【正文】 interpretations I and J are elementarily equivalent (I ? J) if and only if |=I ? ? |=J ? and |=J ? ? |=I ? for any sentence ?. ? Example: |I|= set of real numbers |J|= set of rational numbers (有理數(shù) ) I(R): greater than relation on reals J(R): greater than relation on rationals I and J are elementarily equivalent despite their universes are different. if ?:R(5,3) then |=I ? also |=J ? Logical Foundations of Artificial Intelligence Chapter 2: 54 169。 School of CIT, Beijing JiaoTong University Steps of knowledge representation (1) conceptualizing the application area (2) select a vocabulary of object constants, function constants and relation constants. (3) associate these constants with the objects, functions and relations in our conceptualization. (4) write sentences ? Start with an idea of a conceptualization, write more and more sentences to make it precise. Logical Foundations of Artificial Intelligence Chapter 2: 55 169。 School of CIT, Beijing JiaoTong University 167。 Circuits Examples ? Conceptualizing the application area – f1: a circuit – x1, x2: xor gates – a1,a2: and gates – o1: or gate – 20 Ports: ? Three inputs and two outputs of f1 ? Two inputs and one output of each gate Logical Foundations of Artificial Intelligence Chapter 2: 56 169。 School of CIT, Beijing JiaoTong University Select a vocabulary Logical Foundations of Artificial Intelligence Chapter 2: 57 169。 School of CIT, Beijing JiaoTong University Connection Representation ? Structure description of the circuit Logical Foundations of Artificial Intelligence Chapter 2: 58 169。 School of CIT, Beijing JiaoTong University State representation ? Associate a port with a value ? Example: – Inputs are 1,0,1 – Outputs are 0,1 Logical Foundations of Artificial Intelligence Chapter 2: 59 169。 School of CIT, Beijing JiaoTong University General behavior Logical Foundations of Artificial Intelligence Chapter 2: 60 169。 School of CIT, Beijing JiaoTong University Expansion of conception ? The above sentences only describe the digital structure and behavior of the circuit. ? To express a gate is malfunctioning – Adding additional relations ? To express a connection is malfunctioning – Describe connections as objects (12 new objects) – Extending the binary connectivity relation into a ternary relation Port1,Port2,Conn. Logical Foundations of Artificial Intelligence Chapter 2: 61 169。 School of CIT, Beijing JiaoTong University 167。 Natural Language Example ? The universe of discourse is the set of all plants. Logical Foundations of Artificial Intelligence Chapter 2: 62 169。 School of CIT, Beijing JiaoTong University Negation and quantity ? Representation of negation ? Representation of Quantity Logical Foundations of Artificial Intelligence Chapter 2: 63 169。 School of CIT, Beijing JiaoTong University 167。 Specialized Languages ? Predicate Calculus cannot represent natural language properly. ? Specialized Languages – Binary table – Semantic – Frame Logical Foundations of Artificial Intelligence Chapter 2: 64 169。 School of CIT, Beijing JiaoTong University Binary Table ? Excellent for expressing information about binary functions。 does not work for other type of information. ?I(?iI,?jI)=?ijI ? Example: Logical Foundations of Artificial Intelligence Chapter 2: 65 169。 School of CIT, Beijing JiaoTong University 蘊(yùn)含式的真值表 P Q P?Q T T T T F F F T T F F T ?公式 P?Q ?只有當(dāng) P為真且 Q為假時不滿足 ?即： P為假時Q為任何公式都能從 P推出 Q Logical Foundations of Artificial Intelligence Chapter 2: 66 169。 School of CIT, Beijing JiaoTong University 羅素與教皇 ? 據(jù)說大邏輯學(xué)家羅素告訴一位哲學(xué)家假命題蘊(yùn)涵任何命題后，那位哲學(xué)家頗為震驚，他說：“尊意莫非由 2加 2等于 5能推出你是教皇？”羅素答曰：“正是?！闭軐W(xué)家問：“你能證明這一點(diǎn)么？”羅素答：“當(dāng)然能?！? Logical Foundations of Artificial Intelligence Chapter 2: 67 169。 School of CIT, Beijing JiaoTong University 羅素與教皇 ? （ 1）假定 2＋ 2＝ 5； ? （ 2）由等式兩側(cè)減去 2，得出 2＝ 3； ? （ 3）易位后得出 3＝ 2； ? （ 4）由兩側(cè)減去 1，得出 2＝ 1. ? 請看：教皇與我是二人。既然 2等于 1，教皇與我是一人。因此我是教皇。 Logical Foundations of Artificial Intelligence Chapter 2: 68 169。 School of CIT, Beijing JiaoTong University 作業(yè) ? P42P43： 10 Logical Foundations of Artificial Intelligence Chapter 2: 69 169。 School of CIT, Beijing JiaoTong University 演講完畢，謝謝觀看！