21 research outputs found
Ideograph: A Language for Expressing and Manipulating Structured Data
We introduce Ideograph, a language for expressing and manipulating structured
data. Its types describe kinds of structures, such as natural numbers, lists,
multisets, binary trees, syntax trees with variable binding, directed
multigraphs, and relational databases. Fully normalized terms of a type
correspond exactly to members of the structure, analogous to a Church-encoding.
Moreover, definable operations over these structures are guaranteed to respect
the structures' equivalences. In this paper, we give the syntax and semantics
of the non-polymorphic subset of Ideograph, and we demonstrate how it can
represent and manipulate several interesting structures.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421
Taylor subsumes Scott, Berry, Kahn and Plotkin
The speculative ambition of replacing the old theory of program approximation based on syntactic continuity with the theory of resource consumption based on Taylor expansion and originating from the differential γ-calculus is nowadays at hand. Using this resource sensitive theory, we provide simple proofs of important results in γ-calculus that are usually demonstrated by exploiting Scott's continuity, Berry's stability or Kahn and Plotkin's sequentiality theory. A paradigmatic example is given by the Perpendicular Lines Lemma for the Böhm tree semantics, which is proved here simply by induction, but relying on the main properties of resource approximants: strong normalization, confluence and linearity
New Minimal Linear Inferences in Boolean Logic Independent of Switch and Medial
A linear inference is a valid inequality of Boolean algebra in which each variable occurs at most once on each side. Equivalently, it is a linear rewrite rule on Boolean terms that constitutes a valid implication. Linear inferences have played a significant role in structural proof theory, in particular in models of substructural logics and in normalisation arguments for deep inference proof systems.
Systems of linear logic and, later, deep inference are founded upon two particular linear inferences, switch : x ? (y ? z) ? (x ? y) ? z, and medial : (w ? x) ? (y ? z) ? (w ? y) ? (x ? z). It is well-known that these two are not enough to derive all linear inferences (even modulo all valid linear equations), but beyond this little more is known about the structure of linear inferences in general. In particular despite recurring attention in the literature, the smallest linear inference not derivable under switch and medial ("switch-medial-independent") was not previously known.
In this work we leverage recently developed graphical representations of linear formulae to build an implementation that is capable of more efficiently searching for switch-medial-independent inferences. We use it to find two "minimal" 8-variable independent inferences and also prove that no smaller ones exist; in contrast, a previous approach based directly on formulae reached computational limits already at 7 variables. One of these new inferences derives some previously found independent linear inferences. The other exhibits structure seemingly beyond the scope of previous approaches we are aware of; in particular, its existence contradicts a conjecture of Das and Strassburger
Resourceful program synthesis from graded linear types
Linear types provide a way to constrain programs by specifying that some values must be used exactly once. Recent work on graded modal types augments and refines this notion, enabling fine-grained, quantitative specification of data use in programs. The information provided by graded modal types appears to be useful for type-directed program synthesis, where these additional constraints can be used to prune the search space of candidate programs. We explore one of the major implementation challenges of a synthesis algorithm in this setting: how does the synthesis algorithm efficiently ensure that resource constraints are satisfied throughout program generation? We provide two solutions to this resource management problem, adapting Hodas and Miller’s input-output model of linear context management to a graded modal linear type theory. We evaluate the performance of both approaches via their implementation as a program synthesis tool for the programming language Granule, which provides linear and graded modal typing
Planar Realizability via Left and Right Applications
We introduce a class of applicative structures called bi-BDI-algebras. Bi-BDI-algebras are generalizations of partial combinatory algebras and BCI-algebras, and feature two sorts of applications (left and right applications). Applying the categorical realizability construction to bi-BDI-algebras, we obtain monoidal bi-closed categories of assemblies (as well as of modest sets). We further investigate two kinds of comonadic applicative morphisms on bi-BDI-algebras as non-symmetric analogues of linear combinatory algebras, which induce models of exponential and exchange modalities on non-symmetric linear logics
Client-Server Sessions in Linear Logic
We introduce coexponentials, a new set of modalities for Classical Linear
Logic. As duals to exponentials, the coexponentials codify a distributed form
of the structural rules of weakening and contraction. This makes them a
suitable logical device for encapsulating the pattern of a server receiving
requests from an arbitrary number of clients on a single channel. Guided by
this intuition we formulate a system of session types based on Classical Linear
Logic with coexponentials, which is suited to modelling client-server
interactions. We also present a session-typed functional programming language
for server-client programming, which we translate to our system of
coexponentials
Enumerating Independent Linear Inferences
A linear inference is a valid inequality of Boolean algebra in which each
variable occurs at most once on each side. Equivalently, it is a linear rewrite
rule on Boolean terms that constitutes a valid implication. Linear inferences
have played a significant role in structural proof theory, in particular in
models of substructural logics and in normalisation arguments for deep
inference proof systems.
In this work we leverage recently developed graphical representations of
linear formulae to build an implementation that is capable of more efficiently
searching for switch-medial-independent inferences. We use it to find four
`minimal' 8-variable independent inferences and also prove that no smaller ones
exist; in contrast, a previous approach based directly on formulae reached
computational limits already at 7 variables. Two of these new inferences derive
some previously found independent linear inferences. The other two (which are
dual) exhibit structure seemingly beyond the scope of previous approaches we
are aware of; in particular, their existence contradicts a conjecture of Das
and Strassburger.
We were also able to identify 10 minimal 9-variable linear inferences
independent of all the aforementioned inferences, comprising 5 dual pairs, and
present applications of our implementation to recent `graph logics'.Comment: 33 pages, 3 figure
Categorical Realizability for Non-symmetric Closed Structures
In categorical realizability, it is common to construct categories of
assemblies and categories of modest sets from applicative structures. These
categories have structures corresponding to the structures of applicative
structures. In the literature, classes of applicative structures inducing
categorical structures such as Cartesian closed categories and symmetric
monoidal closed categories have been widely studied. In this paper, we expand
these correspondences between categories with structure and applicative
structures by identifying the classes of applicative structures giving rise to
closed multicategories, closed categories, monoidal bi-closed categories as
well as (non-symmetric) monoidal closed categories. These applicative
structures are planar in that they correspond to appropriate planar lambda
calculi by combinatory completeness. These new correspondences are tight: we
show that, when a category of assemblies has one of the structures listed
above, the based applicative structure is in the corresponding class. In
addition, we introduce planar linear combinatory algebras by adopting linear
combinatory algebras of Abramsky, Hagjverdi and Scott to our planar setting,
that give rise to categorical models of the linear exponential modality and the
exchange modality on the non-symmetric multiplicative intuitionistic linear
logic
A System of Interaction and Structure III: The Complexity of BV and Pomset Logic
Pomset logic and BV are both logics that extend multiplicative linear logic
(with Mix) with a third connective that is self-dual and non-commutative.
Whereas pomset logic originates from the study of coherence spaces and proof
nets, BV originates from the study of series-parallel orders, cographs, and
proof systems. Both logics enjoy a cut-admissibility result, but for neither
logic can this be done in the sequent calculus. Provability in pomset logic can
be checked via a proof net correctness criterion and in BV via a deep inference
proof system. It has long been conjectured that these two logics are the same.
In this paper we show that this conjecture is false. We also investigate the
complexity of the two logics, exhibiting a huge gap between the two. Whereas
provability in BV is NP-complete, provability in pomset logic is
-complete. We also make some observations with respect to possible
sequent systems for the two logics