scholarly journals On Quantifying Literals in Boolean Logic and its Applications to Explainable AI

2021 ◽  
Vol 72 ◽  
pp. 285-328
Author(s):  
Adnan Darwiche ◽  
Pierre Marquis
Keyword(s):  
Boolean Logic ◽  
Fine Grained ◽  
Explainable Ai ◽  
Boolean Formulas ◽  
Universal Quantification ◽  
Existential Quantification

Quantified Boolean logic results from adding operators to Boolean logic for existentially and universally quantifying variables. This extends the reach of Boolean logic by enabling a variety of applications that have been explored over the decades. The existential quantification of literals (variable states) and its applications have also been studied in the literature. In this paper, we complement this by introducing and studying universal literal quantification and its applications, particularly to explainable AI. We also provide a novel semantics for quantification, discuss the interplay between variable/literal and existential/universal quantification, and identify some classes of Boolean formulas and circuits on which quantification can be done efficiently. Literal quantification is more fine-grained than variable quantification as the latter can be defined in terms of the former, leading to a refinement of quantified Boolean logic with literal quantification as its primitive.

scholarly journals Decidability of Definability

2013 ◽  
Vol 78 (4) ◽  
pp. 1036-1054 ◽  
Author(s):  
Manuel Bodirsky ◽  
Michael Pinsker ◽  
Todor Tsankov
Keyword(s):  
Random Graph ◽  
Topological Dynamics ◽  
Homogeneous Structure ◽  
First Order ◽  
Computational Problem ◽  
Random Tournament ◽  
Universal Quantification ◽  
Countably Infinite ◽  
Existential Quantification

AbstractFor a fixed countably infinite structure Γ with finite relational signature τ, we study the following computational problem: input are quantifier-free τ-formulas ϕ0, ϕ1, …, ϕn that define relations R0, R1, …, Rn over Γ. The question is whether the relation R0 is primitive positive definable from R1, …, Rn, i.e., definable by a first-order formula that uses only relation symbols for R1, …, Rn, equality, conjunctions, and existential quantification (disjunction, negation, and universal quantification are forbidden).We show decidability of this problem for all structures Γ that have a first-order definition in an ordered homogeneous structure Δ with a finite relational signature whose age is a Ramsey class and determined by finitely many forbidden substructures. Examples of structures Γ with this property are the order of the rationals, the random graph, the homogeneous universal poset, the random tournament, all homogeneous universal C-relations, and many more. We also obtain decidability of the problem when we replace primitive positive definability by existential positive, or existential definability. Our proof makes use of universal algebraic and model theoretic concepts, Ramsey theory, and a recent characterization of Ramsey classes in topological dynamics.

A remark on Scott's interpolation theorem for Lω1ω

1977 ◽  
Vol 42 (1) ◽  
pp. 63-63 ◽  
Author(s):  
Nobuyoshi Motoháshi
Keyword(s):  
Atomic Formula ◽  
Interpolation Theorem ◽  
Infinitary Logic ◽  
Closed Formula ◽  
Free Variables ◽  
Universal Quantification ◽  
Infinite Sets ◽  
Existential Quantification

In [1], H. Africk proved that Scott's interpolation theorem does not hold in the infinitary logic Lω1ω. In this paper we shall show that there is an interpolation theorem in Lω1ω which can be considered as an extension of Scott's interpolation theorem in Lω1ω by using a technique developed in Motohashi [2] and [3]. We use the terminology in [1]. Therefore {Ri; i ∈ J} is the set of predicate symbols in our language. Now let us divide the set of all the free variables into mutually disjoint infinite sets {VI; I ⊆ J}. Suppose that ℱ ⊆ (J). Then a formula in Lω1ω is said to be an ℱ′-formula if it is obtained from atomic formula of the form Ri(X1, …, Xn) for some I ∈ i ∈ I and X1, …, Xn in V1,, by applying ¬ (negation), ∧ (countable conjunction), ∨ (countable disjunction), → (implication), ∀ (universal quantification), and ∃ (existential quantification). Notice that every ℱ-sentence in [1] is an ℱ′. sentence (ℱ′-closed formula) in our sense.Then we have the following theorem which is an immediate consequence of the interpolation theorem in [2].Theorem. Let A and ? be sentences. There is an ℱ′-sentence C such that A→C and C→B are provable iff whenever and are ℱ-isomorphic structures and satisfies A, then satisfies B.

scholarly journals A QSAT Benchmark Based on Vertex-Folkman Problems (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13881-13882
Author(s):  
David E. Narváez
Keyword(s):  
Graph Theory ◽  
Symmetry Breaking ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Fine Grained ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Level Of Difficulty

The purpose of this paper is to draw attention to a particular family of quantified Boolean formulas (QBFs) stemming from encodings of some vertex Folkman problems in extremal graph theory. We argue that this family of formulas is interesting for QSAT research because it is both conceptually simple and parametrized in a way that allows for a fine-grained diversity in the level of difficulty of its instances. Additionally, when coupled with symmetry breaking, the formulas in this family exhibit backbones (unique satisfying assignments) at the top-level existential variables. This benchmark is thus suitable for addressing questions regarding the connection between the existence of backbones and the hardness of QBFs.

scholarly journals A refinement calculus for logic programs

2002 ◽  
Vol 2 (4-5) ◽  
pp. 425-460 ◽  
Author(s):  
IAN HAYES ◽  
ROBERT COLVIN ◽  
DAVID HEMER ◽  
PAUL STROOPER ◽  
RAY NICKSON
Keyword(s):  
Wide Spectrum ◽  
Tool Support ◽  
Logic Programs ◽  
Refinement Calculus ◽  
Partial Functions ◽  
Prototype Tool ◽  
Stepwise Development ◽  
Declarative Semantics ◽  
Universal Quantification ◽  
Existential Quantification

Existing refinement calculi provide frameworks for the stepwise development of imperative programs from specifications. This paper presents a refinement calculus for deriving logic programs. The calculus contains a wide-spectrum logic programming language, including executable constructs such as sequential conjunction, disjunction, and existential quantification, as well as specification constructs such as general predicates, assumptions and universal quantification. A declarative semantics is defined for this wide-spectrum language based on executions. Executions are partial functions from states to states, where a state is represented as a set of bindings. The semantics is used to define the meaning of programs and specifications, including parameters and recursion. To complete the calculus, a notion of correctness-preserving refinement over programs in the wide-spectrum language is defined and refinement laws for developing programs are introduced. The refinement calculus is illustrated using example derivations and prototype tool support is discussed.

Application of an image management system in microscopy

1992 ◽  
Vol 50 (2) ◽  
pp. 1052-1053
Author(s):  
Richard S. Chemock
Keyword(s):  
Data Storage ◽  
Cost Savings ◽  
Image Data ◽  
Analytical Techniques ◽  
Operational Mode ◽  
Fine Grained ◽  
Image Recording ◽  
Primary Output ◽  
Capacity Data ◽  
Image Management System

One of the most common tasks in a typical analysis lab is the recording of images. Many analytical techniques (TEM, SEM, and metallography for example) produce images as their primary output. Until recently, the most common method of recording images was by using film. Current PS/2R systems offer very large capacity data storage devices and high resolution displays, making it practical to work with analytical images on PS/2s, thereby sidestepping the traditional film and darkroom steps. This change in operational mode offers many benefits: cost savings, throughput, archiving and searching capabilities as well as direct incorporation of the image data into reports.The conventional way to record images involves film, either sheet film (with its associated wet chemistry) for TEM or PolaroidR film for SEM and light microscopy. Although film is inconvenient, it does have the highest quality of all available image recording techniques. The fine grained film used for TEM has a resolution that would exceed a 4096x4096x16 bit digital image.

Use of fractals to describe microstructures of porous thick-film platinum electrodes

1992 ◽  
Vol 50 (1) ◽  
pp. 314-315
Author(s):  
Steven D. Toteda
Keyword(s):  
Fractal Dimension ◽  
Power Plants ◽  
Electrical Response ◽  
Cubic Phase ◽  
Platinum Electrodes ◽  
Fine Grained ◽  
Impedance Response ◽  
Platinum Particles ◽  
Energy Surfaces ◽  
Highly Porous

Zirconia oxygen sensors, in such applications as power plants and automobiles, generally utilize platinum electrodes for the catalytic reaction of dissociating O2 at the surface. The microstructure of the platinum electrode defines the resulting electrical response. The electrode must be porous enough to allow the oxygen to reach the zirconia surface while still remaining electrically continuous. At low sintering temperatures, the platinum is highly porous and fine grained. The platinum particles sinter together as the firing temperatures are increased. As the sintering temperatures are raised even further, the surface of the platinum begins to facet with lower energy surfaces. These microstructural changes can be seen in Figures 1 and 2, but the goal of the work is to characterize the microstructure by its fractal dimension and then relate the fractal dimension to the electrical response. The sensors were fabricated from zirconia powder stabilized in the cubic phase with 8 mol% percent yttria. Each substrate was sintered for 14 hours at 1200°C. The resulting zirconia pellets, 13mm in diameter and 2mm in thickness, were roughly 97 to 98 percent of theoretical density. The Engelhard #6082 platinum paste was applied to the zirconia disks after they were mechanically polished ( diamond). The electrodes were then sintered at temperatures ranging from 600°C to 1000°C. Each sensor was tested to determine the impedance response from 1Hz to 5,000Hz. These frequencies correspond to the electrode at the test temperature of 600°C.

Amorphous silica coating on α-alumina particles

1995 ◽  
Vol 53 ◽  
pp. 210-211
Author(s):  
J. W. Mellowes ◽  
C. M. Chun ◽  
I. A. Aksay
Keyword(s):  
Amorphous Silica ◽  
Heterogeneous Nucleation ◽  
Homogeneous Nucleation ◽  
Silica Coating ◽  
Full Density ◽  
Optimal Conditions ◽  
Fine Grained ◽  
Alumina Particles ◽  
Complete Mullitization ◽  
Powder Compacts

Mullite (3Al2O32SiO2) can be fabricated by transient viscous sintering using composite particles which consist of inner cores of a-alumina and outer coatings of amorphous silica. Powder compacts prepared with these particles are sintered to almost full density at relatively low temperatures (~1300°C) and converted to dense, fine-grained mullite at higher temperatures (>1500°C) by reaction between the alumina core and the silica coating. In order to achieve complete mullitization, optimal conditions for coating alumina particles with amorphous silica must be achieved. Formation of amorphous silica can occur in solution (homogeneous nucleation) or on the surface of alumina (heterogeneous nucleation) depending on the degree of supersaturation of the solvent in which the particles are immersed. Successful coating of silica on alumina occurs when heterogeneous nucleation is promoted and homogeneous nucleation is suppressed. Therefore, one key to successful coating is an understanding of the factors such as pH and concentration that control silica nucleation in aqueous solutions. In the current work, we use TEM to determine the optimal conditions of this processing.

Interfacial structures of dissimilar materials joined by capacitor-discharge welding

1994 ◽  
Vol 52 ◽  
pp. 534-535
Author(s):  
C. P. Doğan ◽  
R. D. Wilson ◽  
J. A. Hawk
Keyword(s):  
Rapid Solidification ◽  
Fusion Zone ◽  
Dissimilar Materials ◽  
Solidification Process ◽  
Weld Interface ◽  
Capacitor Discharge ◽  
Fine Grained ◽  
Iron Aluminum ◽  
Conductive Ceramics ◽  
Capacitor Discharge Welding

Capacitor Discharge Welding is a rapid solidification technique for joining conductive materials that results in a narrow fusion zone and almost no heat affected zone. As a result, the microstructures and properties of the bulk materials are essentially continuous across the weld interface. During the joining process, one of the materials to be joined acts as the anode and the other acts as the cathode. The anode and cathode are brought together with a concomitant discharge of a capacitor bank, creating an arc which melts the materials at the joining surfaces and welds them together (Fig. 1). As the electrodes impact, the arc is extinguished, and the molten interface cools at rates that can exceed 106 K/s. This process results in reduced porosity in the fusion zone, a fine-grained weldment, and a reduced tendency for hot cracking.At the U.S. Bureau of Mines, we are currently examining the possibilities of using capacitor discharge welding to join dissimilar metals, metals to intermetallics, and metals to conductive ceramics. In this particular study, we will examine the microstructural characteristics of iron-aluminum welds in detail, focussing our attention primarily on interfaces produced during the rapid solidification process.

TEM/AEM characterization of fine-grained clay minerals in very-low-grade rocks: Evaluation of contamination by empa involving celadonite family minerals

1996 ◽  
Vol 54 ◽  
pp. 694-695
Author(s):  
Gejing Li ◽  
D. R. Peacor ◽  
D. S. Coombs ◽  
Y. Kawachi
Keyword(s):  
Electron Microscopy ◽  
Volcanic Rocks ◽  
Analytical Electron Microscopy ◽  
Metamorphic Rocks ◽  
New Mineral ◽  
Chemical Compositions ◽  
Low Grade ◽  
Fine Grained ◽  
Depth Analysis ◽  
Mineral Aggregates

Recent advances in transmission electron microscopy (TEM) and analytical electron microscopy (AEM) have led to many new insights into the structural and chemical characteristics of very finegrained, optically homogeneous mineral aggregates in sedimentary and very low-grade metamorphic rocks. Chemical compositions obtained by electron microprobe analysis (EMPA) on such materials have been shown by TEM/AEM to result from beam overlap on contaminant phases on a scale below resolution of EMPA, which in turn can lead to errors in interpretation and determination of formation conditions. Here we present an in-depth analysis of the relation between AEM and EMPA data, which leads also to the definition of new mineral phases, and demonstrate the resolution power of AEM relative to EMPA in investigations of very fine-grained mineral aggregates in sedimentary and very low-grade metamorphic rocks.Celadonite, having end-member composition KMgFe3+Si4O10(OH)2, and with minor substitution of Fe2+ for Mg and Al for Fe3+ on octahedral sites, is a fine-grained mica widespread in volcanic rocks and volcaniclastic sediments which have undergone low-temperature alteration in the oceanic crust and in burial metamorphic sequences.

Correlation between dislocation, grain boundary and interface of duplex SS in stress corrosion cracking(SCC)

1990 ◽  
Vol 48 (4) ◽  
pp. 928-929
Author(s):  
Wang Zheng-fang ◽  
Z.F. Wang
Keyword(s):  
Stainless Steel ◽  
Thermal Cycle ◽  
Secondary Phase ◽  
Corrosion Cracking ◽  
Coarse Grained ◽  
Test Environment ◽  
Fine Grained ◽  
Evaluation Test ◽  
Welding Thermal Cycle ◽  
Micro Analysis

The main purpose of this study highlights on the evaluation of chloride SCC resistance of the material,duplex stainless steel,OOCr18Ni5Mo3Si2 (18-5Mo) and its welded coarse grained zone(CGZ).18-5Mo is a dual phases (A+F) stainless steel with yield strength:512N/mm2 .The proportion of secondary Phase(A phase) accounts for 30-35% of the total with fine grained and homogeneously distributed A and F phases(Fig.1).After being welded by a specific welding thermal cycle to the material,i.e. Tmax=1350°C and t8/5=20s,microstructure may change from fine grained morphology to coarse grained morphology and from homogeneously distributed of A phase to a concentration of A phase(Fig.2).Meanwhile,the proportion of A phase reduced from 35% to 5-10°o.For this reason it is known as welded coarse grained zone(CGZ).In association with difference of microstructure between base metal and welded CGZ,so chloride SCC resistance also differ from each other.Test procedures:Constant load tensile test(CLTT) were performed for recording Esce-t curve by which corrosion cracking growth can be described, tf,fractured time,can also be recorded by the test which is taken as a electrochemical behavior and mechanical property for SCC resistance evaluation. Test environment:143°C boiling 42%MgCl2 solution is used.Besides, micro analysis were conducted with light microscopy(LM),SEM,TEM,and Auger energy spectrum(AES) so as to reveal the correlation between the data generated by the CLTT results and micro analysis.

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