cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 13
Chapter 13
Predicate Logic Proof System QRS
Part 1:
Predicate Languages
Part 2:
Proof System QRS
Chapter 13
Part 1: Predicate Languages
Predicate Languages
Predicate Langu
Chapter 11 (Part 2) Gentzen Sequent Calculus GL
The proof system GL for the classical propositional logic is a version of the original Gentzen (1934) systems LK.
A constructive proof of the completeness theorem for the system GL is very similar to the pro
CHAPTER 11: Automated Proof Systems (3) RS: Counter Models Generated by Decomposition Trees RS: Proof of COMPLETENESS THEOREM
1
Counter-model generated by the decomposition tree.
Example: Given a formula A:
(a b) c) (a c) and its decomposition tree TA.
(a
Chapter 12: Gentzen Sequent Calculus for Intuitionistic Logic
Part 1: LI System
The proof system LI was published by Gentzen in 1935 as a particular case of his proof system LK for the classical logic.
We discussed a version of the original Gentzens syste
Chapter 12: Gentzen Sequent Calculus for Intuitionistic Logic
PART 2: Examples of proof search decomposition trees in LI
Search for proofs in LI is a much more complicated process then the one in classical logic systems RS or GL.
Proof search procedure co
Gentzen Sequent Calculus for Intuitionistic Logic
PART 3: Proof Search Heuristics Before we dene a heuristic method of searching for proof in LI lets make some observations. Observation 1 : the logical rules of LI are similar to those in Gentzen type clas
CHAPTER 4 CLASSICAL PROPOSITIONAL SEMANTICS
1
Language
There are several propositional languages that are routinely called classical propositional logic languages. It is due to the functional dependency of classical connectives discussed briey in chapter
CHAPTER 4 SOME EXTENSIONAL SEMANTICS
Many valued logics in general and 3-valued logics in particular is an old object of study which had its beginning in the work of Lukasiewicz (1920). He was the rst to dene a 3- valued semantics for a language L, of cla
CHAPTER 6 CLASSICAL TAUTOLOGIES AND LOGICAL EQUIVALENCES
We present and discuss here a set of most widely use classical tautologies and logical equivalences. We also discuss the denability of classical connectives and as a consequence, the equivalence cla
Chapter 11: Automated Proof Systems (2)
RS: DECOMPOSITION TREES
The process of searching for the proof of a formula A in RS consists of building a certain tree, called a decomposition tree whose root is the formula A, nodes correspond to sequences which a
Chapter 11: Automated Proof Systems (1)
SYSTEM RS OVERVIEW
Hilbert style systems are easy to dene and admit a simple proof of the Completeness Theorem but they are dicult to use.
Automated systems are less intuitive then the Hilbert-style systems, but the
Chapter 8: Hilbert Systems, Deduction Theorem
Introduction
Hilbert Systems The Hilbert proof systems are based on a language with implication and contain a Modus Ponens rule as a rule of inference.
Modus Ponens is the oldest of all known rules of inferenc
CHAPTER 8
System H2 and Formal Proofs
Hilbert System H2 The system H1 is sound and strong enough to prove the Deduction Theorem, but it is not complete.
We extend now its set of logical axioms to a complete set of axioms, i.e. we dene a system H2 that is
Chapter 9: Completeness Theorem: Proof 1
We consider a sound proof system (under classical semantics)
S = ( Lcfw_,,
AL,
M P ),
such that the formulas listed below are provable in S .
S S
(A (B A),
(A (B C ) (A B ) (A C ),
S S S
(A (A B ), (A A) A),
(B A)
Chapter 9 Completeness Theorem (Part 1) Proof 1 and Examples
We consider a sound proof system (under classical semantics)
S = ( Lcfw_,,
AL,
M P ),
such that the formulas listed below are provable in S .
1. (A (B A),
2. (A (B C ) (A B ) (A C ),
3. (B A) (B
Chapter 9 Completeness Theorem: Proof 2 A Counter- Model Existence Method
We prove now the Completeness Theorem by proving the opposite implication:
If
A, then |= A
We will show now how one can dene of a counter-model for A from the fact that A is not pro
Chapter 10: Introduction to Intuitionistic Logic
PART 1: INTRODUCTION
The intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism.
Intuitionism was originated by L. E. J. Brouwe
Chapter 10: Introduction to Intuitionistic Logic
PART 2: Hilbert Proof System for propositional intuitionistic logic. Language is a propositional language L = Lcfw_, with the set of formulas denoted by F . Axioms A1 A2 A3 (A B ) (B C ) (A C ), (A (A B ),
CHAPTER 7 GENERAL PROOF SYSTEMS
1
Introduction
Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes theorems being its nal products. The starting points
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem
The Hilbert proof systems are systems based on a language with implication and contain a Modus Ponens rule as a rule of inference. They are usually called Hilbert style formalizations. We w
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 8
Chapter 8
HILBERT PROOF SYSTEMS for CLASSICAL PROPOSITIONAL
LOGIC
PART 1: Hilbert Proof Systems
PART 2: Formal Proofs
PART 3: Deduction Theorem
Hilbert Proof Systems
Hilber
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 9
Chapter 9
TWO PROOFS OF COMPLETENESS THEOREM
PART 1: Introduction
PART 2: System S Definition and Proof of the Main Lemma
PART 3: Proof 1: Constructive Proof of Completenes
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 3
Chapter 3
Propositional Languages
PART 1: Propositional Languages: Intuitive Introduction
PART 2: Propositional Languages: Formal Definitions
PART 1: Propositional Language
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 5a
Short REVIEW Chapters1 -5
DEFINITIONS: Chapter 3 and Chapter 4
Here I repeat for you some basic DEFINITIONS from
Chapters 3, 4 and 5
You have to prepare them for MIDTERM 1
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 4
Chapter 4
Classical Propositional Semantics
Semantics- General Principles
Given a propositional language L = LCON
Symbols for connectives of L always have some intuitive
me
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 10
Chapter 10
CLASSICAL AUTOMATED PROOF SYSTEMS
PART 1: RS SYSTEM
PART 2: RS1, RS2, RS3 SYSTEMS
PART 3: GENTZEN SYSTEMS
CLASSICAL AUTOMATED PROOF SYSTEMS
Hilbert style system
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 7
Chapter 7
GENERAL PROOF SYSTEMS
PART 1: Introduction- Intuitive definitions
PART 2: Formal Definition of a Proof System
PART 3: Formal Proofs and Simple Examples
PART 4: Co
cse541
LOGIC FOR COMPUTER SCIENCE
Professor Anita Wasilewska
Spring 2015
LECTURE 12
Chapter 12
Gentzen Sequent Calculus LI for Intuitionistic Logic
Original Gentzen System LI for Intuitionistic Logic
Part 1
Definition of Gentzen System LI
The proof system