405 research outputs found
The first-order theory of lexicographic path orderings is undecidable
We show, under some assumption on the signature, that the *This formula not viewable on a Text-Browser* fragment of the theory of any lexicographic path ordering is undecidable. This applies to partial and to total precedences. Our result implies in particular that the simplification rule of ordered completion is undecidable
Ordering constraints on trees
We survey recent results about ordering constraints on trees and discuss their applications. Our main interest lies in the family of recursive path orderings which enjoy the properties of being total, well-founded and compatible with the tree constructors. The paper includes some new results, in particular the undecidability of the theory of lexicographic path orderings in case of a non-unary signature
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
Higher-Order Termination: from Kruskal to Computability
Termination is a major question in both logic and computer science. In logic,
termination is at the heart of proof theory where it is usually called strong
normalization (of cut elimination). In computer science, termination has always
been an important issue for showing programs correct. In the early days of
logic, strong normalization was usually shown by assigning ordinals to
expressions in such a way that eliminating a cut would yield an expression with
a smaller ordinal. In the early days of verification, computer scientists used
similar ideas, interpreting the arguments of a program call by a natural
number, such as their size. Showing the size of the arguments to decrease for
each recursive call gives a termination proof of the program, which is however
rather weak since it can only yield quite small ordinals. In the sixties, Tait
invented a new method for showing cut elimination of natural deduction, based
on a predicate over the set of terms, such that the membership of an expression
to the predicate implied the strong normalization property for that expression.
The predicate being defined by induction on types, or even as a fixpoint, this
method could yield much larger ordinals. Later generalized by Girard under the
name of reducibility or computability candidates, it showed very effective in
proving the strong normalization property of typed lambda-calculi..
Scattered one-counter languges have rank less than
A linear ordering is called context-free if it is the lexicographic ordering
of some context-free language and is called scattered if it has no dense
subordering. Each scattered ordering has an associated ordinal, called its
rank. It is known that scattered context-free (regular, resp.) orderings have
rank less than (, resp).
In this paper we confirm the conjecture that one-counter languages have rank
less than
Superposition as a logical glue
The typical mathematical language systematically exploits notational and
logical abuses whose resolution requires not just the knowledge of domain
specific notation and conventions, but not trivial skills in the given
mathematical discipline. A large part of this background knowledge is expressed
in form of equalities and isomorphisms, allowing mathematicians to freely move
between different incarnations of the same entity without even mentioning the
transformation. Providing ITP-systems with similar capabilities seems to be a
major way to improve their intelligence, and to ease the communication between
the user and the machine. The present paper discusses our experience of
integration of a superposition calculus within the Matita interactive prover,
providing in particular a very flexible, "smart" application tactic, and a
simple, innovative approach to automation.Comment: In Proceedings TYPES 2009, arXiv:1103.311
Quasi-interpretations a way to control resources
International audienceThis paper presents in a reasoned way our works on resource analysis by quasi- interpretations. The controlled resources are typically the runtime, the runspace or the size of a result in a program execution. Quasi-interpretations allow analyzing system complexity. A quasi-interpretation is a numerical assignment, which provides an upper bound on computed func- tions and which is compatible with the program operational semantics. Quasi- interpretation method offers several advantages: (i) It provides hints in order to optimize an execution, (ii) it gives resource certificates, and (iii) finding quasi- interpretations is decidable for a broad class which is relevant for feasible com- putations. By combining the quasi-interpretation method with termination tools (here term orderings), we obtained several characterizations of complexity classes starting from Ptime and Pspace
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