29,676 research outputs found
Tree Regular Model Checking for Lattice-Based Automata
Tree Regular Model Checking (TRMC) is the name of a family of techniques for
analyzing infinite-state systems in which states are represented by terms, and
sets of states by Tree Automata (TA). The central problem in TRMC is to decide
whether a set of bad states is reachable. The problem of computing a TA
representing (an over- approximation of) the set of reachable states is
undecidable, but efficient solutions based on completion or iteration of tree
transducers exist. Unfortunately, the TRMC framework is unable to efficiently
capture both the complex structure of a system and of some of its features. As
an example, for JAVA programs, the structure of a term is mainly exploited to
capture the structure of a state of the system. On the counter part, integers
of the java programs have to be encoded with Peano numbers, which means that
any algebraic operation is potentially represented by thousands of applications
of rewriting rules. In this paper, we propose Lattice Tree Automata (LTAs), an
extended version of tree automata whose leaves are equipped with lattices. LTAs
allow us to represent possibly infinite sets of interpreted terms. Such terms
are capable to represent complex domains and related operations in an efficient
manner. We also extend classical Boolean operations to LTAs. Finally, as a
major contribution, we introduce a new completion-based algorithm for computing
the possibly infinite set of reachable interpreted terms in a finite amount of
time.Comment: Technical repor
Graphical Reasoning in Compact Closed Categories for Quantum Computation
Compact closed categories provide a foundational formalism for a variety of
important domains, including quantum computation. These categories have a
natural visualisation as a form of graphs. We present a formalism for
equational reasoning about such graphs and develop this into a generic proof
system with a fixed logical kernel for equational reasoning about compact
closed categories. Automating this reasoning process is motivated by the slow
and error prone nature of manual graph manipulation. A salient feature of our
system is that it provides a formal and declarative account of derived results
that can include `ellipses'-style notation. We illustrate the framework by
instantiating it for a graphical language of quantum computation and show how
this can be used to perform symbolic computation.Comment: 21 pages, 9 figures. This is the journal version of the paper
published at AIS
A theorem prover-based analysis tool for object-oriented databases
We present a theorem-prover based analysis tool for object-oriented database systems with integrity constraints. Object-oriented database specifications are mapped to higher-order logic (HOL). This allows us to reason about the semantics of database operations using a mechanical theorem prover such as Isabelle or PVS. The tool can be used to verify various semantics requirements of the schema (such as transaction safety, compensation, and commutativity) to support the advanced transaction models used in workflow and cooperative work. We give an example of method safety analysis for the generic structure editing operations of a cooperative authoring system
Synthesising Graphical Theories
In recent years, diagrammatic languages have been shown to be a powerful and
expressive tool for reasoning about physical, logical, and semantic processes
represented as morphisms in a monoidal category. In particular, categorical
quantum mechanics, or "Quantum Picturalism", aims to turn concrete features of
quantum theory into abstract structural properties, expressed in the form of
diagrammatic identities. One way we search for these properties is to start
with a concrete model (e.g. a set of linear maps or finite relations) and start
composing generators into diagrams and looking for graphical identities.
Naively, we could automate this procedure by enumerating all diagrams up to a
given size and check for equalities, but this is intractable in practice
because it produces far too many equations. Luckily, many of these identities
are not primitive, but rather derivable from simpler ones. In 2010, Johansson,
Dixon, and Bundy developed a technique called conjecture synthesis for
automatically generating conjectured term equations to feed into an inductive
theorem prover. In this extended abstract, we adapt this technique to
diagrammatic theories, expressed as graph rewrite systems, and demonstrate its
application by synthesising a graphical theory for studying entangled quantum
states.Comment: 10 pages, 22 figures. Shortened and one theorem adde
Tensors, !-graphs, and non-commutative quantum structures
Categorical quantum mechanics (CQM) and the theory of quantum groups rely
heavily on the use of structures that have both an algebraic and co-algebraic
component, making them well-suited for manipulation using diagrammatic
techniques. Diagrams allow us to easily form complex compositions of
(co)algebraic structures, and prove their equality via graph rewriting. One of
the biggest challenges in going beyond simple rewriting-based proofs is
designing a graphical language that is expressive enough to prove interesting
properties (e.g. normal form results) about not just single diagrams, but
entire families of diagrams. One candidate is the language of !-graphs, which
consist of graphs with certain subgraphs marked with boxes (called !-boxes)
that can be repeated any number of times. New !-graph equations can then be
proved using a powerful technique called !-box induction. However, previously
this technique only applied to commutative (or cocommutative) algebraic
structures, severely limiting its applications in some parts of CQM and
(especially) quantum groups. In this paper, we fix this shortcoming by offering
a new semantics for non-commutative !-graphs using an enriched version of
Penrose's abstract tensor notation.Comment: In Proceedings QPL 2014, arXiv:1412.810
- …