Equivalence-preserving Transformations Research Articles

Super-reparametrizations of weighted CSPs: properties and optimization perspective

The notion of reparametrizations of Weighted CSPs (WCSPs) (also known as equivalence-preserving transformations of WCSPs) is well-known and finds its use in many algorithms to approximate or bound the optimal WCSP value. In contrast, the concept of super-reparametrizations (which are changes of the weights that keep or increase the WCSP objective for every assignment) was already proposed but never studied in detail. To fill this gap, we present a number of theoretical properties of super-reparametrizations and compare them to those of reparametrizations. Furthermore, we propose a framework for computing upper bounds on the optimal value of the (maximization version of) WCSP using super-reparametrizations. We show that it is in principle possible to employ arbitrary (under some technical conditions) constraint propagation rules to improve the bound. For arc consistency in particular, the method reduces to the known Virtual AC (VAC) algorithm. We implemented the method for singleton arc consistency (SAC) and compared it to other strong local consistencies in WCSPs on a public benchmark. The results show that the bounds obtained from SAC are superior for many instance groups.

Proofs and Certificates for Max-SAT

Current Max-SAT solvers are able to efficiently compute the optimal value of an input instance but they do not provide any certificate of its validity. In this paper, we present a tool, called MS-Builder, which generates certificates for the Max-SAT problem in the particular form of a sequence of equivalence-preserving transformations. To generate a certificate, MS-Builder iteratively calls a SAT oracle to get a SAT resolution refutation which is handled and adapted into a sound refutation for Max-SAT. In particular, we prove that the size of the computed Max-SAT refutation is linear with respect to the size of the initial refutation if it is semi-read-once, tree-like regular, tree-like or semi-tree-like. Additionally, we propose an extendable tool, called MS-Checker, able to verify the validity of any Max-SAT certificate using Max-SAT inference rules. Both tools are evaluated on the unweighted and weighted benchmark instances of the 2020 Max-SAT Evaluation.

A mathematical perspective of ultrasound image representations and image quality criteria

What is an ultrasound “image”? Where an engineer or physicist may see a quantitative map of physical properties, a physician may see qualitative anatomical features and pathologies. In the past, each image type was evaluated appropriately within its respective context: quantitative image quality was measured with criteria like the contrast-to-noise ratio (CNR), and qualitative images underwent dynamic range transformations to optimize human perception of contrast. Recent non-linear beamforming methods have blurred this line: the produced images are transformed like qualitative images but evaluated with quantitative criteria like CNR. Here, we identify the mathematical flaws in reasoning and suggest rigorous alternatives.An image assigns values to spatial coordinates. Let X denote the set of all possible image values. The mathematical structure endowed on X determines the set of equivalence-preserving transformations. For instance, qualitative images treat X as a general topological space, where images are equivalent under homeomorphisms of X (dynamic range transformations). Quantitative images treat X as a metric space and have fewer equivalence-preserving transformations. These structures determine when criteria like contrast, CNR, generalized CNR, and spatial resolution can be used. To extrapolate these criteria to other contexts, one must use relevant structure-preserving isomorphisms (e.g., histogram matching to preserve information content).

Tractability-preserving transformations of global cost functions

Graphical model processing is a central problem in artificial intelligence. The optimization of the combined cost of a network of local cost functions federates a variety of famous problems including CSP, SAT and Max-SAT but also optimization in stochastic variants such as Markov Random Fields and Bayesian networks. Exact solving methods for these problems typically include branch and bound and local inference-based bounds.In this paper we are interested in understanding when and how dynamic programming based optimization can be used to efficiently enforce soft local consistencies on Global Cost Functions, defined as parameterized families of cost functions of unbounded arity. Enforcing local consistencies in cost function networks is performed by applying so-called Equivalence Preserving Transformations (EPTs) to the cost functions. These EPTs may transform global cost functions and make them intractable to optimize.We identify as tractable projection-safe those global cost functions whose optimization is and remains tractable after applying the EPTs used for enforcing arc consistency. We also provide new classes of cost functions that are tractable projection-safe thanks to dynamic programming.We show that dynamic programming can either be directly used inside filtering algorithms, defining polynomially DAG-filterable cost functions, or emulated by arc consistency filtering on a Berge-acyclic network of bounded-arity cost functions, defining Berge-acyclic network-decomposable cost functions. We give examples of such cost functions and we provide a systematic way to define decompositions from existing decomposable global constraints.These two approaches to enforcing consistency in global cost functions are then embedded in a solver for extensive experiments that confirm the feasibility and efficiency of our proposal.

Handling equivalence classes of Optimality-Theoretic tableaux

Many Optimality-Theoretic tableaux contain exactly the same information, and equivalence-preserving operations on them have been an object of study for some two decades. This paper shows that several of the operations proposed in the earlier literature together are actually enough to express any possible equivalence-preserving transformation. Moreover, every equivalence class of comparative tableaux (equivalently, of sets of Elementary Ranking Conditions, or ERC sets) has a unique and computable normal form that can be derived using those elementary operations in polynomial time. Any equivalence-preserving operation on comparative tableaux (ERC sets) is thus computable, and normal form tableaux may therefore represent their equivalence classes without loss of generality.

Encoding Nested Boolean Functions as Quantified Boolean Formulas

In this paper, we consider the problem of compactly representing nested instantiations of propositional subformulas with different arguments as quantified Boolean formulas (QBF). We develop a generic QBF encoding pattern which combines and generalizes existing QBF encoding techniques for simpler types of redundancy. We obtain an equivalence-preserving transformation in linear time from the PSPACE-complete language of nested Boolean functions (NBF), also called Boolean programs, to prenex QBF. A transformation in the other direction from QBF to NBF is also possible in at most quadratic time by simulating quantifier expansion.

Finding Multiple Equivalence-Preserving Transformations in Combinational Circuits through Incremental-SAT

This paper introduces an approach to effectively exploit incremental SAT in order to search for multiple equivalence-preserving transformations of combinational circuits. Typical applications, such as redundancy removal with observability and external care conditions, adequate abstractions and other optimizations used in a state-of-the-art SAT-based model checker, can reap benefits from the proposed strategies. Our techniques exploit SAT incrementality, by iteratively refining the set of candidate transformations with a counter-example driven analysis, until an unsatisfiable point is reached. The key point of our technique is the ability to address satisfiable instances first, where SAT solvers are generally much faster than with unsatisfiable runs. We also discuss partitioning and problem reduction issues, that are fundamental in order to provide a scalable approach. Experimental results show the effectiveness of the proposed strategies.

A formal model of real-time program compilation

Program compilation can be formally defined as a sequence of equivalence-preserving transformations, or refinements, from high-level language programs to assembler code. Recent models also incorporate timing properties, but the resulting formalisms are intimidatingly complex. Here we take advantage of a new, simple model of real-time refinement, based on predicate transformer semantics, to present a straightforward compilation formalism that incorporates real-time constraints.

Representing and Reasoning on Database Conceptual Schemas

In this paper we address the problem of reasoning about database conceptual schemas by exploiting the possibility of using a description logic. We develop an approach by using as a foundation an entity-relationship model that displays features such as ISA, disjointness and cardinality constraints. We propose an equivalence-preserving transformation of entity relationship schemas into terminologies in a description logic. This equivalence, based on the measure of information capacity, ensures that the semantics of entity-relationship schemas is accurately captured by the corresponding terminologies. As a consequence, reasoning on entity-relationship schemas is appropriately reduced to reasoning on terminologies in a description logic.

Optimizing general chain programs

The goal of knowledge compilation is to transform programs in order to speed up their evaluation. In Inductive Logic Programming, two major approaches to speed-up learning exist: Approaches that intertwine the learning and the optimization process and approaches that separate these two processes. We follow the latter approach and present a new equivalence-preserving transformation method for programs with ordered clauses. It eliminates redundancies that make forward inference procedures slow. We introduce general chain rules, a specific class of ordered clauses, whose syntactical features are exploited in a new forward inference method. The comparison of the time needed by this method to evaluate the transformed program with the time needed by a standard forward inference procedure for the original program confirms the expected speed-up.

A note on assumptions about Skolem functions

Skolemization is not an equivalence preserving transformation. For the purposes of refutational theorem proving it is sufficient that skolemization preserves satisfiability and unsatisfiability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae considerably. Examples for cases where this occurs systematically are the relational translation from modal logics to predicate logic and the relativization of first-order logics with sorts.

Towards restructuring and normalization of types in databases

The equivalence-preserving transformation and normalization of types in object-oriented databases are discussed. Specifically, a normal form of types based on set-theoretic equivalence is proposed, rewrite rules which transform types into normal forms are presented, and the uniqueness of normal form and the completeness of rewrite rules are proved. The emphasis of this work is on normal forms and corresponding rewrite rules. It provides a new formal approach for the study of restructuring of database schema and other manipulations in object-oriented databases.

Verified functions for generating signed-binary arithmetic hardware

Formally verified metafunctions which synthesize array multipliers and inner product hardware of arbitrary size and structure are presented. The metafunctions operate on signed-binary inputs in general and two's-complement in particular, and are higher order. They are shown to be equivalence-preserving transformations and correctly produce multipliers and inner product hardware of arbitrary size and structure. All the metafunctions, their associated correctness theorems, and their correctness proofs are machine executable within the higher order logic (HOL) theorem prover. The function expressions produced by the metafunctions can be used as hardware synthesis descriptions or as comparison functions for Boolean-comparison-based systems. In addition to the definitions written in higher-order logic, the major definitions are written in a more informal functional programming language-like notation which should facilitate translation of the synthesis functions to other hardware description languages.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Queries and query processing in object-oriented database systems

Object-oriented database mangement systems (OODBMS) combine the data abstraction and computational models of object-oriented programming languages with the query and performance capabilities of database management systems. A concise, formal data model for OODBMS has not been universally accepted, preventing detailed investigation of various system issues such as query processing. We define a data model that captures the essence of classification-based object-oriented systems and formalize concepts such as object identity, inheritence, and methods. The main topic of the paper is the presentation of a query processing methodology complete with an object calculus to object algebra translation are discussed in detail. The paper concludes with a discussion of equivalence-preserving transformation rules for object algebra expressions.