Representing Proof-Planning in the ρ-Calculus
以形状为话题英语对话作文
以形状为话题英语对话作文在网络上搜索了一些范文，然后我将它们融合并加入了一些原创的内容，希望你会喜欢。
Title: Exploring the Fascinating World of Shapes。
Introduction:Shapes are everywhere around us, from the natural world to the man-made environment. They play a fundamental role in our perception of the world and are essential in various fields such as mathematics, art, and architecture. In this essay, we will delve into the captivating realm of shapes, exploring their significance, characteristics, and applications.Body:1. Understanding Shapes:Shapes are defined as the external form or appearance of an object. They can be categorized into basic shapes such as circles, squares, triangles, and rectangles, as well as complex shapes like polygons and irregular forms. Each shape has distinct properties that define its characteristics, including the number of sides, angles, and symmetry.2. Importance of Shapes in Mathematics:In mathematics, shapes are studied extensively in geometry. They serve as the foundation for understanding spatial relationships, measurement, and calculation. Basic geometric shapes are used to teach children fundamental mathematical concepts such as addition, subtraction, multiplication, and division. Moreover, advanced mathematical principles such as calculus and topology rely on the concept of shapes to solve complex problems.3. Shapes in Art and Design:Shapes are integral to the world of art and design.Artists use shapes to create visual compositions, convey emotions, and communicate ideas. Whether it's the geometric patterns in Islamic art, the abstract shapes in modern art movements like cubism, or the organic forms in nature-inspired designs, shapes play a crucial role in artistic expression. Furthermore, architects and designers utilize shapes in architectural plans, interior design, and product development to achieve balance, harmony, and functionality.4. Symbolism of Shapes:Shapes often carry symbolic meanings across different cultures and contexts. For example, the circle represents unity, wholeness, and eternity in many cultures, while the triangle symbolizes stability, balance, and progression. In religious symbolism, the cross is a significant shape representing faith and salvation. Understanding the symbolic significance of shapes enhances our appreciation of cultural traditions and rituals.5. Applications of Shapes in Everyday Life:Shapes have practical applications in various aspects of everyday life. From the shape of traffic signs guiding us on the road to the packaging of products on supermarket shelves, shapes influence our daily experiences. In technology, shapes are used in graphic design, user interface design, and industrial design to create visually appealing and user-friendly products. Moreover, in medicine and biology, understanding the shapes of molecules andcells is crucial for research and diagnosis.6. The Beauty of Geometric Patterns:Geometric patterns, characterized by repetitive shapes and symmetrical arrangements, captivate our senses with their mesmerizing beauty. From the intricate designs of Islamic architecture to the kaleidoscopic patterns found in nature, geometric shapes create visually stunning compositions that evoke a sense of harmony and order. Whether it's the hexagonal honeycomb structure of a beehive or the spiraling patterns of a seashell, geometric shapes abound in the natural world, inspiring artists, mathematicians, and designers alike.Conclusion:In conclusion, shapes are not merely geometric entities but integral components of our perception, expression, and understanding of the world. From the mathematical abstractions of Euclidean geometry to the artistic creations of renowned painters, shapes transcend disciplinary boundaries, enriching our lives with their beauty, significance, and versatility. By exploring the fascinating world of shapes, we gain insights into the interconnectedness of art, science, and culture, unlocking new avenues of creativity, knowledge, and appreciation.。
漫画老师作文2篇英语
漫画老师作文2篇英语Title: The Influence of Comic Books on Education。
Essay 1:Comic books have long been considered a form of entertainment rather than educational tools. However, the reality is that comic books can play a significant role in education. From teaching complex concepts to fostering a love of reading, comic books offer numerous benefits in the realm of education.Firstly, comic books can serve as a gateway to literacy for reluctant readers. The combination of visual imagery and text makes comic books more accessible and less intimidating than traditional novels. Reluctant readers are often drawn to the vibrant illustrations and engaging storylines found in comic books, which can motivate them to spend more time reading. As they become more comfortable with reading comic books, they may gradually transition tomore text-heavy materials, thereby improving their literacy skills.Moreover, comic books have the potential to make complex subjects more comprehensible. By visually representing abstract concepts and ideas, comic books can help students better understand difficult topics such as science, history, and mathematics. For example, graphic novels like "The Manga Guide to Calculus" and "Logicomix: An Epic Search for Truth" use comic book format to elucidate complex mathematical and philosophical conceptsin a way that is both engaging and accessible to readers.Additionally, comic books can promote critical thinking and analysis skills. Readers must actively interpret the visual and textual information presented in comic panels, deciphering meaning from both the dialogue and the artwork. This process encourages readers to think critically about the story's themes, characters, and plot developments. Furthermore, comic books often employ techniques such as symbolism, foreshadowing, and visual metaphors, which require readers to analyze the text on multiple levels.Furthermore, comic books have the potential to foster cultural awareness and empathy. Many comic books explore themes of diversity, social justice, and human rights, exposing readers to different perspectives and experiences. By immersing themselves in the lives of fictional characters from diverse backgrounds, readers can develop a greater understanding and appreciation for the complexities of the world around them.In conclusion, comic books have a valuable role to play in education. From promoting literacy and comprehension to fostering critical thinking and empathy, comic books offer numerous benefits for students of all ages. Byincorporating comic books into educational curricula, educators can harness the power of visual storytelling to enhance learning outcomes and engage students in meaningful ways.Essay 2:The integration of comic books into the classroom hassparked a debate among educators regarding their educational value. While some argue that comic books are merely a form of entertainment with little educational merit, others believe that comic books can be powerful educational tools that enhance learning outcomes and engage students in meaningful ways.One of the primary benefits of using comic books in education is their ability to engage students in the learning process. The visual nature of comic books captures students' attention and makes learning more enjoyable and interactive. By presenting information in a visually stimulating format, comic books can help students stay focused and retain information more effectively than traditional textbooks or lectures.Moreover, comic books can cater to diverse learning styles and abilities. For visual learners, the combination of images and text in comic books can reinforce understanding and facilitate comprehension. Meanwhile, auditory learners can benefit from reading aloud the dialogue and narration found in comic panels. Additionally,students with learning disabilities or language barriers may find comic books more accessible than traditional educational materials, allowing them to participate more fully in classroom activities.Furthermore, comic books can serve as a springboard for interdisciplinary learning. The integration of art, literature, history, and science in comic books provides opportunities for teachers to incorporate multiple subjects into their lessons. For example, a history teacher could use a graphic novel set during a particular historical period to supplement their curriculum, providing students with a more immersive and engaging learning experience.Additionally, comic books can promote creativity and self-expression among students. Encouraging students to create their own comic books allows them to apply their knowledge and imagination in a creative way. Through the process of writing, illustrating, and editing their own comics, students can develop essential skills such as storytelling, visual communication, and problem-solving.In conclusion, comic books have the potential to be valuable educational tools that enhance learning outcomes and engage students in meaningful ways. By leveraging the visual nature of comic books, educators can create dynamic and interactive learning experiences that cater to diverse learning styles and abilities. As the educational landscape continues to evolve, comic books may emerge as a powerful tool for promoting student engagement, creativity, and academic achievement.。
高中参数方程知识点总结
高中参数方程知识点总结Parameter equations are an important topic in high school mathematics, often covered in Algebra II or Pre-Calculus courses. These equations involve representing a curve or shape using parameters such as t, which allows for a different approach to graphing compared to traditional Cartesian coordinates. By using parameter equations, students can better understand and analyze curves and functions in a unique way that helps enhance their problem-solving skills.参数方程是高中数学中一个重要的课题，通常在代数II或预微积分课程中讨论。
这些方程涉及使用诸如t之类的参数来表示曲线或形状，这种方法与传统的笛卡尔坐标系相比有所不同。
通过使用参数方程，学生可以更好地理解和分析曲线和函数，以独特的方式来增强他们的问题解决能力。
One key benefit of parameter equations is their ability to describe more complex shapes and curves that may be difficult to represent using only Cartesian coordinates. For example, an ellipse can be represented using parameter equations as x = acos(t) and y = bsin(t), where a and b are the semi-major and semi-minor axes of the ellipse.This allows students to visualize and analyze such shapes in a new light, providing a deeper understanding of the geometry involved.参数方程的一个关键好处是它们能够描述更复杂的形状和曲线，这些形状和曲线可能很难仅通过笛卡尔坐标系来表示。
数字零的作文
数字零的作文Numbers are an essential part of our daily lives, representing quantity and order. 数字在我们日常生活中扮演着至关重要的角色，代表着数量和顺序。
Among all the numbers, the number zero stands out as both unique and intriguing. 在所有数字中，数字零以其独特性和引人入胜的特点而脱颖而出。
Zero is the only number that represents the absence of quantity, yet it plays a crucial role in mathematics and science. 零是唯一一个代表数量不存在的数字，然而在数学和科学中却起着至关重要的作用。
In mathematics, zero serves as a neutral element in operations such as addition and multiplication, allowing for the concept of zero to be used in various calculations and equations. 在数学中，零在加法和乘法等运算中起到了中性元素的作用，使得零的概念可以被应用在各种计算和方程中。
The concept of zero was revolutionary in the history of mathematics, as it paved the way for the development of algebra, calculus, and other advanced branches of mathematics. 零的概念在数学史上是具有革命性的，它为代数、微积分和其他高级数学分支的发展铺平了道路。
切线方程和导数的关系
切线方程和导数的关系Tangent lines and derivatives are closely related in calculus. The equation of a tangent line at a point on a curve is determined by the derivative of the function representing that curve at that particular point. This relationship between tangent lines and derivatives allows us to make connections between the rate of change of a function and the slopes of lines tangent to the curve it represents.切线方程和导数在微积分中密切相关。
在曲线上某一点处的切线方程由该曲线的导数在那个特定点处确定。
切线和导数之间的这种关系使我们能够在函数的变化率和曲线的切线斜率之间建立联系。
When we find the derivative of a function at a given point, we are essentially determining the slope of the tangent line to the curve at that point. This slope represents the instantaneous rate of change of the function at that specific point. By knowing the derivative, we can find the equation of the tangent line at the point and use it to approximate the behavior of the curve near that point.当我们在给定点求函数的导数时，实质上是确定该点处曲线的切线斜率。
数学上的高阶无穷小 英文
数学上的高阶无穷小英文Mathematics of Higher-Order InfinitesimalsThe realm of mathematics is a vast and intricate landscape, where the exploration of concepts often leads to unexpected and profound discoveries. One such intriguing area within the mathematical universe is the study of higher-order infinitesimals. These infinitesimal quantities, which are smaller than any finite number yet larger than zero, have captivated the minds of mathematicians and philosophers alike, unveiling a deeper understanding of the nature of infinity and the fundamental principles that govern the universe.At the heart of this inquiry lies the notion of differentiation, a foundational concept in calculus that allows us to analyze the rate of change of a function at a specific point. Traditionally, the study of differentiation has been limited to first-order infinitesimals, where the derivative represents the slope of the tangent line to a curve at a given point. However, as mathematicians delved deeper into this field, they recognized the potential for exploring higher-order derivatives and the corresponding infinitesimal quantities.The concept of higher-order infinitesimals emerged from the work ofrenowned mathematicians such as Gottfried Wilhelm Leibniz, Isaac Newton, and Augustin-Louis Cauchy, among others. These pioneers of calculus recognized that the study of higher-order derivatives could unlock a deeper understanding of the behavior of functions and the underlying mathematical structures.One of the key insights that drove the exploration of higher-order infinitesimals was the realization that the traditional approach to differentiation, which relied on the concept of limits, could be expanded to encompass more complex scenarios. By considering the behavior of functions as they approached a specific point, mathematicians were able to define higher-order derivatives, each representing a distinct rate of change.The second-order derivative, for instance, describes the rate of change of the rate of change, providing valuable information about the curvature of a function. Similarly, third-order derivatives shed light on the rate of change of the curvature, and so on, leading to a hierarchy of higher-order infinitesimals.The significance of higher-order infinitesimals extends far beyond the realm of pure mathematics. In the fields of physics and engineering, the study of these infinitesimal quantities has proven invaluable in modeling complex systems and predicting their behavior. For example, in the analysis of mechanical vibrations,higher-order derivatives are used to capture the nuances of oscillatory motion, enabling more accurate predictions and the design of more efficient systems.Furthermore, the exploration of higher-order infinitesimals has opened up new avenues of research in areas such as optimization theory, where the study of higher-order derivatives can lead to more efficient algorithms for solving complex optimization problems. In the realm of quantum mechanics, the application of higher-order infinitesimals has contributed to a deeper understanding of the behavior of subatomic particles and the fundamental forces that govern the universe.Despite the profound implications of higher-order infinitesimals, their study is not without its challenges. The conceptual and technical complexities involved in working with these infinitesimal quantities have often posed significant barriers to their widespread adoption and understanding. Mathematicians and scientists have had to grapple with issues such as the convergence of infinite series, the properties of higher-order derivatives, and the interpretation of these quantities in the context of various mathematical and physical theories.Nevertheless, the pursuit of knowledge and the desire to unravel the mysteries of the universe have driven mathematicians and scientiststo overcome these challenges, continuously expanding the boundaries of our understanding. The study of higher-order infinitesimals has become a testament to the power of human ingenuity and the relentless pursuit of deeper insights into the fundamental nature of our world.As we delve deeper into the realm of higher-order infinitesimals, we are confronted with the realization that the universe is not merely a collection of discrete entities, but rather a tapestry of interconnected processes and relationships. The study of these infinitesimal quantities has the potential to reveal the underlying patterns and structures that govern the behavior of complex systems, from the subatomic to the cosmic scale.In conclusion, the mathematics of higher-order infinitesimals represents a captivating and ever-evolving field of study, one that continues to challenge and inspire mathematicians, scientists, and thinkers alike. By exploring these infinitesimal quantities, we gain a deeper understanding of the fundamental principles that govern the universe, opening up new avenues for discovery and the advancement of human knowledge. As we continue to push the boundaries of our understanding, the study of higher-order infinitesimals promises to uncover even more profound insights, shaping the future of our scientific and technological landscape.。
人工智能与神经网络 试题3
2-4 试说明怎样把一棵与或解树用来表达图2.28所示的电网络阻抗的计算。
单独的R 、L 或C 可分别用R 、j ωL 或1/j ωC 来计算，这个事实用作本原问题。
后继算符应以复合并联和串联阻抗的规则为基础。
图 2.28My RoomRoomISAMy BedMy Desk My chairBedDeskChairLength Width Height 4 m3 m3 mININ IN ISA ISA ISA BookON YelloCOLOURZR 1Z 2Z 3Z 21Z 22Z 211C 1R 2C 3L 1R 3 C 2 L 2Z 32Z 31C 4L 3R 4L 42-5 试用四元数列结构表示四圆盘梵塔问题，并画出求解该问题的与或图。
用四元数列(nA, nB, nC, nD) 来表示状态，其中nA表示A盘落在第nA号柱子上，nB表示B盘落在第nB号柱子上，nC表示C盘落在第nC号柱子上，nD表示D盘落在第nD号柱子上。
初始状态为1111，目标状态为33333-3 化为子句形有哪些步骤?请结合例子说明之。
任一谓词演算公式可以化成一个子句集。
其变换过程由下列九个步骤组成：(1)消去蕴涵符号将蕴涵符号化为析取和否定符号(2)减少否定符号的辖域每个否定符号最多只用到一个谓词符号上，并反复应用狄·摩根定律(3)对变量标准化对哑元改名以保证每个量词有其自己唯一的哑元(4)消去存在量词引入Skolem函数，消去存在量词如果要消去的存在量词不在任何一个全称量词的辖域内，那么我们就用不含变量的Skolem函数即常量。
(5)化为前束形把所有全称量词移到公式的左边，并使每个量词的辖域包括这个量词后面公式的整个部分。
前束形= (前缀) (母式)前缀= 全称量词串母式= 无量词公式(6)把母式化为合取范式反复应用分配律，将母式写成许多合取项的合取的形式，而每一个合取项是一些谓词公式和(或)谓词公式的否定的析取(7)消去全称量词消去前缀，即消去明显出现的全称量词(8)消去连词符号(合取)用{合取项1,合取项2}替换明显出现的合取符号(9)更换变量名称更换变量符号的名称，使一个变量符号不出现在一个以上的子句中3-4 如何通过消解反演求取问题的答案?给出一个公式集S和目标公式L，通过反证或反演来求证目标公式L，其证明步骤如下：(1)否定L，得～L；(2)把～L添加到S中去；(3)把新产生的集合｛～L，S｝化成子句集；(4)应用消解原理，力图推导出一个表示矛盾的空子句NIL。
人工智能原理 北京大学 8 PartIVPlanningChapter8Classicand (8.2.1)
School of Electronic and Computer EngineeringPeking UniversityWang WenminArtificial IntelligenceClassic PlanningContents☐8.2.1. Planning as State-Space Search☐8.2.2. Planning Graphs☐8.2.3. Other Classical Planning Approaches☐1) Forward state-space search 前向状态空间搜索⏹starting in the initial state,从初始状态开始,⏹using the problem’s actions,运用该问题的动作，⏹search forward for a member of the goal states.朝着一个目标状态向前搜索。
☐2) Backward relevant-states search 后向状态空间搜索⏹starting at the set of states representing the goal,从表示该目标的状态集开始，⏹using the inverse of the actions,运用反向的动作，⏹search backward for the initial state.朝着初始状态向后搜索。
Heuristics for planning 规划的启发法☐Think of a search problem as a graph 将搜索问题视为一个图⏹where the nodes are states and the edges are actions,to find a path connectingthe initial state to a goal state.其中节点表示状态、边为动作，寻找一条连接初始状态至某个目标状态的路径。
AP Calculus BC AP微积分BC考试内容授课大纲
AP Calculus BC AP微积分BC考试内容授课大纲AP微积分BC包含AB的内容，难度也远高于AB，大多数学校也更愿意接受BC的成绩。
一般需要花费1年的时间来完成。
微积分是除了部分文科和艺术类专业外，绝大多数专业都要学的基础课程，专业适用面很广。
AP微积分BC考试将测试你对课程单元所涵盖的数学概念的理解，以及你解决问题所用的适当公式和程序的能力，以及用正确的符号交流工作的能力。
部分考试允许使用绘图计算器。
注意：你不能在同一年内同时参加AP微积分AB和微积分BC考试。
考试时长：3hrs 15mins考试时间：Tue, May 4, 2021, 8 AM Local考试分布：Section 1:Multiple Choice多选题，45道题，时长1hr45mins，占比50%Section 2:Free Response自由回答，6道题，时常1hr30mins，占比50%在此之前学生最好掌握以下的知识点：代数、几何、三角学、解析几何、初等函数的课程，特别是理解线性、多项式、有理函数、指数函数、对数函数、三角函数、反三角函数、分段函数，以及数列、级数、极方程。
你应该知道如何画这些函数的图和解。
你还应该熟悉一般函数的代数变换、组合、综合和逆运算。
课程安排：1、Unit 1: 极限和连续性You’ll start to explore how limits will allow you to solve problems involving change学习用极限解决变化问题，并更好地理解函数的数学推理。
Topics may include:•How limits help us to handle change at an instant 极限如何帮助我们处理瞬间的变化•Definition and properties of limits in variousrepresentations各种表示法中极限的定义和性质•Definitions of continuity of a function at apoint and over a domain函数在一点上和区间上连续性的定义•Asymptotes and limits at infinity渐近线和无穷极限•Reasoning using the Squeeze theorem and theIntermediate Value Theorem用夹逼定理和介值定理进行推理考试占比4%–7%Unit 2:微分：定义与基本性质Unit 2: Differentiation: Definition and Fundamental PropertiesTopics may include:•Defining the derivative of a function at a point and as a function定义在一点处函数的导数和导函数•Connecting differentiability and continuity可微性与连续性•Determining derivatives for elementary functions微分运算法则•Applying differentiation rules微分运算法则考试占比4%–7%Unit 3: Differentiation: Composite, Implicit, and Inverse Functions微分：复合函数、隐函数和反函数掌握使用链式法则，学习新的微分技巧和高阶导数Topics may include:· The chain rule for differentiating posite functions 微分复合函数的链式法则· Implicit differentiation隐函数微分· Differentiation of general and particular inverse functions区分一般反函数和特殊反函数· Determining higher-order derivatives of functions 函数的高阶导数求法考试占比4%–7%Unit 4: Contextual Applications of Differentiation微分的实境应用利用导数设置和解决瞬时变化率的实际问题，利用数学推理求不定式的极限。
数学的英语知识点总结
数学的英语知识点总结Basic Operations:The basic operations in mathematics include addition, subtraction, multiplication, and division. Addition is the process of combining two or more numbers to find their total. Subtraction is the process of finding the difference between two numbers. Multiplication is the process of repeated addition, and division is the process of splitting a number into equal parts.Algebra:Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. It involves solving equations, finding unknown quantities, and representing relationships between variables. Algebra is used in various fields such as physics, engineering, and computer science.Geometry:Geometry is the branch of mathematics that deals with shapes, sizes, and properties of space. It includes topics such as points, lines, angles, triangles, circles, and solid figures. Geometry is used in architecture, art, and navigation.Calculus:Calculus is the branch of mathematics that deals with rates of change and accumulation. It includes topics such as derivatives, integrals, limits, and differential equations. Calculus is used in physics, engineering, economics, and finance.Real-World Applications:Mathematics is used in various real-world applications such as finance, engineering, and science. In finance, mathematics is used to calculate interest rates, risk, and investment returns. In engineering, mathematics is used to design and analyze structures, machines, and systems. In science, mathematics is used to model natural phenomena, analyze data, and make predictions.In conclusion, mathematics is a fundamental subject that is essential for understanding the world around us. It includes basic operations, algebra, geometry, and calculus, and is used in various real-world applications. Understanding mathematics is important for developing critical thinking and problem-solving skills, and for pursuing careers in science, technology, engineering, and mathematics.。
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
RepresentingProof-Planningintheρ-CalculusBenjaminWack1SergeAutexier2,3AndreasMeier31LORIA&UniversityH.Poincar´e,Nancy,France,wack@loria.fr
2SaarlandUniversity,Saarbr¨ucken,Germany,autexier@ags.uni-sb.de
3DFKIGmbH,Saarbr¨ucken,Germany,{ameier,autexier}@dfki.de
Introduction.Proofplanningisatheoremprovingtechniqueinwhichproofsarenotconceivedimmediatelybyusingthelow-levelcalculusrulesbutatahigherlevelofabstractionthathighlightsthemainideasandtheoutlineofaproofandde-emphasizesminorlogicalormathematicalma-nipulationsonformulae.Hence,proofplanningconsiderablydiffersfromtraditionalproofsearchstrategiesinautomatedtheoremproving,andacomparisonofproofplanningwithtraditionalstrategiesisdifficult.WedevelopedamodellingofasimpleproofplanningprocedureasusedintheΩmegaproofdevelopmentsystem[4]intheρ-calculus[3].Thismodellingprovidesafirstversionofaformalframeworkwhichallowsforthecomparisionofproofplanningwithotherproofsearchstrategies.
ProofPlanning.Proofplanning[1]plansaproofattheabstractlevelofmethods,i.e.,tacticsenrichedbyexplicitpre-andpostconditions.Mathematicallymotivatedheuristicshowtoproceedareencodedinthecontrolknowledgeneededtosearchforthesequenceorhierarchyofmethodsthatresultsinasolutionplan.Knowledge-basedproofplanning[5]declarativelyrepresentscontrolknowledgeascontrolrules,whichareevaluatedatchoicepointsintheplanningprocess(choiceofmethod,choiceofgoal,etc.).Asimpleproofplanningproceduresearchesatthelevelofmethods,i.e.,itsearchesforapplicablemethodsandintroducestheinstantiatedmethodsintheproofplanunderconstructionuntilalltasksareclosed,i.e.therearenoopengoalsleft.Inanutshell,thiscanbedescribedbythefollowingalgorithm,whereAisanagenda,i.e.,thelistofcurrenttasks,CLisalistofgivencontrolrules,andMLisalistofgivenmethods:
WhileA=∅do1.SelecttaskTfromAusingthecontrolrulesCL;2.SelectmethodMfromMLusingthecontrolrulesCL;3.ApplymethodMtoTwhichresultsinalistofactions,i.e.,instantiatedmethods,denotedbyM(−→p1),...M(−→pn);4.SelectanactionM(−→pi)usingcontrolrulesfromCLandapplyittoTtoobtainalistofnewtasksTL;5.A:=A−{T}∪TL.
Modellingintheρ-calculus.Theρ-calculusisaformalismintegratingboththeλ-calculusandrewriting.Itstermsaregivenbythegrammar
T::=X|f|TT|T→T|T;TThebasetermsarevariablesXandconstantsf.Explicitapplicationofatermtoanotherisavailableasintheλ-calculus,butitisalsousedtobuildalgebraicterms:atermf(t1,...,tn)isrepresentedby(((ft1)...)tn).Themainabstractionmechanismisthearrow→,soaλ-abstractionλx.Mcanberepresentedasx→Mbutarewritingrulel→rcanalsoberepresented,usingtherepresentationofalgebraicterms.AstructureA;Bcanbeusedtorepresentasetofresults.
Animportantfeatureoftheρ-calculusisthatthereductionofatermimpliestheresolutionofcertainmatchingequations,moduloanarbitrarytheoryT:
(P→T)U→ρTθifPθ=TU
1Forthepurposeofmodelingproof-planning,itisconvenienttousethefollowingpropertiesinthetheoryT:
•asymbol“,”whichisassociative(i.e.,(a,b),c=a,(b,c))andhasaneutralelementnil(a,nil=a=nil,a)sothatwecaneasilyrepresentandmanipulatelists;
•matchingisnon-linear:theequationf(X,X)f(T,U)hasasolutionifandonlyifT=TU;Letusalsorecallthat,astheλ-calculusisembeddedintheρ-calculus,fixedpointsaregrantedforfree.Basedonthat,thesimpleproofplanningprocedureisthenmodeledintheρ-calculusasfollows:
•propositionsarerepresentedbyalgebraicterms;•ataskisanalgebraictermcontainingalabel,ajustification,agoalpropositionandalistofpremises;
•anagendaisalistoftasks;•amethodisarewriteruletakingatask,checkingsomeapplicabilityconditions(bypatternmatching)andreturningalistofnewtasks;
•thecontrolrulesthatdrivetheselectionofacurrenttask,methodsandnewtasksarealsorepresentedbyrewriterules,whichmainlyperformsimpleoperationsonlists,suchasinsertion,deletionandreordering.
Oncethelistoperationsareproperlyexpressedintheρ-calculus,themainalgorithmcanthenbeexpressedbyasinglefixedpoint(accountingforthemainloop).
ComparisonwithElanTheρ-calculuswasfirstintroducedasasemanticstotherule-basedlanguageElan[2].Inthisworkwesawthatthebasiccomponentsofthecalculus(patternmatch-ing,explicitfunctionmanipulation)canbeusedquitedifferentlyformodelingvariousparadigms.Forinstance,fixedpointsareusedasanunderlyingmechanismforencodingthemainloopoftheproofplanneraswellasthetraversalofaterminElan;however,theyarealsousedforrepre-sentingtherepeat*strategyofElan,whichhasnocounterpartinproofplanning.Conversely,patternmatchingcanbeusedinbothformalismsasawaytocheckapplicabilityofaruleoramethod,butitisonlyinproofplanningthatthehigher-levelcontroldiscriminatesontheobjectitistreatingforselectingamethod,whereasinElanstrategiesarekeptassimpleaspossible.