64 research outputs found
Unary profile of lambda terms with restricted De Bruijn indices
In this paper we present an average-case analysis of closed lambda terms with restricted values of De Bruijn indices in the model where each occurrence of a variable contributes one to the size. Given a fixed integer k, a lambda term in which all De Bruijn indices are bounded by k has the following shape: It starts with k De Bruijn levels, forming the so-called hat of the term, to which some number of k-colored Motzkin trees are attached. By means of analytic combinatorics, we show that the size of this hat is constant on average and that the average number of De Bruijn levels of k-colored Motzkin trees of size n is asymptotically Θ(√ n). Combining these two facts, we conclude that the maximal non-empty De Bruijn level in a lambda term with restrictions on De Bruijn indices and of size n is, on average, also of order √ n. On this basis, we provide the average unary profile of such lambda terms
Analyticity results for the cumulants in a random matrix model
The generating function of the cumulants in random matrix models, as well as
the cumulants themselves, can be expanded as asymptotic (divergent) series
indexed by maps. While at fixed genus the sums over maps converge, the sums
over genera do not. In this paper we obtain alternative expansions both for the
generating function and for the cumulants that cure this problem. We provide
explicit and convergent expansions for the cumulants, for the remainders of
their perturbative expansion (in the size of the maps) and for the remainders
of their topological expansion (in the genus of the maps). We show that any
cumulant is an analytic function inside a cardioid domain in the complex plane
and we prove that any cumulant is Borel summable at the origin
A framework for relating, implementing and verifying argumentation models and their translations
Computational argumentation theory deals with the formalisation of argument structure, conflict between arguments and domain-specific constructs, such as proof standards, epistemic probabilities or argument schemes. However, despite these practical components, there is a lack of implementations and implementation methods available for most structured models of argumentation and translations between them.
This thesis addresses this problem, by constructing a general framework for relating, implementing and formally verifying argumentation models and translations between them, drawing from dependent type theory and the Curry-Howard correspondence. The framework provides mathematical tools and programming methodologies to implement argumentation models, allowing programmers and argumentation theorists to construct implementations that are closely related to the mathematical definitions. It furthermore provides tools that, without much effort on the programmer's side, can automatically construct counter-examples to desired properties, while finally providing methodologies that can prove formal correctness of the implementation in a theorem prover.
The thesis consists of various use cases that demonstrate the general approach of the framework. The Carneades argumentation model, Dung's abstract argumentation frameworks and a translation between them, are implemented in the functional programming language Haskell. Implementations of formal properties of the translation are provided together with a formalisation of AFs in the theorem prover, Agda. The result is a verified pipeline, from the structured model Carneades into existing efficient SAT-based implementations of Dung's AFs. Finally, the ASPIC+ model for argumentation is generalised to incorporate content orderings, weight propagation and argument accrual. The framework is applied to provide a translation from this new model into Dung's AFs, together with a complete implementation
A framework for relating, implementing and verifying argumentation models and their translations
Computational argumentation theory deals with the formalisation of argument structure, conflict between arguments and domain-specific constructs, such as proof standards, epistemic probabilities or argument schemes. However, despite these practical components, there is a lack of implementations and implementation methods available for most structured models of argumentation and translations between them.
This thesis addresses this problem, by constructing a general framework for relating, implementing and formally verifying argumentation models and translations between them, drawing from dependent type theory and the Curry-Howard correspondence. The framework provides mathematical tools and programming methodologies to implement argumentation models, allowing programmers and argumentation theorists to construct implementations that are closely related to the mathematical definitions. It furthermore provides tools that, without much effort on the programmer's side, can automatically construct counter-examples to desired properties, while finally providing methodologies that can prove formal correctness of the implementation in a theorem prover.
The thesis consists of various use cases that demonstrate the general approach of the framework. The Carneades argumentation model, Dung's abstract argumentation frameworks and a translation between them, are implemented in the functional programming language Haskell. Implementations of formal properties of the translation are provided together with a formalisation of AFs in the theorem prover, Agda. The result is a verified pipeline, from the structured model Carneades into existing efficient SAT-based implementations of Dung's AFs. Finally, the ASPIC+ model for argumentation is generalised to incorporate content orderings, weight propagation and argument accrual. The framework is applied to provide a translation from this new model into Dung's AFs, together with a complete implementation
Relaxation and its Role in Vision
It is argued that a visual system, especially one
which handles imperfect data, needs a way of selecting
the best consistent combination from among the many interrelated,
locally plausible hypotheses about how parts
or aspects of the visual input may be interpreted. A
method is presented in which each hypothesis is given a
supposition value between 0 and 1. A parallel relaxation
I
operator, based on the plausibilities of hypotheses and
the logical relations between them, is then used to modify
the supposition values, and the process is repeated
until the best consistent set of hypotheses have supposition
values of approximately 1, and the rest have values
of approximately 0.
The method is incorporated in a program which can
interpret configurations of overlapping rectangles as
puppets. For this task it is possible to formulate all
the potentially relevant hypotheses before using relaxation
to select the best consistent set. For more complex
tasks, it is necessary to use relaxation on the locally
plausible interpretations to guide the search for locally
less obvious ones. Ways of doing this are discussed.
Finally, an implemented system is presented which
allows the user to specify schemas and inference rules,
and uses relaxation to control the building of a network
of instances of the schemas, when presented with data
about some instances and relations between the
Implicit automata in typed -calculi II: streaming transducers vs categorical semantics
We characterize regular string transductions as programs in a linear
-calculus with additives. One direction of this equivalence is proved
by encoding copyless streaming string transducers (SSTs), which compute regular
functions, into our -calculus. For the converse, we consider a
categorical framework for defining automata and transducers over words, which
allows us to relate register updates in SSTs to the semantics of the linear
-calculus in a suitable monoidal closed category. To illustrate the
relevance of monoidal closure to automata theory, we also leverage this notion
to give abstract generalizations of the arguments showing that copyless SSTs
may be determinized and that the composition of two regular functions may be
implemented by a copyless SST. Our main result is then generalized from strings
to trees using a similar approach. In doing so, we exhibit a connection between
a feature of streaming tree transducers and the multiplicative/additive
distinction of linear logic.
Keywords: MSO transductions, implicit complexity, Dialectica categories,
Church encodingsComment: 105 pages, 24 figure
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