Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction Computational Logic Lecture 1

Published byAmbrose Jennings Modified over 6 years ago

Similar presentations

Presentation on theme: "Introduction Computational Logic Lecture 1"— Presentation transcript:

1 Introduction Computational Logic Lecture 1
Michael Genesereth Autumn 2011

2 Computational Logic q(b,c) p(a,b) x.y.(p(x,y)  q(x,y)) x.p(x,d)
p(b,d) p(c,b) p(c,d) Computational Logic is an advanced topic in Computer Science concerned with the processing of information encoded in the form of logical statements. For example, statements like “If A is true and B is true, then either C is true or D is true”. Aristotle, leibnitz, babbage, 20th century

3 Mathematics Group Axioms  Theorem Tasks: Proof Checking
Proof Generation

4 Some Successes Various Theorems Other
4 color theorem (Appel and Haken) the limit of a sum is the sum of the limits the Bolzano-Weierstrass Theorem the Fundamental Theorem of calculus Euler's identity Gauss' law of quadratic reciprocity the undecidability of the halting problem Godel's incompleteness theorem (Shankar) Other Thousands of Problems for Theorem Provers (TPTP) CADE ATP Systems Competition (CASC)

5 Software Engineering Program Partial Specification: Tasks:
Program Verification Proof of Termination Complexity Analysis Partial Evaluation L sorter M

6 Hardware Engineering Circuit: Premises: Conclusion: x  y  c
z s c o a b Applications: Simulation Configuration Diagnosis Test Generation

7 Constraint Satisfaction Systems

8 Deductive Database Systems
Database Tables Queries query(X,Z) :- parent(X,Y) & parent(Y,Z) Constraints illegal :- parent(X,X) illegal :- parent(X,Y) & parent(Y,X)

9 Data Integration Infomaster Integrated Search Side-by-side Comparison
General Data Supplier 1 Supplier 2 Satisfaction Ratings An "information broker" is a system that provides integrated access to distributed, heterogeneous information sources, thus giving its users the illusion of a centralized, homogeneous information system. Supplier 3 Manufacturer 1 Supplier 4 Manufacturer 2 Product analysis

10 Logical Spreadsheets Computerized spreadsheets are a great success. They are often touted in newspapers and magazine articles as the first "killer app" for personal computers. Over the years, they have proven their worth time and again. Today, they are used for managing enterprises of all sorts - from one-person projects to multi-institutional conglomerates. The power of computerized spreadsheets derives in large part from two primary features - the automatic calculation of the values on spreadsheets and the use of mathematical formulas in computing those values. The automatic calculation of values helps the user check his work on data entry and editing. The use of mathematical formulas simplifies the task of setting up those calculations and makes spreadsheet technology accessible to a broad class of users, including those with no background in programming.

11 Examples of Non-Functional Constraints
Scheduling Start times must be before end times Room 104 may not be scheduled after 5:00 pm Only senior managers can reserve the third floor conference room Travel Reservations The number of lap infants in a group on a flight must not exceed the number of adults. Academic Programs Students must take at least 2 math courses

12 Regulations and Business Rules
Using the language of logic, it is possible to define new relations. Office mates are people who share an office. officemate(X,Y) :- office(X,Z)  office(Y,Z) This includes the property of legality / illegality. Managers and subordinates may not be office mates. illegal :- manages(X,Y)  officemate(X,Y)

13 Michigan Lease Termination Clause
The University may terminate this lease when the Lessee, having made application and executed this lease in advance of enrollment, is not eligible to enroll or fails to enroll in the University or leaves the University at any time prior to the expiration of this lease, or for violation of any provisions of this lease, or for violation of any University regulation relative to resident Halls, or for health reasons, by providing the student with written notice of this termination 30 days prior to the effective data of termination; unless life, limb, or property would be jeopardized, the Lessee engages in the sales of purchase of controlled substances in violation of federal, state or local law, or the Lessee is no longer enrolled as a student, or the Lessee engages in the use or possession of firearms, explosives, inflammable liquids, fireworks, or other dangerous weapons within the building, or turns in a false alarm, in which cases a maximum of 24 hours notice would be sufficient.

14 Logical Version A  A1  A2 B A  A4 B A  A5 B A  A6 B
B  B1 B  B2 B  B3 B  B4 B  B5 Computer Science and Law are philosophically related.

15 Elements

16 Formal Mathematics Algebra 1. Formal language for encoding information
2. Legal transformations 3. Automation Logic

17 Algebra Problem Xavier is three times as old as Yolanda. Xavier's age and Yolanda's age add up to twelve. How old are Xavier and Yolanda?

18 Algebra Solution Xavier is three times as old as Yolanda. Xavier's age and Yolanda's age add up to twelve. How old are Xavier and Yolanda? Automation: Saint, Sin, Reduce, Macsyma, Mathematica

19 Logic Problem If Mary loves Pat, then Mary loves Quincy. If it is Monday, then Mary loves Pat or Quincy. If it is Monday, does Mary love Quincy? If it is Monday, does Mary love Pat? Mary loves only one person at a time. If it is Monday, does Mary love Pat?

20 Formalization Simple Sentences: Mary loves Pat. Mary loves Quincy.
It is Monday. Premises: If Mary loves pat, Mary loves Quincy. If it Monday, Mary loves Pat or Quincy. Mary loves one person at a time. Questions: Does Mary love Pat? Does Mary love Qunicy?

21 Rule of Inference Propositional Resolution
NB: If pi is the same as sj, it is okay to drop the two symbols, with the proviso that only one such pair may be dropped. NB: If a constant is repeated on the left or the right, all but one of the occurrences can be deleted.

22 Examples

23 Logic Problem Revisited
If Mary loves Pat, then Mary loves Quincy. If it is Monday, then Mary loves Pat or Quincy. If it is Monday, does Mary love Quincy?

24 Logic Problem Concluded
Mary loves only one person at a time. If it is Monday, does Mary love Pat?

25 Automated Reasoning

26 Logic Technology Languages Knowledge Interchange Format (KIF) - ANSI
Common Logic - W3C Some Popular Automated Reasoning Systems Otter / Snark / Vampire / … PTTP / Epilog Knowledge Bases Definitions (Bachelor is an unmarried adult male.) Physical Laws (e.g. PV=nRT) Laws (e.g necessary if earnings > $10,000.)

27 Study Guide

28 Multiple Logics Propositional Logic
If it is raining, the ground is wet. Relational Logic If x is a parent of y, then y is a child of x. Modal Logic John believes that all men are mortal.

29 Common Topics Common Topics Syntax - expressions in the language
Semantics - meaning of expressions Computational Matters Contrasts Expressiveness - operators, variables, terms, ... Computational Hierarchy - polynomial? decidable? Tradeoffs - expressiveness versus computability

30 Meta We will frequently write sentences about sentences.
Sentence: When it rains, it pours. Metasentence: That sentence contains two verbs. We will frequently prove things about proofs. Proofs: formal Metaproofs: informal

31 Mike took it twice!

32

Download ppt "Introduction Computational Logic Lecture 1"

Similar presentations

1 Knowledge Representation Introduction KR and Logic.

1 Knowledge Representation Introduction KR and Logic.
Brief Introduction to Logic. Outline Historical View Propositional Logic : Syntax Propositional Logic : Semantics Satisfiability Natural Deduction : Proofs.

Brief Introduction to Logic. Outline Historical View Propositional Logic : Syntax Propositional Logic : Semantics Satisfiability Natural Deduction : Proofs.
Logic: The Big Picture Propositional logic: atomic statements are facts –Inference via resolution is sound and complete (though likely computationally.

Logic: The Big Picture Propositional logic: atomic statements are facts –Inference via resolution is sound and complete (though likely computationally.
Methods of Proof Chapter 7, second half.. Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules: Legitimate (sound)

Methods of Proof Chapter 7, second half.. Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules: Legitimate (sound)
Methods of Proof Chapter 7, Part II. Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules: Legitimate (sound) generation.

Methods of Proof Chapter 7, Part II. Proof methods Proof methods divide into (roughly) two kinds: Application of inference rules: Legitimate (sound) generation.
Logic Concepts Lecture Module 11.

Logic Concepts Lecture Module 11.
Knowledge Representation Methods

Knowledge Representation Methods
Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language.

Logic in general Logics are formal languages for representing information such that conclusions can be drawn Syntax defines the sentences in the language.
CS1001 Lecture 22. Overview Mechanizing Reasoning Mechanizing Reasoning G ö del ’ s Incompleteness Theorem G ö del ’ s Incompleteness Theorem.

CS1001 Lecture 22. Overview Mechanizing Reasoning Mechanizing Reasoning G ö del ’ s Incompleteness Theorem G ö del ’ s Incompleteness Theorem.
Brief Introduction to Logic. Outline Historical View Propositional Logic : Syntax Propositional Logic : Semantics Satisfiability Natural Deduction : Proofs.

Brief Introduction to Logic. Outline Historical View Propositional Logic : Syntax Propositional Logic : Semantics Satisfiability Natural Deduction : Proofs.
Methods of Proof Chapter 7, second half.

Methods of Proof Chapter 7, second half.
Proof by Deduction. Deductions and Formal Proofs A deduction is a sequence of logic statements, each of which is known or assumed to be true A formal.

Proof by Deduction. Deductions and Formal Proofs A deduction is a sequence of logic statements, each of which is known or assumed to be true A formal.
Lecture 24: Gödel’s Proof CS150: Computer Science

Lecture 24: Gödel’s Proof CS150: Computer Science
Propositional Logic Reasoning correctly computationally Chapter 7 or 8.

Propositional Logic Reasoning correctly computationally Chapter 7 or 8.
Class 36: Proofs about Unprovability David Evans University of Virginia cs1120.

Class 36: Proofs about Unprovability David Evans University of Virginia cs1120.
Computational Logic both as a descriptive theory of human thinking and as a normative theory of how humans can think and communicate more effectively The.

Computational Logic both as a descriptive theory of human thinking and as a normative theory of how humans can think and communicate more effectively The.
Inference is a process of building a proof of a sentence, or put it differently inference is an implementation of the entailment relation between sentences.

Inference is a process of building a proof of a sentence, or put it differently inference is an implementation of the entailment relation between sentences.
Propositional Resolution Computational LogicLecture 4 Michael Genesereth Spring 2005.

Propositional Resolution Computational LogicLecture 4 Michael Genesereth Spring 2005.
David Evans CS200: Computer Science University of Virginia Computer Science Class 24: Gödel’s Theorem.

David Evans CS200: Computer Science University of Virginia Computer Science Class 24: Gödel’s Theorem.
1 Introduction to Logic Programming. 2 Human Logic Humans are information processors, We acquire information about the world and use this information.

1 Introduction to Logic Programming. 2 Human Logic Humans are information processors, We acquire information about the world and use this information.

Similar presentations

Ads by Google