6,235 research outputs found
On linear rewriting systems for Boolean logic and some applications to proof theory
Linear rules have played an increasing role in structural proof theory in
recent years. It has been observed that the set of all sound linear inference
rules in Boolean logic is already coNP-complete, i.e. that every Boolean
tautology can be written as a (left- and right-)linear rewrite rule. In this
paper we study properties of systems consisting only of linear inferences. Our
main result is that the length of any 'nontrivial' derivation in such a system
is bound by a polynomial. As a consequence there is no polynomial-time
decidable sound and complete system of linear inferences, unless coNP=NP. We
draw tools and concepts from term rewriting, Boolean function theory and graph
theory in order to access some required intermediate results. At the same time
we make several connections between these areas that, to our knowledge, have
not yet been presented and constitute a rich theoretical framework for
reasoning about linear TRSs for Boolean logic.Comment: 27 pages, 3 figures, special issue of RTA 201
Proving Looping and Non-Looping Non-Termination by Finite Automata
A new technique is presented to prove non-termination of term rewriting. The
basic idea is to find a non-empty regular language of terms that is closed
under rewriting and does not contain normal forms. It is automated by
representing the language by a tree automaton with a fixed number of states,
and expressing the mentioned requirements in a SAT formula. Satisfiability of
this formula implies non-termination. Our approach succeeds for many examples
where all earlier techniques fail, for instance for the S-rule from combinatory
logic
Deduction modulo theory
This paper is a survey on Deduction modulo theor
Automated verification of termination certificates
In order to increase user confidence, many automated theorem provers provide
certificates that can be independently verified. In this paper, we report on
our progress in developing a standalone tool for checking the correctness of
certificates for the termination of term rewrite systems, and formally proving
its correctness in the proof assistant Coq. To this end, we use the extraction
mechanism of Coq and the library on rewriting theory and termination called
CoLoR
Deterministic Automata for Unordered Trees
Automata for unordered unranked trees are relevant for defining schemas and
queries for data trees in Json or Xml format. While the existing notions are
well-investigated concerning expressiveness, they all lack a proper notion of
determinism, which makes it difficult to distinguish subclasses of automata for
which problems such as inclusion, equivalence, and minimization can be solved
efficiently. In this paper, we propose and investigate different notions of
"horizontal determinism", starting from automata for unranked trees in which
the horizontal evaluation is performed by finite state automata. We show that a
restriction to confluent horizontal evaluation leads to polynomial-time
emptiness and universality, but still suffers from coNP-completeness of the
emptiness of binary intersections. Finally, efficient algorithms can be
obtained by imposing an order of horizontal evaluation globally for all
automata in the class. Depending on the choice of the order, we obtain
different classes of automata, each of which has the same expressiveness as
CMso.Comment: In Proceedings GandALF 2014, arXiv:1408.556
Efficient Generation of Craig Interpolants in Satisfiability Modulo Theories
The problem of computing Craig Interpolants has recently received a lot of
interest. In this paper, we address the problem of efficient generation of
interpolants for some important fragments of first order logic, which are
amenable for effective decision procedures, called Satisfiability Modulo Theory
solvers.
We make the following contributions.
First, we provide interpolation procedures for several basic theories of
interest: the theories of linear arithmetic over the rationals, difference
logic over rationals and integers, and UTVPI over rationals and integers.
Second, we define a novel approach to interpolate combinations of theories,
that applies to the Delayed Theory Combination approach.
Efficiency is ensured by the fact that the proposed interpolation algorithms
extend state of the art algorithms for Satisfiability Modulo Theories. Our
experimental evaluation shows that the MathSAT SMT solver can produce
interpolants with minor overhead in search, and much more efficiently than
other competitor solvers.Comment: submitted to ACM Transactions on Computational Logic (TOCL
Open Graphs and Monoidal Theories
String diagrams are a powerful tool for reasoning about physical processes,
logic circuits, tensor networks, and many other compositional structures. The
distinguishing feature of these diagrams is that edges need not be connected to
vertices at both ends, and these unconnected ends can be interpreted as the
inputs and outputs of a diagram. In this paper, we give a concrete construction
for string diagrams using a special kind of typed graph called an open-graph.
While the category of open-graphs is not itself adhesive, we introduce the
notion of a selective adhesive functor, and show that such a functor embeds the
category of open-graphs into the ambient adhesive category of typed graphs.
Using this functor, the category of open-graphs inherits "enough adhesivity"
from the category of typed graphs to perform double-pushout (DPO) graph
rewriting. A salient feature of our theory is that it ensures rewrite systems
are "type-safe" in the sense that rewriting respects the inputs and outputs.
This formalism lets us safely encode the interesting structure of a
computational model, such as evaluation dynamics, with succinct, explicit
rewrite rules, while the graphical representation absorbs many of the tedious
details. Although topological formalisms exist for string diagrams, our
construction is discreet, finitary, and enjoys decidable algorithms for
composition and rewriting. We also show how open-graphs can be parametrised by
graphical signatures, similar to the monoidal signatures of Joyal and Street,
which define types for vertices in the diagrammatic language and constraints on
how they can be connected. Using typed open-graphs, we can construct free
symmetric monoidal categories, PROPs, and more general monoidal theories. Thus
open-graphs give us a handle for mechanised reasoning in monoidal categories.Comment: 31 pages, currently technical report, submitted to MSCS, waiting
review
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