108 research outputs found
Unifying Functional Interpretations: Past and Future
This article surveys work done in the last six years on the unification of
various functional interpretations including G\"odel's dialectica
interpretation, its Diller-Nahm variant, Kreisel modified realizability,
Stein's family of functional interpretations, functional interpretations "with
truth", and bounded functional interpretations. Our goal in the present paper
is twofold: (1) to look back and single out the main lessons learnt so far, and
(2) to look forward and list several open questions and possible directions for
further research.Comment: 18 page
Preserving Filtering Unification by Adding Compatible Operations to Some Heyting Algebras
We show that adding compatible operations to Heyting algebras and to commutative residuated lattices, both satisfying the Stone law ¬x ⋁ ¬¬x = 1, preserves filtering (or directed) unification, that is, the property that for every two unifiers there is a unifier more general then both of them. Contrary to that, often adding new operations to algebras results in changing the unification type. To prove the results we apply the theorems of [9] on direct products of l-algebras and filtering unification. We consider examples of frontal Heyting algebras, in particular Heyting algebras with the successor, and G operations as well as expansions of some commutative integral residuated lattices with successor operations
The complexity of admissible rules of {\L}ukasiewicz logic
We investigate the computational complexity of admissibility of inference
rules in infinite-valued {\L}ukasiewicz propositional logic (\L). It was shown
in [13] that admissibility in {\L} is checkable in PSPACE. We establish that
this result is optimal, i.e., admissible rules of {\L} are PSPACE-complete. In
contrast, derivable rules of {\L} are known to be coNP-complete.Comment: 14 pages, 2 figures; to appear in Journal of Logic and Computatio
Light Logics and the Call-by-Value Lambda Calculus
The so-called light logics have been introduced as logical systems enjoying
quite remarkable normalization properties. Designing a type assignment system
for pure lambda calculus from these logics, however, is problematic. In this
paper we show that shifting from usual call-by-name to call-by-value lambda
calculus allows regaining strong connections with the underlying logic. This
will be done in the context of Elementary Affine Logic (EAL), designing a type
system in natural deduction style assigning EAL formulae to lambda terms.Comment: 28 page
Logic Programming and Logarithmic Space
We present an algebraic view on logic programming, related to proof theory
and more specifically linear logic and geometry of interaction. Within this
construction, a characterization of logspace (deterministic and
non-deterministic) computation is given via a synctactic restriction, using an
encoding of words that derives from proof theory.
We show that the acceptance of a word by an observation (the counterpart of a
program in the encoding) can be decided within logarithmic space, by reducing
this problem to the acyclicity of a graph. We show moreover that observations
are as expressive as two-ways multi-heads finite automata, a kind of pointer
machines that is a standard model of logarithmic space computation
Recommended from our members
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
The Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
A Fibrational Framework for Substructural and Modal Logics
We define a general framework that abstracts the common features of many intuitionistic substructural and modal logics / type theories. The framework is a sequent calculus / normal-form type theory parametrized by a mode theory, which is used to describe the structure of contexts and the structural properties they obey. In this sequent calculus, the context itself obeys standard structural properties, while a term, drawn from the mode theory, constrains how the context can be used. Product types, implications, and modalities are defined as instances of two general connectives, one positive and one negative, that manipulate these terms. Specific mode theories can express a range of substructural and modal connectives, including non-associative, ordered, linear, affine, relevant, and cartesian products and implications; monoidal and non-monoidal functors, (co)monads and adjunctions; n-linear variables; and bunched implications. We prove cut (and identity) admissibility independently of the mode theory, obtaining it for many different logics at once. Further, we give a general equational theory on derivations / terms that, in addition to the usual beta/eta-rules, characterizes when two derivations differ only by the placement of structural rules. Additionally, we give an equivalent semantic presentation of these ideas, in which a mode theory corresponds to a 2-dimensional cartesian multicategory, the framework corresponds to another such multicategory with a functor to the mode theory, and the logical connectives make this into a bifibration. Finally, we show how the framework can be used both to encode existing existing logics / type theories and to design new ones
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