9 research outputs found
How Fast Can We Play Tetris Greedily With Rectangular Pieces?
Consider a variant of Tetris played on a board of width and infinite
height, where the pieces are axis-aligned rectangles of arbitrary integer
dimensions, the pieces can only be moved before letting them drop, and a row
does not disappear once it is full. Suppose we want to follow a greedy
strategy: let each rectangle fall where it will end up the lowest given the
current state of the board. To do so, we want a data structure which can always
suggest a greedy move. In other words, we want a data structure which maintains
a set of rectangles, supports queries which return where to drop the
rectangle, and updates which insert a rectangle dropped at a certain position
and return the height of the highest point in the updated set of rectangles. We
show via a reduction to the Multiphase problem [P\u{a}tra\c{s}cu, 2010] that on
a board of width , if the OMv conjecture [Henzinger et al., 2015]
is true, then both operations cannot be supported in time
simultaneously. The reduction also implies polynomial bounds from the 3-SUM
conjecture and the APSP conjecture. On the other hand, we show that there is a
data structure supporting both operations in time on
boards of width , matching the lower bound up to a factor.Comment: Correction of typos and other minor correction
Proceedings of the 2022 XCSP3 Competition
This document represents the proceedings of the 2022 XCSP3 Competition. The
results of this competition of constraint solvers were presented at FLOC
(Federated Logic Conference) 2022 Olympic Games, held in Haifa, Israel from
31th July 2022 to 7th August, 2022.Comment: arXiv admin note: text overlap with arXiv:1901.0183
Proof-relevant resolution : the foundations of constructive proof automation
Dependent type theory is an expressive programming language. This language
allows to write programs that carry proofs of their properties. This in turn gives
high confidence in such programs, making the software trustworthy. Yet, the trustworthiness comes for a price: type inference involves an increasing number of proof
obligations.
Automation of this process becomes necessary for any system with dependent
types that aims to be usable in practice. At the same time, implementation of automation in a verified manner is prohibitively complex. Sometimes, external solvers
are used to aid the automation. These solvers may be based on classical logic and
may not be themselves verified, thus compromising the guarantees provided by constructive nature of type theory. In this thesis, we explore the idea of proof relevant
resolution that allows automation of type inference in type theory in a verifiable and
constructive manner, hence to restore the confidence in programs and the trustworthiness of software.
Technical content of this thesis is threefold. First, we propose a novel framework for proof-relevant resolution. We take two constructive logics, Horn-clause
and hereditary Harrop formulae logics as a starting point. We formulate the standard big-step operational semantics of these logics. We expose their Curry-Howard
nature by treating formulae of these logics as types and proofs as terms thus developing a theory of proof-relevant resolution. We develop small-step operational
semantics of proof-relevant resolution and prove it sound with respect to the big-step
operational semantics.
Secondly, we demonstrate our approach on an example of type inference in Logical Framework (LF). We translate a type-inference problem in LF into resolution
in proof-relevant Horn-clause logic. Such resolution provides, besides an answer
substitution to logic variables, a proof term that captures the resolution tree. We
interpret the proof term as a derivation of well-formedness judgement of the object in the original problem. This allows for a straightforward implementation of
type checking of the resolved solution since type checking is reduced to verifying the derivation captured by the proof term. The theoretical development is substantiated
by an implementation.
Finally, we demonstrate that our approach allows to reason about semantic properties of code. Type class resolution has been well-known to be a proof-relevant fragment of Horn-clause logic, and recently its coinductive extensions were introduced.
In this thesis, we show that all of these extensions amalgamate with the theoretical
framework we introduce. Our novel result here is exposing that the coinductive
extensions are actually based on hereditary Harrop logic, rather than Horn-clause
logic. We establish a number of soundness and completeness results for them. We
also discuss soundness of program transformation that are allowed by proof-relevant
presentation of type class resolution
Random Testing For Language Design
Property-based random testing can facilitate formal verification, exposing errors early on in the proving process and guiding users towards correct specifications and implementations. However, effective random testing often requires users to write custom generators for well-distributed random data satisfying complex logical predicates, a task which can be tedious and error prone.
In this work, I aim to reduce the cost of property-based testing by making such generators easier to write, read and maintain. I present a domain-specific language, called Luck, in which generators are conveniently expressed by decorating predicates with lightweight annotations to control both the distribution of generated values and the amount of constraint solving that happens before each variable is instantiated.
I also aim to increase the applicability of testing to formal verification by bringing advanced random testing techniques to the Coq proof assistant. I describe QuickChick, a QuickCheck clone for Coq, and improve it by incorporating ideas explored in the context of Luck
to automatically derive provably correct generators for data constrained by inductive relations.
Finally, I evaluate both QuickChick and Luck in a variety of complex case studies from programming languages literature, such as information-flow abstract machines and type systems for lambda calculi
Automated Deduction – CADE 28
This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions
Automated Reasoning
This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book