7,969 research outputs found
Termination Casts: A Flexible Approach to Termination with General Recursion
This paper proposes a type-and-effect system called Teqt, which distinguishes
terminating terms and total functions from possibly diverging terms and partial
functions, for a lambda calculus with general recursion and equality types. The
central idea is to include a primitive type-form "Terminates t", expressing
that term t is terminating; and then allow terms t to be coerced from possibly
diverging to total, using a proof of Terminates t. We call such coercions
termination casts, and show how to implement terminating recursion using them.
For the meta-theory of the system, we describe a translation from Teqt to a
logical theory of termination for general recursive, simply typed functions.
Every typing judgment of Teqt is translated to a theorem expressing the
appropriate termination property of the computational part of the Teqt term.Comment: In Proceedings PAR 2010, arXiv:1012.455
Complexity Analysis of Precedence Terminating Infinite Graph Rewrite Systems
The general form of safe recursion (or ramified recurrence) can be expressed
by an infinite graph rewrite system including unfolding graph rewrite rules
introduced by Dal Lago, Martini and Zorzi, in which the size of every normal
form by innermost rewriting is polynomially bounded. Every unfolding graph
rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and
Zantema. Although precedence terminating infinite rewrite systems cover all the
primitive recursive functions, in this paper we consider graph rewrite systems
precedence terminating with argument separation, which form a subclass of
precedence terminating graph rewrite systems. We show that for any precedence
terminating infinite graph rewrite system G with a specific argument
separation, both the runtime complexity of G and the size of every normal form
in G can be polynomially bounded. As a corollary, we obtain an alternative
proof of the original result by Dal Lago et al.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.06818. arXiv admin note:
text overlap with arXiv:1404.619
Termination Casts: A Flexible Approach to Termination With General Recursion (Technical Appendix)
This paper proposes a type-and-effect system called Teq↓, which distinguishes terminating terms and total functions from possibly diverging terms and partial functions, for a lambda calculus with general recursion and equality types. The central idea is to include a primitive type-form Terminates t , expressing that term t is terminating; and then allow terms t to be coerced from possibly diverging to total, using a proof of Terminates t. We call such coercions termination casts, and show how to implement terminating recursion using them. For the meta-theory of the system, we describe a translation from Teq↓ to a logical theory of termination for general recursive, simply typed functions. Every typing judgment of Teq↓ is translated to a theorem expressing the appropriate termination property of the computational part of the Teq↓ term
Affine functions and series with co-inductive real numbers
We extend the work of A. Ciaffaglione and P. Di Gianantonio on mechanical
verification of algorithms for exact computation on real numbers, using
infinite streams of digits implemented as co-inductive types. Four aspects are
studied: the first aspect concerns the proof that digit streams can be related
to the axiomatized real numbers that are already axiomatized in the proof
system (axiomatized, but with no fixed representation). The second aspect
re-visits the definition of an addition function, looking at techniques to let
the proof search mechanism perform the effective construction of an algorithm
that is correct by construction. The third aspect concerns the definition of a
function to compute affine formulas with positive rational coefficients. This
should be understood as a testbed to describe a technique to combine
co-recursion and recursion to obtain a model for an algorithm that appears at
first sight to be outside the expressive power allowed by the proof system. The
fourth aspect concerns the definition of a function to compute series, with an
application on the series that is used to compute Euler's number e. All these
experiments should be reproducible in any proof system that supports
co-inductive types, co-recursion and general forms of terminating recursion,
but we performed with the Coq system [12, 3, 14]
Process algebra with strategic interleaving
In process algebras such as ACP (Algebra of Communicating Processes),
parallel processes are considered to be interleaved in an arbitrary way. In the
case of multi-threading as found in contemporary programming languages,
parallel processes are actually interleaved according to some interleaving
strategy. An interleaving strategy is what is called a process-scheduling
policy in the field of operating systems. In many systems, for instance
hardware/software systems, we have to do with both parallel processes that may
best be considered to be interleaved in an arbitrary way and parallel processes
that may best be considered to be interleaved according to some interleaving
strategy. Therefore, we extend ACP in this paper with the latter form of
interleaving. The established properties of the extension concerned include an
elimination property, a conservative extension property, and a unique expansion
property.Comment: 19 pages, this version is a revision of the published versio
Efficient Algorithms for Asymptotic Bounds on Termination Time in VASS
Vector Addition Systems with States (VASS) provide a well-known and
fundamental model for the analysis of concurrent processes, parameterized
systems, and are also used as abstract models of programs in resource bound
analysis. In this paper we study the problem of obtaining asymptotic bounds on
the termination time of a given VASS. In particular, we focus on the
practically important case of obtaining polynomial bounds on termination time.
Our main contributions are as follows: First, we present a polynomial-time
algorithm for deciding whether a given VASS has a linear asymptotic complexity.
We also show that if the complexity of a VASS is not linear, it is at least
quadratic. Second, we classify VASS according to quantitative properties of
their cycles. We show that certain singularities in these properties are the
key reason for non-polynomial asymptotic complexity of VASS. In absence of
singularities, we show that the asymptotic complexity is always polynomial and
of the form , for some integer , where is the
dimension of the VASS. We present a polynomial-time algorithm computing the
optimal . For general VASS, the same algorithm, which is based on a complete
technique for the construction of ranking functions in VASS, produces a valid
lower bound, i.e., a such that the termination complexity is .
Our results are based on new insights into the geometry of VASS dynamics, which
hold the potential for further applicability to VASS analysis.Comment: arXiv admin note: text overlap with arXiv:1708.0925
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