint definitions
We introduce fixpoint definitions, a rule-based reformulation of fixpoint
constructs. The logic FO(FD), an extension of classical logic with fixpoint
definitions, is defined. We illustrate the relation between FO(FD) and FO(ID),
which is developed as an integration of two knowledge representation paradigms.
The satisfiability problem for FO(FD) is investigated by first reducing FO(FD)
to difference logic and then using solvers for difference logic. These
reductions are evaluated in the computation of models for FO(FD) theories
representing fairness conditions and we provide potential applications of
FO(FD).Comment: Presented at ICLP 2010. 16 pages, 1 figur
Reasoning on Schemata of Formulae
A logic is presented for reasoning on iterated sequences of formulae over
some given base language. The considered sequences, or "schemata", are defined
inductively, on some algebraic structure (for instance the natural numbers, the
lists, the trees etc.). A proof procedure is proposed to relate the
satisfiability problem for schemata to that of finite disjunctions of base
formulae. It is shown that this procedure is sound, complete and terminating,
hence the basic computational properties of the base language can be carried
over to schemata
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
A Modal Logic for Subject-Oriented Spatial Reasoning
We present a modal logic for representing and reasoning about space seen from the subject\u27s perspective. The language of our logic comprises modal operators for the relations "in front", "behind", "to the left", and "to the right" of the subject, which introduce the intrinsic frame of reference; and operators for "behind an object", "between the subject and an object", "to the left of an object", and "to the right of an object", employing the relative frame of reference. The language allows us to express nominals, hybrid operators, and a restricted form of distance operators which, as we demonstrate by example, makes the logic interesting for potential applications. We prove that the satisfiability problem in the logic is decidable and in particular PSpace-complete
A decidable weakening of Compass Logic based on cone-shaped cardinal directions
We introduce a modal logic, called Cone Logic, whose formulas describe
properties of points in the plane and spatial relationships between them.
Points are labelled by proposition letters and spatial relations are induced by
the four cone-shaped cardinal directions. Cone Logic can be seen as a weakening
of Venema's Compass Logic. We prove that, unlike Compass Logic and other
projection-based spatial logics, its satisfiability problem is decidable
(precisely, PSPACE-complete). We also show that it is expressive enough to
capture meaningful interval temporal logics - in particular, the interval
temporal logic of Allen's relations "Begins", "During", and "Later", and their
transposes
A Theory of Sampling for Continuous-time Metric Temporal Logic
This paper revisits the classical notion of sampling in the setting of
real-time temporal logics for the modeling and analysis of systems. The
relationship between the satisfiability of Metric Temporal Logic (MTL) formulas
over continuous-time models and over discrete-time models is studied. It is
shown to what extent discrete-time sequences obtained by sampling
continuous-time signals capture the semantics of MTL formulas over the two time
domains. The main results apply to "flat" formulas that do not nest temporal
operators and can be applied to the problem of reducing the verification
problem for MTL over continuous-time models to the same problem over
discrete-time, resulting in an automated partial practically-efficient
discretization technique.Comment: Revised version, 43 pages
An Effective Fixpoint Semantics for Linear Logic Programs
In this paper we investigate the theoretical foundation of a new bottom-up
semantics for linear logic programs, and more precisely for the fragment of
LinLog that consists of the language LO enriched with the constant 1. We use
constraints to symbolically and finitely represent possibly infinite
collections of provable goals. We define a fixpoint semantics based on a new
operator in the style of Tp working over constraints. An application of the
fixpoint operator can be computed algorithmically. As sufficient conditions for
termination, we show that the fixpoint computation is guaranteed to converge
for propositional LO. To our knowledge, this is the first attempt to define an
effective fixpoint semantics for linear logic programs. As an application of
our framework, we also present a formal investigation of the relations between
LO and Disjunctive Logic Programming. Using an approach based on abstract
interpretation, we show that DLP fixpoint semantics can be viewed as an
abstraction of our semantics for LO. We prove that the resulting abstraction is
correct and complete for an interesting class of LO programs encoding Petri
Nets.Comment: 39 pages, 5 figures. To appear in Theory and Practice of Logic
Programmin
Fast LTL Satisfiability Checking by SAT Solvers
Satisfiability checking for Linear Temporal Logic (LTL) is a fundamental step
in checking for possible errors in LTL assertions. Extant LTL satisfiability
checkers use a variety of different search procedures. With the sole exception
of LTL satisfiability checking based on bounded model checking, which does not
provide a complete decision procedure, LTL satisfiability checkers have not
taken advantage of the remarkable progress over the past 20 years in Boolean
satisfiability solving. In this paper, we propose a new LTL
satisfiability-checking framework that is accelerated using a Boolean SAT
solver. Our approach is based on the variant of the \emph{obligation-set
method}, which we proposed in earlier work. We describe here heuristics that
allow the use of a Boolean SAT solver to analyze the obligations for a given
LTL formula. The experimental evaluation indicates that the new approach
provides a a significant performance advantage
Model Checking Linear Logic Specifications
The overall goal of this paper is to investigate the theoretical foundations
of algorithmic verification techniques for first order linear logic
specifications. The fragment of linear logic we consider in this paper is based
on the linear logic programming language called LO enriched with universally
quantified goal formulas. Although LO was originally introduced as a
theoretical foundation for extensions of logic programming languages, it can
also be viewed as a very general language to specify a wide range of
infinite-state concurrent systems.
Our approach is based on the relation between backward reachability and
provability highlighted in our previous work on propositional LO programs.
Following this line of research, we define here a general framework for the
bottom-up evaluation of first order linear logic specifications. The evaluation
procedure is based on an effective fixpoint operator working on a symbolic
representation of infinite collections of first order linear logic formulas.
The theory of well quasi-orderings can be used to provide sufficient conditions
for the termination of the evaluation of non trivial fragments of first order
linear logic.Comment: 53 pages, 12 figures "Under consideration for publication in Theory
and Practice of Logic Programming
Efficient Certified Resolution Proof Checking
We present a novel propositional proof tracing format that eliminates complex
processing, thus enabling efficient (formal) proof checking. The benefits of
this format are demonstrated by implementing a proof checker in C, which
outperforms a state-of-the-art checker by two orders of magnitude. We then
formalize the theory underlying propositional proof checking in Coq, and
extract a correct-by-construction proof checker for our format from the
formalization. An empirical evaluation using 280 unsatisfiable instances from
the 2015 and 2016 SAT competitions shows that this certified checker usually
performs comparably to a state-of-the-art non-certified proof checker. Using
this format, we formally verify the recent 200 TB proof of the Boolean
Pythagorean Triples conjecture
- âŠ