11 research outputs found
Randomisation and Derandomisation in Descriptive Complexity Theory
We study probabilistic complexity classes and questions of derandomisation
from a logical point of view. For each logic L we introduce a new logic BPL,
bounded error probabilistic L, which is defined from L in a similar way as the
complexity class BPP, bounded error probabilistic polynomial time, is defined
from PTIME. Our main focus lies on questions of derandomisation, and we prove
that there is a query which is definable in BPFO, the probabilistic version of
first-order logic, but not in Cinf, finite variable infinitary logic with
counting. This implies that many of the standard logics of finite model theory,
like transitive closure logic and fixed-point logic, both with and without
counting, cannot be derandomised. Similarly, we present a query on ordered
structures which is definable in BPFO but not in monadic second-order logic,
and a query on additive structures which is definable in BPFO but not in FO.
The latter of these queries shows that certain uniform variants of AC0
(bounded-depth polynomial sized circuits) cannot be derandomised. These results
are in contrast to the general belief that most standard complexity classes can
be derandomised. Finally, we note that BPIFP+C, the probabilistic version of
fixed-point logic with counting, captures the complexity class BPP, even on
unordered structures
Non-Definability Results for Randomised First-Order Logic
We investigate the expressive power of randomised first-order logic
(BPFO) on restricted classes of structures. While BPFO is
stronger than FO in general, even on structures with a built-in
addition relation, we show that BPFO is not stronger than FO
on structures with a unary vocabulary, nor on the class of
equivalence relations. The same techniques can be applied to show
that evenness of a linear order, and therefore graph connectivity,
can not be defined in BPFO. Finally, we show that there is an
FO[<]-definable query on word structures which can not be
defined in BPFO[+1]
A Recursion-Theoretic Characterization of the Probabilistic Class PP
Probabilistic complexity classes, despite capturing the notion of feasibility, have escaped any treatment by the tools of so-called implicit-complexity. Their inherently semantic nature is of course a barrier to the characterization of classes like BPP or ZPP, but not all classes are semantic. In this paper, we introduce a recursion-theoretic characterization of the probabilistic class PP, using recursion schemata with pointers
A Recursion-Theoretic Characterization of the Probabilistic Class PP
Probabilistic complexity classes, despite capturing the notion of feasibility, have escaped any treatment by the tools of so-called implicit-complexity. Their inherently semantic nature is of course a barrier to the characterization of classes like BPP or ZPP, but not all classes are semantic. In this paper, we introduce a recursion-theoretic characterization of the probabilistic class PP, using recursion schemata with pointers
Enumerating Error Bounded Polytime Algorithms Through Arithmetical Theories
We consider a minimal extension of the language of arithmetic, such that the
bounded formulas provably total in a suitably-defined theory \`a la Buss
(expressed in this new language) precisely capture polytime random functions.
Then, we provide two new characterizations of the semantic class BPP obtained
by internalizing the error-bound check within a logical system: the first
relies on measure-sensitive quantifiers, while the second is based on standard
first-order quantification. This leads us to introduce a family of effectively
enumerable subclasses of BPP, called BPP_T and consisting of languages captured
by those probabilistic Turing machines whose underlying error can be proved
bounded in the theory T. As a paradigmatic example of this approach, we
establish that polynomial identity testing is in BPP_T where
T= is a well-studied theory based on bounded
induction
Generalisation and expressiveness for over-parameterised neural networks
Over-parameterised modern neural networks owe their success to two fundamental properties: expressive power and generalisation capability. The former refers to the model's ability to fit a large variety of data sets, while the latter enables the network to extrapolate patterns from training examples and apply them to previously unseen data. This thesis addresses a few challenges related to these two key properties.
The fact that over-parameterised networks can fit any data set is not always indicative of their practical expressiveness. This is the object of the first part of this thesis, where we delve into how the input information can get lost when propagating through a deep architecture, and we propose as an easily implementable possible solution the introduction of suitable scaling factors and residual connections.
The second part of this thesis focuses on generalisation. The reason why modern neural networks can generalise well to new data without overfitting, despite being over-parameterised, is an open question that is currently receiving considerable attention in the research community. We explore this subject from information-theoretic and PAC-Bayesian viewpoints, proposing novel learning algorithms and generalisation bounds
Towards a logical foundation of randomized computation
This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations.
The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy.
In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence.
Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss
Randomisation and Derandomisation in Descriptive Complexity Theory
We study probabilistic complexity classes and questions of derandomisation
from a logical point of view. For each logic L we introduce a new logic BPL,
bounded error probabilistic L, which is defined from L in a similar way as the
complexity class BPP, bounded error probabilistic polynomial time, is defined
from PTIME. Our main focus lies on questions of derandomisation, and we prove
that there is a query which is definable in BPFO, the probabilistic version of
first-order logic, but not in Cinf, finite variable infinitary logic with
counting. This implies that many of the standard logics of finite model theory,
like transitive closure logic and fixed-point logic, both with and without
counting, cannot be derandomised. Similarly, we present a query on ordered
structures which is definable in BPFO but not in monadic second-order logic,
and a query on additive structures which is definable in BPFO but not in FO.
The latter of these queries shows that certain uniform variants of AC0
(bounded-depth polynomial sized circuits) cannot be derandomised. These results
are in contrast to the general belief that most standard complexity classes can
be derandomised. Finally, we note that BPIFP+C, the probabilistic version of
fixed-point logic with counting, captures the complexity class BPP, even on
unordered structures
RANDOMISATION AND DERANDOMISATION IN DESCRIPTIVE COMPLEXITY THEORY
Vol. 7 (3:14) 2011, pp. 1–2