812 research outputs found
Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach
We present an extension of the second-order logic AF2 with iso-style
inductive and coinductive definitions specifically designed to extract programs
from proofs a la Krivine-Parigot by means of primitive (co)recursion
principles. Our logic includes primitive constructors of least and greatest
fixed points of predicate transformers, but contrary to the common approach, we
do not restrict ourselves to positive operators to ensure monotonicity, instead
we use the Mendler-style, motivated here by the concept of monotonization of an
arbitrary operator on a complete lattice. We prove an adequacy theorem with
respect to a realizability semantics based on saturated sets and
saturated-valued functions and as a consequence we obtain the strong
normalization property for the proof-term reduction, an important feature which
is absent in previous related work.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Finite Countermodel Based Verification for Program Transformation (A Case Study)
Both automatic program verification and program transformation are based on
program analysis. In the past decade a number of approaches using various
automatic general-purpose program transformation techniques (partial deduction,
specialization, supercompilation) for verification of unreachability properties
of computing systems were introduced and demonstrated. On the other hand, the
semantics based unfold-fold program transformation methods pose themselves
diverse kinds of reachability tasks and try to solve them, aiming at improving
the semantics tree of the program being transformed. That means some
general-purpose verification methods may be used for strengthening program
transformation techniques. This paper considers the question how finite
countermodels for safety verification method might be used in Turchin's
supercompilation method. We extract a number of supercompilation sub-algorithms
trying to solve reachability problems and demonstrate use of an external
countermodel finder for solving some of the problems.Comment: In Proceedings VPT 2015, arXiv:1512.0221
Interaction Trees: Representing Recursive and Impure Programs in Coq
"Interaction trees" (ITrees) are a general-purpose data structure for
representing the behaviors of recursive programs that interact with their
environments. A coinductive variant of "free monads," ITrees are built out of
uninterpreted events and their continuations. They support compositional
construction of interpreters from "event handlers", which give meaning to
events by defining their semantics as monadic actions. ITrees are expressive
enough to represent impure and potentially nonterminating, mutually recursive
computations, while admitting a rich equational theory of equivalence up to
weak bisimulation. In contrast to other approaches such as relationally
specified operational semantics, ITrees are executable via code extraction,
making them suitable for debugging, testing, and implementing software
artifacts that are amenable to formal verification.
We have implemented ITrees and their associated theory as a Coq library,
mechanizing classic domain- and category-theoretic results about program
semantics, iteration, monadic structures, and equational reasoning. Although
the internals of the library rely heavily on coinductive proofs, the interface
hides these details so that clients can use and reason about ITrees without
explicit use of Coq's coinduction tactics.
To showcase the utility of our theory, we prove the termination-sensitive
correctness of a compiler from a simple imperative source language to an
assembly-like target whose meanings are given in an ITree-based denotational
semantics. Unlike previous results using operational techniques, our
bisimulation proof follows straightforwardly by structural induction and
elementary rewriting via an equational theory of combinators for control-flow
graphs.Comment: 28 pages, 4 pages references, published at POPL 202
Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study
A biform theory is a combination of an axiomatic theory and an algorithmic
theory that supports the integration of reasoning and computation. These are
ideal for formalizing algorithms that manipulate mathematical expressions. A
theory graph is a network of theories connected by meaning-preserving theory
morphisms that map the formulas of one theory to the formulas of another
theory. Theory graphs are in turn well suited for formalizing mathematical
knowledge at the most convenient level of abstraction using the most convenient
vocabulary. We are interested in the problem of whether a body of mathematical
knowledge can be effectively formalized as a theory graph of biform theories.
As a test case, we look at the graph of theories encoding natural number
arithmetic. We used two different formalisms to do this, which we describe and
compare. The first is realized in , a version of Church's
type theory with quotation and evaluation, and the second is realized in Agda,
a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds,
Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer
Science, Vol. 10383, pp. 9-24, Springer, 201
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