629 research outputs found
Deciding First-Order Satisfiability when Universal and Existential Variables are Separated
We introduce a new decidable fragment of first-order logic with equality,
which strictly generalizes two already well-known ones -- the
Bernays-Sch\"onfinkel-Ramsey (BSR) Fragment and the Monadic Fragment. The
defining principle is the syntactic separation of universally quantified
variables from existentially quantified ones at the level of atoms. Thus, our
classification neither rests on restrictions on quantifier prefixes (as in the
BSR case) nor on restrictions on the arity of predicate symbols (as in the
monadic case). We demonstrate that the new fragment exhibits the finite model
property and derive a non-elementary upper bound on the computing time required
for deciding satisfiability in the new fragment. For the subfragment of prenex
sentences with the quantifier prefix the
satisfiability problem is shown to be complete for NEXPTIME. Finally, we
discuss how automated reasoning procedures can take advantage of our results.Comment: Extended version of our LICS 2016 conference paper, 23 page
Lower Complexity Bounds for Lifted Inference
One of the big challenges in the development of probabilistic relational (or
probabilistic logical) modeling and learning frameworks is the design of
inference techniques that operate on the level of the abstract model
representation language, rather than on the level of ground, propositional
instances of the model. Numerous approaches for such "lifted inference"
techniques have been proposed. While it has been demonstrated that these
techniques will lead to significantly more efficient inference on some specific
models, there are only very recent and still quite restricted results that show
the feasibility of lifted inference on certain syntactically defined classes of
models. Lower complexity bounds that imply some limitations for the feasibility
of lifted inference on more expressive model classes were established early on
in (Jaeger 2000). However, it is not immediate that these results also apply to
the type of modeling languages that currently receive the most attention, i.e.,
weighted, quantifier-free formulas. In this paper we extend these earlier
results, and show that under the assumption that NETIME =/= ETIME, there is no
polynomial lifted inference algorithm for knowledge bases of weighted,
quantifier- and function-free formulas. Further strengthening earlier results,
this is also shown to hold for approximate inference, and for knowledge bases
not containing the equality predicate.Comment: To appear in Theory and Practice of Logic Programming (TPLP
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
An Improved Algorithm for Generating Database Transactions from Relational Algebra Specifications
Alloy is a lightweight modeling formalism based on relational algebra. In
prior work with Fisler, Giannakopoulos, Krishnamurthi, and Yoo, we have
presented a tool, Alchemy, that compiles Alloy specifications into
implementations that execute against persistent databases. The foundation of
Alchemy is an algorithm for rewriting relational algebra formulas into code for
database transactions. In this paper we report on recent progress in improving
the robustness and efficiency of this transformation
lim+, delta+, and Non-Permutability of beta-Steps
Using a human-oriented formal example proof of the (lim+) theorem, i.e. that
the sum of limits is the limit of the sum, which is of value for reference on
its own, we exhibit a non-permutability of beta-steps and delta+-steps
(according to Smullyan's classification), which is not visible with
non-liberalized delta-rules and not serious with further liberalized
delta-rules, such as the delta++-rule. Besides a careful presentation of the
search for a proof of (lim+) with several pedagogical intentions, the main
subject is to explain why the order of beta-steps plays such a practically
important role in some calculi.Comment: ii + 36 page
Advanced Proof Viewing in ProofTool
Sequent calculus is widely used for formalizing proofs. However, due to the
proliferation of data, understanding the proofs of even simple mathematical
arguments soon becomes impossible. Graphical user interfaces help in this
matter, but since they normally utilize Gentzen's original notation, some of
the problems persist. In this paper, we introduce a number of criteria for
proof visualization which we have found out to be crucial for analyzing proofs.
We then evaluate recent developments in tree visualization with regard to these
criteria and propose the Sunburst Tree layout as a complement to the
traditional tree structure. This layout constructs inferences as concentric
circle arcs around the root inference, allowing the user to focus on the
proof's structural content. Finally, we describe its integration into ProofTool
and explain how it interacts with the Gentzen layout.Comment: In Proceedings UITP 2014, arXiv:1410.785
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