106 research outputs found
A foundation for synthesising programming language semantics
Programming or scripting languages used in real-world systems are seldom designed
with a formal semantics in mind from the outset. Therefore, the first step for developing well-founded analysis tools for these systems is to reverse-engineer a formal
semantics. This can take months or years of effort.
Could we automate this process, at least partially? Though desirable, automatically reverse-engineering semantics rules from an implementation is very challenging,
as found by Krishnamurthi, Lerner and Elberty. They propose automatically learning
desugaring translation rules, mapping the language whose semantics we seek to a simplified, core version, whose semantics are much easier to write. The present thesis
contains an analysis of their challenge, as well as the first steps towards a solution.
Scaling methods with the size of the language is very difficult due to state space
explosion, so this thesis proposes an incremental approach to learning the translation
rules. I present a formalisation that both clarifies the informal description of the challenge by Krishnamurthi et al, and re-formulates the problem, shifting the focus to the
conditions for incremental learning. The central definition of the new formalisation is
the desugaring extension problem, i.e. extending a set of established translation rules
by synthesising new ones.
In a synthesis algorithm, the choice of search space is important and non-trivial,
as it needs to strike a good balance between expressiveness and efficiency. The rest
of the thesis focuses on defining search spaces for translation rules via typing rules.
Two prerequisites are required for comparing search spaces. The first is a series of
benchmarks, a set of source and target languages equipped with intended translation
rules between them. The second is an enumerative synthesis algorithm for efficiently
enumerating typed programs. I show how algebraic enumeration techniques can be applied to enumerating well-typed translation rules, and discuss the properties expected
from a type system for ensuring that typed programs be efficiently enumerable.
The thesis presents and empirically evaluates two search spaces. A baseline search
space yields the first practical solution to the challenge. The second search space is
based on a natural heuristic for translation rules, limiting the usage of variables so that
they are used exactly once. I present a linear type system designed to efficiently enumerate translation rules, where this heuristic is enforced. Through informal analysis
and empirical comparison to the baseline, I then show that using linear types can speed
up the synthesis of translation rules by an order of magnitude
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Programs as Diagrams: From Categorical Computability to Computable Categories
This is a draft of the textbook/monograph that presents computability theory
using string diagrams. The introductory chapters have been taught as graduate
and undergraduate courses and evolved through 8 years of lecture notes. The
later chapters contain new ideas and results about categorical computability
and some first steps into computable category theory. The underlying
categorical view of computation is based on monoidal categories with program
evaluators, called *monoidal computers*. This categorical structure can be
viewed as a single-instruction diagrammatic programming language called Run,
whose only instruction is called RUN. This version: improved text, moved the
final chapter to the next volume. (The final version will continue lots of
exercises and workouts, but already this version has severely degraded graphics
to meet the size bounds.)Comment: 150 pages, 81 figure
From Saturated Embedding Tests to Explicit Algorithms
Quantifier elimination theorems show that each formula in a certain theory is
equivalent to a formula of a specific form -- usually a quantifier-free one,
sometimes in an extended language. Model theoretic embedding tests are a
frequently used tool for proving such results without providing an explicit
algorithm.
We explain how proof mining methods can be adapted to apply to embedding
tests, and provide two explicit examples, giving algorithms for theories of
algebraic and real closed fields with a distinguished small subgroup
corresponding to the embedding test proofs given by van den Dries and
G\"unaydin
Some results on Kolmogorov-Loveland randomness
Whether Kolmogorov-Loveland randomness is equal to the Martin-Löf randomness is a well known open question in the field of algorithmic information theory. Randomness of infinite binary sequences can be defined in terms of betting strategies, a string is non-random if a computable betting strategy wins unbounded capital by successive betting on the sequence.
For Martin-Löf randomness, a betting strategy makes a bet by splitting a set of sequences into any two clopen sets, and placing a portion of capital on one of them as a wager. Kolmogorov-Loveland betting strategies are more restricted, they bet on a value of the bit at some position they choose, which splits a set of sequences into two clopen sets, the sequences that have 0 at the chosen position and the sequences that have 1.
In this thesis we consider betting strategies that when making a bet are restricted to split a set of sequences into two sets of equal uniform Lebesgue measure. We call this generalization of Kolmogorov-Loveland betting strategies the half-betting strategies. We show that there is a pair of such betting strategies such that for every non-Martin-Löf random sequence one of them wins unbounded capital (the pair is universal).
Next, we define a finite betting game where the betting strategies bet on finite binary strings, and show that in this game Kolmogorov-Loveland betting strategies cannot increase capital by more than an arbitrary small amount on all strings on which the unrestricted betting strategy achieves arbitrary large capital.
We also look at another relaxation of Kolmogorov-Loveland betting, where a betting strategy is allowed to access bits of the sequence within a set of positions a bounded number of times. We show that if this bound is less than â„“ - log â„“ for the first â„“ positions then a pair of such betting strategies cannot be universal. Furthermore, we show that, at least for some universal betting strategies, this bound is exponential
Relative order and spectrum in free and related groups
We consider a natural generalization of the concept of order of an element in a group:
an element g ¿ G is said to have order k in a subgroup H (resp., in a coset Hu) of a group G if k is the first strictly positive integer such that gk ¿ H (resp., gk ¿ Hu). We study this notion and its algorithmic properties in the realm of free groups and some related families.
Both positive and negative (algorithmic) results emerge in this setting. On the positive
side, among other results, we prove that the order of elements, the set of orders (called spectrum), and the set of preorders (i.e., the set of elements of a given order) w.r.t. finitely generated subgroups are always computable in free and free times free-abelian groups.
On the negative side, we provide examples of groups and subgroups having essentially any subset of natural numbers as relative spectrum; in particular, non-recursive and even non-recursively enumerable sets of natural numbers. Also, we take advantage of Mikhailova’s construction to see that the spectrum membership problem is unsolvable for direct products of nonabelian free groups.The first named author was partially supported by MINECO grant PID2019-107444GA-I00 and the Basque Government grant IT974-16. The second named author acknowledges partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/ FEDER, UE), and also from the Graduate School of Mathematics through the MarÃa de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). The
third named author was partially supported by (Polish) Narodowe Centrum Nauki, grant
UMO-2018/31/G/ST1/02681.Peer ReviewedPostprint (author's final draft
Computability and l2-Betti Numbers
In Chapter 1, we will introduce L2 -Betti numbers after covering the preliminaries for this definition. We will also show an algebraic characterisation of L2 -Betti numbers:
For a group G, all L2 -Betti numbers arising from G are given as dimRG ker(·A) for some self-adjoint A ∈ Mn×n(ZG) (see Section 1.2.4). In Section 1.3, we will cover Atiyah’s conjecture and Lück’s approximation theorem.
Chapter 2 is dedicated to the introduction of computability concepts. After a ‘naive’ introduction into this subject, we will define different computability classes such as EC (effectively computable), LC (left-computable) and RC (right-computable). We will then take a look at some results on right-computability of topological invariants (Section 2.3).
The main part of this thesis is Chapter 3. We will start with a survey on some known computability results on L2 -Betti numbers (Section 3.1). We will then discuss right-, left- and effective computability of L2 -Betti numbers under different assumptions.
Finally, in Chapter 4, we will discuss an implementation of some of the main results in the Lean Theorem Prover. This formally verifies some of these results. The .lean files used for this can be found on a git repository online. More information on how to install these files can be found in Section 4.2
Studying Hilbert's 10th problem via explicit elliptic curves
N.Garc\'ia-Fritz and H.Pasten showed that Hilbert's 10th problem is
unsolvable in the ring of integers of number fields of the form
for positive proportions of primes and
. We improve their proportions and extend their results to the case of
number fields of the form , where
belongs to an explicit family of positive square-free integers. We achieve this
by using multiple elliptic curves, and replace their Iwasawa theory arguments
by a more direct method.Comment: Comments very welcome
Computable Stone spaces
We investigate computable metrizability of Polish spaces up to homeomorphism.
In this paper we focus on Stone spaces. We use Stone duality to construct the
first known example of a computable topological Polish space not homeomorphic
to any computably metrized space. In fact, in our proof we construct a
right-c.e. metrized Stone space which is not homeomorphic to any computably
metrized space. Then we introduce a new notion of effective categoricity for
effectively compact spaces and prove that effectively categorical Stone spaces
are exactly the duals of computably categorical Boolean algebras. Finally, we
prove that, for a Stone space , the Banach space has a
computable presentation if, and only if, is homeomorphic to a computably
metrized space. This gives an unexpected positive partial answer to a question
recently posed by McNicholl.Comment: 16 page
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