5,323 research outputs found
Intuitionistic fixed point logic
The logical system IFP introduced in this paper supports program extraction from proofs, unifying theoretical and practical advantages: Based on first-order logic and powerful strictly positive inductive and coinductive definitions, IFP support abstract axiomatic mathematics with a large amount of classical logic. The Haskell-like target programming language has a denotational and an operational semantics which are linked through a computational adequacy theorem that extends to infinite data. Program extraction is fully verified and highly optimised, thus extracted programs are guaranteed to be correct and free of junk. A case study in exact real number computation underpins IFP's effectiveness
Intuitionistic Fixed Point Logic
We study the system IFP of intuitionistic fixed point logic, an extension of
intuitionistic first-order logic by strictly positive inductive and coinductive
definitions. We define a realizability interpretation of IFP and use it to
extract computational content from proofs about abstract structures specified
by arbitrary classically true disjunction free formulas. The interpretation is
shown to be sound with respect to a domain-theoretic denotational semantics and
a corresponding lazy operational semantics of a functional language for
extracted programs. We also show how extracted programs can be translated into
Haskell. As an application we extract a program converting the signed digit
representation of real numbers to infinite Gray-code from a proof of inclusion
of the corresponding coinductive predicates.Comment: 65 page
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
Reasoning on Assembly Code using Linear Logic
We present a logic for reasoning on assembly code. The logic is an extension of intuitionistic linear logic with greatest fixed points, pointer assertions for reasoning about the heap, and modalities for reasoning about program execution. One of the modality corresponds to the step relation of the semantics of an assembly code interpreter. Safety is defined as the greatest fixed point of this modal operator. We can deal with first class code pointers, in a modular way, by defining an indexed model of the logic
Extracting nondeterministic concurrent programs
We introduce an extension of intuitionistic fixed point logic by a modal operator facilitating the extraction of nondeterministic concurrent programsfrom proofs. We apply this extension to program extraction in computable analysis, more precisely, to computing with Tsuiki's infinite Gray code for real numbers
Fixed-point elimination in the Intuitionistic Propositional Calculus (extended version)
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the alge-
braic models of the Intuitionistic Propositional Calculus-always exist, even
when these algebras are not complete as lattices. The reason is that these
extremal fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equiv- alent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed- point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal.Comment: extended version of arXiv:1601.0040
Deciding regular grammar logics with converse through first-order logic
We provide a simple translation of the satisfiability problem for regular
grammar logics with converse into GF2, which is the intersection of the guarded
fragment and the 2-variable fragment of first-order logic. This translation is
theoretically interesting because it translates modal logics with certain frame
conditions into first-order logic, without explicitly expressing the frame
conditions.
A consequence of the translation is that the general satisfiability problem
for regular grammar logics with converse is in EXPTIME. This extends a previous
result of the first author for grammar logics without converse. Using the same
method, we show how some other modal logics can be naturally translated into
GF2, including nominal tense logics and intuitionistic logic.
In our view, the results in this paper show that the natural first-order
fragment corresponding to regular grammar logics is simply GF2 without extra
machinery such as fixed point-operators.Comment: 34 page
- …