97 research outputs found
ML4PG in Computer Algebra verification
ML4PG is a machine-learning extension that provides statistical proof hints
during the process of Coq/SSReflect proof development. In this paper, we use
ML4PG to find proof patterns in the CoqEAL library -- a library that was
devised to verify the correctness of Computer Algebra algorithms. In
particular, we use ML4PG to help us in the formalisation of an efficient
algorithm to compute the inverse of triangular matrices
A Type-Theoretic Approach to Structural Resolution
Structural resolution (or S-resolution) is a newly proposed alternative to
SLD-resolution that allows a systematic separation of derivations into
term-matching and unification steps. Productive logic programs are those for
which term-matching reduction on any query must terminate. For productive
programs with coinductive meaning, finite term-rewriting reductions can be seen
as measures of observation in an infinite derivation. Ability of handling
corecursion in a productive way is an attractive computational feature of
S-resolution.
In this paper, we make first steps towards a better conceptual understanding
of operational properties of S-resolution as compared to SLD-resolution. To
this aim, we propose a type system for the analysis of both SLD-resolution and
S-resolution.
We formulate S-resolution and SLD-resolution as reduction systems, and show
their soundness relative to the type system. One of the central methods of this
paper is realizability transformation, which makes logic programs productive
and non-overlapping. We show that S-resolution and SLD-resolution are only
equivalent for programs with these two properties.Comment: LOPSTR 201
Operational Semantics of Resolution and Productivity in Horn Clause Logic
This paper presents a study of operational and type-theoretic properties of
different resolution strategies in Horn clause logic. We distinguish four
different kinds of resolution: resolution by unification (SLD-resolution),
resolution by term-matching, the recently introduced structural resolution, and
partial (or lazy) resolution. We express them all uniformly as abstract
reduction systems, which allows us to undertake a thorough comparative analysis
of their properties. To match this small-step semantics, we propose to take
Howard's System H as a type-theoretic semantic counterpart. Using System H, we
interpret Horn formulas as types, and a derivation for a given formula as the
proof term inhabiting the type given by the formula. We prove soundness of
these abstract reduction systems relative to System H, and we show completeness
of SLD-resolution and structural resolution relative to System H. We identify
conditions under which structural resolution is operationally equivalent to
SLD-resolution. We show correspondence between term-matching resolution for
Horn clause programs without existential variables and term rewriting.Comment: Journal Formal Aspect of Computing, 201
Productive Corecursion in Logic Programming
Logic Programming is a Turing complete language. As a consequence, designing
algorithms that decide termination and non-termination of programs or decide
inductive/coinductive soundness of formulae is a challenging task. For example,
the existing state-of-the-art algorithms can only semi-decide coinductive
soundness of queries in logic programming for regular formulae. Another, less
famous, but equally fundamental and important undecidable property is
productivity. If a derivation is infinite and coinductively sound, we may ask
whether the computed answer it determines actually computes an infinite
formula. If it does, the infinite computation is productive. This intuition was
first expressed under the name of computations at infinity in the 80s. In
modern days of the Internet and stream processing, its importance lies in
connection to infinite data structure processing.
Recently, an algorithm was presented that semi-decides a weaker property --
of productivity of logic programs. A logic program is productive if it can give
rise to productive derivations. In this paper we strengthen these recent
results. We propose a method that semi-decides productivity of individual
derivations for regular formulae. Thus we at last give an algorithmic
counterpart to the notion of productivity of derivations in logic programming.
This is the first algorithmic solution to the problem since it was raised more
than 30 years ago. We also present an implementation of this algorithm.Comment: Paper presented at the 33nd International Conference on Logic
Programming (ICLP 2017), Melbourne, Australia, August 28 to September 1, 2017
16 pages, LaTeX, no figure
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