Type Preservation and Normalization: Two Consequences
|
|
|
- Ginger Hensley
- 6 years ago
- Views:
Transcription
1 Type Preservation and Normalization: Two Consequences Joshua D. Guttman Worcester Polytechnic Institute 14 Sep 2010 Guttman ( WPI ) Normalization 14 Sep 10 1 / 32
2 Today s Goal Theorem Every normal derivation has the subformula property Guttman ( WPI ) Normalization 14 Sep 10 2 / 32
3 Subformula Property A derivation d proving Γ s : ϕ has the subformula property iff, for every t : ψ appearing in d, either ψ is a subformula of ϕ, or ψ is a subformula of χ, where some v : χ Γ Guttman ( WPI ) Normalization 14 Sep 10 3 / 32
4 Subformula Property A derivation d proving Γ s : ϕ has the subformula property iff, for every t : ψ appearing in d, either ψ is a subformula of ϕ, or ψ is a subformula of χ, where some v : χ Γ Subformula is transitive, and: is a subformula of every ϕ ϕ is a subformula of ϕ ϕ, ψ are subformulas of: ϕ ψ, ϕ ψ, ϕ ψ Guttman ( WPI ) Normalization 14 Sep 10 3 / 32
5 Today s Goal Theorem Every normal derivation has the subformula property Corollary The derivation rules are consistent: there is no derivation s : Guttman ( WPI ) Normalization 14 Sep 10 4 / 32
6 Proving Consistency From the Subformula Property Corollary The derivation rules are consistent: there is no derivation s :. Proof. s : is not an instance of an axiom. Guttman ( WPI ) Normalization 14 Sep 10 5 / 32
7 Proving Consistency From the Subformula Property Corollary The derivation rules are consistent: there is no derivation s :. Proof. s : is not an instance of an axiom. It may be derived by the rule, if s = emp(t), but then there is no progress, as the premise is again of the form t :. Guttman ( WPI ) Normalization 14 Sep 10 5 / 32
8 Proving Consistency From the Subformula Property Corollary The derivation rules are consistent: there is no derivation s :. Proof. s : is not an instance of an axiom. It may be derived by the rule, if s = emp(t), but then there is no progress, as the premise is again of the form t :. Every other rule requires using a formula that is not a subformula of. Guttman ( WPI ) Normalization 14 Sep 10 5 / 32
9 Two Computational Theorems Type Preservation and Normalization Theorem (Type Preservation) If s t and Γ s : ϕ, then also Γ t : ϕ. Theorem (Normal Form) If Γ s : ϕ, then there is a normal form t such that s t. Guttman ( WPI ) Normalization 14 Sep 10 6 / 32
10 Local reduction rules fst( s, s ) r s scd( s, s ) r s cases( lft, s, t, r) r t s cases( rgt, s, t, r) r r s (λv. s) t r s[t/v] (β) Guttman ( WPI ) Normalization 14 Sep 10 7 / 32
11 Reducing Intro/Elim Pair:. s. Γ s : ϕ. t. Γ t : ψ Γ s, t : ϕ ψ Γ fst( s, t ): ϕ. s. Γ s : ϕ Guttman ( WPI ) Normalization 14 Sep 10 8 / 32
12 Reducing Intro/Elim Pair:. s. Γ, x : ϕ s : ψ Γ λx. s : ϕ ψ Γ (λx. s) t : ψ. t. Γ t : ϕ. s[t/x]. Γ s[t/x]: ψ Guttman ( WPI ) Normalization 14 Sep 10 9 / 32
13 Compile-time reduction rules fst(cases(s, t, r)) r cases(s, fst t, fst r) scd(cases(s, t, r)) r cases(s, scd t, scd r) cases(cases(s, t, r), u, w) r cases(s, λv. cases(t(v), u, w), λv. cases(r(v), u, w)) (u (cases(s, t, r))) r cases(s, u t, u r) Guttman ( WPI ) Normalization 14 Sep / 32
14 Reducing a cases/elim pair: Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ fst(cases(s, λx. t, λy. r)): χ 1 Guttman ( WPI ) Normalization 14 Sep / 32
15 Reducing a cases/elim pair: Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ fst(cases(s, λx. t, λy. r)): χ 1 Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, x : ϕ fst(t): χ 1 Γ, y : ψ r : χ 1 χ 2 Γ, y : ϕ fst(r): χ 1 Γ cases(s, λx. fst(t), λy. fst(r)): χ 1 Guttman ( WPI ) Normalization 14 Sep / 32
16 Reducing a cases/elim pair: Γ, z : χ 1 u : ρ Γ, x : ϕ t : χ 1 χ 2 Γ s : ϕ ψ Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ, v : χ 2 w : ρ Γ f : ρ Guttman ( WPI ) Normalization 14 Sep / 32
17 Reducing a cases/elim pair: Omitted: Analogous deriv. of Γ, y : ψ g 2 : ρ Γ, z : χ 1 u : ρ Γ, x : ϕ t : χ 1 χ 2 Γ s : ϕ ψ Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ, v : χ 2 w : ρ Γ f : ρ Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, x : ϕ, v : χ 2 w : ρ Γ, x : ϕ, z : χ 1 u : ρ Γ, x : ϕ g 1 : ρ Γ f : ρ Guttman ( WPI ) Normalization 14 Sep / 32
18 Reducing a cases/elim pair: Omitted: Analogous deriv. of Γ, y : ψ g 2 : ρ Γ, z : χ 1 u : ρ Γ, x : ϕ t : χ 1 χ 2 Γ s : ϕ ψ Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ, v : χ 2 w : ρ Γ f : ρ Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, x : ϕ, v : χ 2 w : ρ Γ, x : ϕ, z : χ 1 u : ρ Γ, x : ϕ g 1 : ρ Γ f : ρ f = cases(cases(s, λx. t, λy. r), λz. u, λv. w) g 1 = cases(t, λz. u, λv. w) g 2 = cases(r, λz. u, λv. w) f = cases(u, g 1, g 2 ) Guttman ( WPI ) Normalization 14 Sep / 32
19 The Reduction Relation s r t s t s t C[s] C[t] s s s t t u s u Guttman ( WPI ) Normalization 14 Sep / 32
20 Contexts C[x] Replace any s, t, u with an x to make a context C[x]: C[x] ::= x C [x], t s, C [x] fst(c [x]) scd(c [x]) (λv. C [x]) (C [x] t) (s C [x]) lft, C [x] rgt, C [x] cases(c [x], t, r) cases(s, C [x], r) cases(s, t, C [x]) Guttman ( WPI ) Normalization 14 Sep / 32
21 Two Computational Theorems Type Preservation and Normalization Theorem (Type Preservation) If s t and Γ s : ϕ, then also Γ t : ϕ. Theorem (Normal Form) If Γ s : ϕ, then there is a normal form t such that s t. Guttman ( WPI ) Normalization 14 Sep / 32
22 A Corollary: Normal Derivations Corollary 1 If ϕ is derivable from Γ, then there is a normal derivation t such that Γ t : ϕ 2 If additionally Γ =, then t is closed (i.e. no free variables) Guttman ( WPI ) Normalization 14 Sep / 32
23 A Corollary: Normal Derivations Corollary 1 If ϕ is derivable from Γ, then there is a normal derivation t such that Γ t : ϕ 2 If additionally Γ =, then t is closed (i.e. no free variables) Proof. 1. If ϕ is derivable from Γ, then for some s, Γ s : ϕ. By normal form, s t for some normal t. By type preservation, Γ t : ϕ. Guttman ( WPI ) Normalization 14 Sep / 32
24 A Corollary: Normal Derivations Corollary 1 If ϕ is derivable from Γ, then there is a normal derivation t such that Γ t : ϕ 2 If additionally Γ =, then t is closed (i.e. no free variables) Proof. 1. If ϕ is derivable from Γ, then for some s, Γ s : ϕ. By normal form, s t for some normal t. By type preservation, Γ t : ϕ. 2. Via the Context Lemma, which says: If Γ s : ϕ, then fv(s) dom(γ). Guttman ( WPI ) Normalization 14 Sep / 32
25 A Normal Proof p, (p ) q (p ) q p, (p ) q p p, (p ) q p p, (p ) q p, (p ) q q (p ) q p q ((p ) q) (p q) λx. λy. emp(fst(x) y) Guttman ( WPI ) Normalization 14 Sep / 32
26 Another Normal Proof p, (p q) r p p, (p q) r (p q) r p, (p q) r p q p, (p q) r r (p q) r p r ((p q) r) (p r) λy. λx. (y fst, x ) Guttman ( WPI ) Normalization 14 Sep / 32
27 Normal Derivations, 1 Regarding s as a tree with the conclusion at the root If d is a normal derivation, then working upward from any point through major premises, every application of an introduction rule is reached before any application of an elimination rule Guttman ( WPI ) Normalization 14 Sep / 32
28 Normal Derivations, 1 Regarding s as a tree with the conclusion at the root If d is a normal derivation, then working upward from any point through major premises, every application of an introduction rule is reached before any application of an elimination rule All premises in an introduction rule are major The major premise of an elimination rule is the premise containing the connective to be eliminated Guttman ( WPI ) Normalization 14 Sep / 32
29 Major and Minor Premises: Conjunction Γ s : ϕ Γ t : ψ Γ s, t : ϕ ψ Γ s : ϕ ψ Γ fst(s): ϕ Γ s : ϕ ψ Γ scd(s): ψ Guttman ( WPI ) Normalization 14 Sep / 32
30 Major and Minor Premises: Implication Γ, x : ϕ s : ψ Γ λx. s : ϕ ψ Γ s : ϕ ψ Γ t : ϕ Γ (s t): ψ Guttman ( WPI ) Normalization 14 Sep / 32
31 Major and Minor Premises: Disjunction Γ s : ϕ Γ lft, s : ϕ ψ Γ s : ψ Γ rgt, s : ϕ ψ Γ s : ϕ ψ Γ, x : ϕ t : χ Γ, y : ψ r : χ Γ cases(s, λx. t, λy. r): χ Guttman ( WPI ) Normalization 14 Sep / 32
32 Major and Minor Premises: Axiom and Falsehood Γ, x : ϕ x : ϕ Γ x : Γ emp(x): ϕ Guttman ( WPI ) Normalization 14 Sep / 32
33 Normal Derivations, 1 Regarding s as a tree with the conclusion at the root If d is a normal derivation, then working upward from any point through major premises, every application of an introduction rule is reached before any application of an elimination rule All premises in an introduction rule are major The major premise of an elimination rule is the premise containing the connective to be eliminated Guttman ( WPI ) Normalization 14 Sep / 32
34 Normal Derivations, 2 If d is a normal derivation, and p is any upwards path in d if p traverses only elimination rules and p traverses a disjunction elimination inference then it is below any other elimination rule Guttman ( WPI ) Normalization 14 Sep / 32
35 Normal Derivations, 2 If d is a normal derivation, and p is any upwards path in d if p traverses only elimination rules and p traverses a disjunction elimination inference then it is below any other elimination rule By the compile-time rules Guttman ( WPI ) Normalization 14 Sep / 32
36 Reducing a cases/elim pair: Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ fst(cases(s, λx. t, λy. r)): χ 1 Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, x : ϕ fst(t): χ 1 Γ, y : ψ r : χ 1 χ 2 Γ, y : ϕ fst(r): χ 1 Γ cases(s, λx. fst(t), λy. fst(r)): χ 1 Guttman ( WPI ) Normalization 14 Sep / 32
37 Normal Derivations, 3 If d is a normal derivation, and p is any upwards path in d if p traverses only elimination rules then p traverses at most one major premise of a disjunction elimination inference Guttman ( WPI ) Normalization 14 Sep / 32
38 Normal Derivations, 3 If d is a normal derivation, and p is any upwards path in d if p traverses only elimination rules then p traverses at most one major premise of a disjunction elimination inference By the case/elim rule for and the previous claim Guttman ( WPI ) Normalization 14 Sep / 32
39 Reducing a cases/elim pair: Omitted: Analogous deriv. of Γ, y : ψ g 2 : ρ Γ, z : χ 1 u : ρ Γ, x : ϕ t : χ 1 χ 2 Γ s : ϕ ψ Γ, y : ψ r : χ 1 χ 2 Γ cases(s, λx. t, λy. r): χ 1 χ 2 Γ, v : χ 2 w : ρ Γ f : ρ Γ s : ϕ ψ Γ, x : ϕ t : χ 1 χ 2 Γ, x : ϕ, v : χ 2 w : ρ Γ, x : ϕ, z : χ 1 u : ρ Γ, x : ϕ g 1 : ρ Γ f : ρ f = cases(cases(s, λx. t, λy. r), λz. u, λv. w) g 1 = cases(t, λz. u, λv. w) g 2 = cases(r, λz. u, λv. w) f = cases(u, g 1, g 2 ) Guttman ( WPI ) Normalization 14 Sep / 32
40 Normal Derivations, 4 If d is a normal derivation, and p is any upwards path in d if p traverses only introduction rules then each successive right hand side is a subformula of the one below it By the form of the introduction rules Guttman ( WPI ) Normalization 14 Sep / 32
41 Today s Goal Theorem Every normal derivation has the subformula property Guttman ( WPI ) Normalization 14 Sep / 32
42 Why it s true Theorem Every normal derivation d of Γ s : ϕ has the subformula property Proof. 1 Conclusion of an introduction rule: subformula of ϕ Guttman ( WPI ) Normalization 14 Sep / 32
43 Why it s true Theorem Every normal derivation d of Γ s : ϕ has the subformula property Proof. 1 Conclusion of an introduction rule: subformula of ϕ 2 Major premise of an elimination rule: subformula of some ψ in Γ Guttman ( WPI ) Normalization 14 Sep / 32
44 Why it s true Theorem Every normal derivation d of Γ s : ϕ has the subformula property Proof. 1 Conclusion of an introduction rule: subformula of ϕ 2 Major premise of an elimination rule: subformula of some ψ in Γ 3 Minor premise of -elimination: subformula of the major premise Guttman ( WPI ) Normalization 14 Sep / 32
45 Why it s true Theorem Every normal derivation d of Γ s : ϕ has the subformula property Proof. 1 Conclusion of an introduction rule: subformula of ϕ 2 Major premise of an elimination rule: subformula of some ψ in Γ 3 Minor premise of -elimination: subformula of the major premise 4 Minor premise of -elimination: subformula of ϕ Guttman ( WPI ) Normalization 14 Sep / 32
46 Today s Goal Theorem Every normal derivation has the subformula property Corollary The derivation rules are consistent: there is no derivation s : Guttman ( WPI ) Normalization 14 Sep / 32
Consequence Relations and Natural Deduction
Consequence Relations and Natural Deduction Joshua D Guttman Worcester Polytechnic Institute September 16, 2010 Contents 1 Consequence Relations 1 2 A Derivation System for Natural Deduction 3 3 Derivations
Consequence Relations and Natural Deduction
Consequence Relations and Natural Deduction Joshua D. Guttman Worcester Polytechnic Institute September 9, 2010 Contents 1 Consequence Relations 1 2 A Derivation System for Natural Deduction 3 3 Derivations
Büchi Automata and Linear Temporal Logic
Büchi Automata and Linear Temporal Logic Joshua D. Guttman Worcester Polytechnic Institute 18 February 2010 Guttman ( WPI ) Büchi & LTL 18 Feb 10 1 / 10 Büchi Automata Definition A Büchi automaton is a
Proof Tactics, Strategies and Derived Rules. CS 270 Math Foundations of CS Jeremy Johnson
Proof Tactics, Strategies and Derived Rules CS 270 Math Foundations of CS Jeremy Johnson Outline 1. Review Rules 2. Positive subformulas and extraction 3. Proof tactics Extraction, Conversion, Inversion,
Introduction to Intuitionistic Logic
Introduction to Intuitionistic Logic August 31, 2016 We deal exclusively with propositional intuitionistic logic. The language is defined as follows. φ := p φ ψ φ ψ φ ψ φ := φ and φ ψ := (φ ψ) (ψ φ). A
Propositional Logic: Part II - Syntax & Proofs 0-0
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline Syntax of Propositional Formulas Motivating Proofs Syntactic Entailment and Proofs Proof Rules for Natural Deduction Axioms, theories and theorems
Natural Deduction for Propositional Logic
Natural Deduction for Propositional Logic Bow-Yaw Wang Institute of Information Science Academia Sinica, Taiwan September 10, 2018 Bow-Yaw Wang (Academia Sinica) Natural Deduction for Propositional Logic
Natural Deduction. Formal Methods in Verification of Computer Systems Jeremy Johnson
Natural Deduction Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. An example 1. Validity by truth table 2. Validity by proof 2. What s a proof 1. Proof checker 3. Rules of
Propositional Logic: Deductive Proof & Natural Deduction Part 1
Propositional Logic: Deductive Proof & Natural Deduction Part 1 CS402, Spring 2016 Shin Yoo Deductive Proof In propositional logic, a valid formula is a tautology. So far, we could show the validity of
02 Propositional Logic
SE 2F03 Fall 2005 02 Propositional Logic Instructor: W. M. Farmer Revised: 25 September 2005 1 What is Propositional Logic? Propositional logic is the study of the truth or falsehood of propositions or
Chapter 1, Logic and Proofs (3) 1.6. Rules of Inference
CSI 2350, Discrete Structures Chapter 1, Logic and Proofs (3) Young-Rae Cho Associate Professor Department of Computer Science Baylor University 1.6. Rules of Inference Basic Terminology Axiom: a statement
CS Lecture 19: Logic To Truth through Proof. Prof. Clarkson Fall Today s music: Theme from Sherlock
CS 3110 Lecture 19: Logic To Truth through Proof Prof. Clarkson Fall 2014 Today s music: Theme from Sherlock Review Current topic: How to reason about correctness of code Last week: informal arguments
Kleene realizability and negative translations
Q E I U G I C Kleene realizability and negative translations Alexandre Miquel O P. D E. L Ō A U D E L A R April 21th, IMERL Plan 1 Kleene realizability 2 Gödel-Gentzen negative translation 3 Lafont-Reus-Streicher
Appendix A: Separation theorems in IR n
Appendix A: Separation theorems in IR n These notes provide a number of separation theorems for convex sets in IR n. We start with a basic result, give a proof with the help on an auxiliary result and
Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016
![Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016](/thumbs/90/101773263.jpg "Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016")
Rules and derivations in an elementary logic course arxiv:160101483v1 [cslo] 7 Jan 2016 Gilles Dowek Inria, 23 avenue d Italie, CS 81321, 75214 Paris Cedex 13, France gillesdowek@inriafr When teaching
15414/614 Optional Lecture 1: Propositional Logic
15414/614 Optional Lecture 1: Propositional Logic Qinsi Wang Logic is the study of information encoded in the form of logical sentences. We use the language of Logic to state observations, to define concepts,
Lecture Notes on Combinatory Modal Logic
Lecture Notes on Combinatory Modal Logic 15-816: Modal Logic Frank Pfenning Lecture 9 February 16, 2010 1 Introduction The connection between proofs and program so far has been through a proof term assignment
Mathematics for linguists
Mathematics for linguists WS 2009/2010 University of Tübingen January 7, 2010 Gerhard Jäger Mathematics for linguists p. 1 Inferences and truth trees Inferences (with a finite set of premises; from now
Propositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
Propositional Calculus - Deductive Systems
Propositional Calculus - Deductive Systems Moonzoo Kim CS Division of EECS Dept. KAIST moonzoo@cs.kaist.ac.kr http://pswlab.kaist.ac.kr/courses/cs402-07 1 Deductive proofs (1/3) Suppose we want to know
Propositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
Propositional Calculus - Soundness & Completeness of H
Propositional Calculus - Soundness & Completeness of H Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á `
Fundamentals of Logic
Fundamentals of Logic No.5 Soundness and Completeness Tatsuya Hagino Faculty of Environment and Information Studies Keio University 2015/5/18 Tatsuya Hagino (Faculty of Environment and InformationFundamentals
Propositional and Predicate Logic - V
Propositional and Predicate Logic - V Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - V WS 2016/2017 1 / 21 Formal proof systems Hilbert s calculus
Equivalences. Proposition 2.8: The following equivalences are valid for all formulas φ, ψ, χ: (φ φ) φ. Idempotency Idempotency Commutativity
Substitution Theorem Proposition 2.7: Let φ 1 and φ 2 be equivalent formulas, and ψ[φ 1 ] p be a formula in which φ 1 occurs as a subformula at position p. Then ψ[φ 1 ] p is equivalent to ψ[φ 2 ] p. 84
The Importance of Being Formal. Martin Henz. February 5, Propositional Logic
The Importance of Being Formal Martin Henz February 5, 2014 Propositional Logic 1 Motivation In traditional logic, terms represent sets, and therefore, propositions are limited to stating facts on sets
Gentzen Sequent Calculus LK
October 8, 2007 Completeness of Gentzen LK Proof for completeness to the proof Start from the root Γ Given a node Φ Ψ, break both Φ and Ψ until they only contain propositional variables Can be done using
Chapter 11: Automated Proof Systems
Chapter 11: Automated Proof Systems SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems are
Proofs. Example of an axiom in this system: Given two distinct points, there is exactly one line that contains them.
Proofs A mathematical system consists of axioms, definitions and undefined terms. An axiom is assumed true. Definitions are used to create new concepts in terms of existing ones. Undefined terms are only
CHAPTER 10. Gentzen Style Proof Systems for Classical Logic
CHAPTER 10 Gentzen Style Proof Systems for Classical Logic Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. By humans, not mentioning
Propositional Logics and their Algebraic Equivalents
Propositional Logics and their Algebraic Equivalents Kyle Brooks April 18, 2012 Contents 1 Introduction 1 2 Formal Logic Systems 1 2.1 Consequence Relations......................... 2 3 Propositional Logic
Proofs. Joe Patten August 10, 2018
Proofs Joe Patten August 10, 2018 1 Statements and Open Sentences 1.1 Statements A statement is a declarative sentence or assertion that is either true or false. They are often labelled with a capital
KE/Tableaux. What is it for?
CS3UR: utomated Reasoning 2002 The term Tableaux refers to a family of deduction methods for different logics. We start by introducing one of them: non-free-variable KE for classical FOL What is it for?
Artificial Intelligence. Propositional logic
Artificial Intelligence Propositional logic Propositional Logic: Syntax Syntax of propositional logic defines allowable sentences Atomic sentences consists of a single proposition symbol Each symbol stands
CSE 20 DISCRETE MATH. Winter
CSE 20 DISCRETE MATH Winter 2017 http://cseweb.ucsd.edu/classes/wi17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
Lecture Notes on Sequent Calculus
Lecture Notes on Sequent Calculus 15-816: Modal Logic Frank Pfenning Lecture 8 February 9, 2010 1 Introduction In this lecture we present the sequent calculus and its theory. The sequent calculus was originally
An Introduction to Proof Theory
An Introduction to Proof Theory Class 1: Foundations Agata Ciabattoni and Shawn Standefer anu lss december 2016 anu Our Aim To introduce proof theory, with a focus on its applications in philosophy, linguistics
2. The Logic of Compound Statements Summary. Aaron Tan August 2017
2. The Logic of Compound Statements Summary Aaron Tan 21 25 August 2017 1 2. The Logic of Compound Statements 2.1 Logical Form and Logical Equivalence Statements; Compound Statements; Statement Form (Propositional
On Sequent Calculi for Intuitionistic Propositional Logic
On Sequent Calculi for Intuitionistic Propositional Logic Vítězslav Švejdar Jan 29, 2005 The original publication is available at CMUC. Abstract The well-known Dyckoff s 1992 calculus/procedure for intuitionistic
Mathematics 114L Spring 2018 D.A. Martin. Mathematical Logic
Mathematics 114L Spring 2018 D.A. Martin Mathematical Logic 1 First-Order Languages. Symbols. All first-order languages we consider will have the following symbols: (i) variables v 1, v 2, v 3,... ; (ii)
If-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables)
If-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables) Herman Geuvers Nijmegen, NL (Joint work with Tonny Hurkens) Types for Proofs
Propositional Calculus - Natural deduction Moonzoo Kim CS Dept. KAIST
Propositional Calculus - Natural deduction Moonzoo Kim CS Dept. KAIST moonzoo@cs.kaist.ac.kr 1 Review Goal of logic To check whether given a formula Á is valid To prove a given formula Á ` Á Syntactic
Chapter 11: Automated Proof Systems (1)
Chapter 11: Automated Proof Systems (1) SYSTEM RS OVERVIEW Hilbert style systems are easy to define and admit a simple proof of the Completeness Theorem but they are difficult to use. Automated systems
Propositional logic. Programming and Modal Logic
Propositional logic Programming and Modal Logic 2006-2007 4 Contents Syntax of propositional logic Semantics of propositional logic Semantic entailment Natural deduction proof system Soundness and completeness
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 17 Tuesday, April 2, 2013 1 There is a strong connection between types in programming languages and propositions
3.2 Reduction 29. Truth. The constructor just forms the unit element,. Since there is no destructor, there is no reduction rule.
32 Reduction 29 32 Reduction In the preceding section, we have introduced the assignment of proof terms to natural deductions If proofs are programs then we need to explain how proofs are to be executed,
The Many Faces of Modal Logic Day 3: Algebraic Semantics
The Many Faces of Modal Logic Day 3: Algebraic Semantics Dirk Pattinson Australian National University, Canberra (Slides based on a NASSLLI 2014 Tutorial and are joint work with Lutz Schröder) LAC 2018
A non-classical refinement of the interpolation property for classical propositional logic
Accepted for publication in Logique & Analyse A non-classical refinement of the interpolation property for classical propositional logic Peter Milne Abstract We refine the interpolation property of the
CS 486: Lecture 2, Thursday, Jan 22, 2009
CS 486: Lecture 2, Thursday, Jan 22, 2009 Mark Bickford January 22, 2009 1 Outline Propositional formulas Interpretations and Valuations Validity and Satisfiability Truth tables and Disjunctive Normal
Propositional Logic. CS 3234: Logic and Formal Systems. Martin Henz and Aquinas Hobor. August 26, Generated on Tuesday 31 August, 2010, 16:54
Propositional Logic CS 3234: Logic and Formal Systems Martin Henz and Aquinas Hobor August 26, 2010 Generated on Tuesday 31 August, 2010, 16:54 1 Motivation In traditional logic, terms represent sets,
Natural deduction for propositional logic via truth tables
Natural deduction for propositional logic via truth tables Herman Geuvers Nijmegen, NL (Joint work with Tonny Hurkens) Bengt Nordström honorary workshop Marstrand, Sweden April 2016 H. Geuvers - April
Learning Goals of CS245 Logic and Computation
Learning Goals of CS245 Logic and Computation Alice Gao April 27, 2018 Contents 1 Propositional Logic 2 2 Predicate Logic 4 3 Program Verification 6 4 Undecidability 7 1 1 Propositional Logic Introduction
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies
Logic as a Tool Chapter 1: Understanding Propositional Logic 1.1 Propositions and logical connectives. Truth tables and tautologies Valentin Stockholm University September 2016 Propositions Proposition:
Classical Propositional Logic
The Language of A Henkin-style Proof for Natural Deduction January 16, 2013 The Language of A Henkin-style Proof for Natural Deduction Logic Logic is the science of inference. Given a body of information,
Type Systems. Lecture 9: Classical Logic. Neel Krishnaswami University of Cambridge
Type Systems Lecture 9: Classical Logic Neel Krishnaswami University of Cambridge Where We Are We have seen the Curry Howard correspondence: Intuitionistic propositional logic Simply-typed lambda calculus
2.5.2 Basic CNF/DNF Transformation
2.5. NORMAL FORMS 39 On the other hand, checking the unsatisfiability of CNF formulas or the validity of DNF formulas is conp-complete. For any propositional formula φ there is an equivalent formula in
AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC
Bulletin of the Section of Logic Volume 45/1 (2016), pp 33 51 http://dxdoiorg/1018778/0138-068045103 Mirjana Ilić 1 AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC Abstract
Denotational semantics: proofs
APPENDIX A Denotational semantics: proofs We show that every closed term M has a computable functional [[M ] as its denotation. A.1. Unification We show that for any two constructor terms one can decide
Two sources of explosion
Two sources of explosion Eric Kao Computer Science Department Stanford University Stanford, CA 94305 United States of America Abstract. In pursuit of enhancing the deductive power of Direct Logic while
Proseminar on Semantic Theory Fall 2013 Ling 720 Propositional Logic: Syntax and Natural Deduction 1
Propositional Logic: Syntax and Natural Deduction 1 The Plot That Will Unfold I want to provide some key historical and intellectual context to the model theoretic approach to natural language semantics,
Sec$on Summary. Mathematical Proofs Forms of Theorems Trivial & Vacuous Proofs Direct Proofs Indirect Proofs
Section 1.7 Sec$on Summary Mathematical Proofs Forms of Theorems Trivial & Vacuous Proofs Direct Proofs Indirect Proofs Proof of the Contrapositive Proof by Contradiction 2 Proofs of Mathema$cal Statements
Introduction to Theoretical Computer Science
Introduction to Theoretical Computer Science Zdeněk Sawa Department of Computer Science, FEI, Technical University of Ostrava 17. listopadu 15, Ostrava-Poruba 708 33 Czech republic February 11, 2018 Z.
Equivalents of Mingle and Positive Paradox
Eric Schechter Equivalents of Mingle and Positive Paradox Abstract. Relevant logic is a proper subset of classical logic. It does not include among itstheoremsanyof positive paradox A (B A) mingle A (A
Equational Logic: Part 2. Roland Backhouse March 6, 2001
1 Equational Logic: Part 2 Roland Backhouse March 6, 2001 Outline 2 We continue the axiomatisation of the propositional connectives. The axioms for disjunction (the logical or of two statements) are added
NOTES (1) FOR MATH 375, FALL 2012
NOTES 1) FOR MATH 375, FALL 2012 1 Vector Spaces 11 Axioms Linear algebra grows out of the problem of solving simultaneous systems of linear equations such as 3x + 2y = 5, 111) x 3y = 9, or 2x + 3y z =
Logic. Introduction to Artificial Intelligence CS/ECE 348 Lecture 11 September 27, 2001
Logic Introduction to Artificial Intelligence CS/ECE 348 Lecture 11 September 27, 2001 Last Lecture Games Cont. α-β pruning Outline Games with chance, e.g. Backgammon Logical Agents and thewumpus World
The Modal Logic of Pure Provability
The Modal Logic of Pure Provability Samuel R. Buss Department of Mathematics University of California, San Diego July 11, 2002 Abstract We introduce a propositional modal logic PP of pure provability in
Branching Bisimilarity with Explicit Divergence
Branching Bisimilarity with Explicit Divergence Ro van Glaeek National ICT Australia, Sydney, Australia School of Computer Science and Engineering, University of New South Wales, Sydney, Australia Bas
Chapter 3 Deterministic planning
Chapter 3 Deterministic planning In this chapter we describe a number of algorithms for solving the historically most important and most basic type of planning problem. Two rather strong simplifying assumptions
CSE 20 DISCRETE MATH. Fall
CSE 20 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Distinguish between a theorem, an axiom, lemma, a corollary, and a conjecture. Recognize direct proofs
Deductive Systems. Lecture - 3
Deductive Systems Lecture - 3 Axiomatic System Axiomatic System (AS) for PL AS is based on the set of only three axioms and one rule of deduction. It is minimal in structure but as powerful as the truth
Inducing syntactic cut-elimination for indexed nested sequents
Inducing syntactic cut-elimination for indexed nested sequents Revantha Ramanayake Technische Universität Wien (Austria) IJCAR 2016 June 28, 2016 Revantha Ramanayake (TU Wien) Inducing syntactic cut-elimination
Propositional and Predicate Logic - II
Propositional and Predicate Logic - II Petr Gregor KTIML MFF UK WS 2016/2017 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - II WS 2016/2017 1 / 16 Basic syntax Language Propositional logic
What is Logic? Introduction to Logic. Simple Statements. Which one is statement?
What is Logic? Introduction to Logic Peter Lo Logic is the study of reasoning It is specifically concerned with whether reasoning is correct Logic is also known as Propositional Calculus CS218 Peter Lo
Logic and Computation
Logic and Computation CS245 Dr. Borzoo Bonakdarpour University of Waterloo (Fall 2012) Resolution in First-order Predicate Logic Logic and Computation p. 1/38 Agenda Resolution in Propositional Logic Prenex
Mathematical Logic and Linguistics
Mathematical Logic and Linguistics Glyn Morrill & Oriol Valentín Department of Computer Science Universitat Politècnica de Catalunya morrill@cs.upc.edu & oriol.valentin@gmail.com BGSMath Course Class 9
Part 1: Propositional Logic
Part 1: Propositional Logic Literature (also for first-order logic) Schöning: Logik für Informatiker, Spektrum Fitting: First-Order Logic and Automated Theorem Proving, Springer 1 Last time 1.1 Syntax
Completeness in the Monadic Predicate Calculus. We have a system of eight rules of proof. Let's list them:
Completeness in the Monadic Predicate Calculus We have a system of eight rules of proof. Let's list them: PI At any stage of a derivation, you may write down a sentence φ with {φ} as its premiss set. TC
Deriving natural deduction rules from truth tables (Extended version)
Deriving natural deduction rules from truth tables (Extended version) Herman Geuvers 1 and Tonny Hurkens 2 1 Radboud University & Technical University Eindhoven, The Netherlands herman@cs.ru.nl Abstract
Propositional Dynamic Logic
Propositional Dynamic Logic Contents 1 Introduction 1 2 Syntax and Semantics 2 2.1 Syntax................................. 2 2.2 Semantics............................... 2 3 Hilbert-style axiom system
On the Role of Implication in Formal Logic 1
On the Role of Implication in Formal Logic 1 Jonathan P. Seldin Department of Mathematics Concordia University Montréal, Québec, Canada seldin@alcor.concordia.ca April 24, 1998 1 This research was supported
1 Overview. 2 A Characterization of Convex Functions. 2.1 First-order Taylor approximation. AM 221: Advanced Optimization Spring 2016
AM 221: Advanced Optimization Spring 2016 Prof. Yaron Singer Lecture 8 February 22nd 1 Overview In the previous lecture we saw characterizations of optimality in linear optimization, and we reviewed the
5. The Logical Framework
5. The Logical Framework (a) Judgements. (b) Basic form of rules. (c) The non-dependent function type and product. (d) Structural rules. (Omitted 2008). (e) The dependent function set and -quantification.
3 Propositional Logic
3 Propositional Logic 3.1 Syntax 3.2 Semantics 3.3 Equivalence and Normal Forms 3.4 Proof Procedures 3.5 Properties Propositional Logic (25th October 2007) 1 3.1 Syntax Definition 3.0 An alphabet Σ consists
First Order Logic vs Propositional Logic CS477 Formal Software Dev Methods
First Order Logic vs Propositional Logic CS477 Formal Software Dev Methods Elsa L Gunter 2112 SC, UIUC egunter@illinois.edu http://courses.engr.illinois.edu/cs477 Slides based in part on previous lectures
arxiv:math/ v1 [math.lo] 5 Mar 2007
![arxiv:math/ v1 [math.lo] 5 Mar 2007](/thumbs/94/118958132.jpg "arxiv:math/ v1 [math.lo] 5 Mar 2007")
Topological Semantics and Decidability Dmitry Sustretov arxiv:math/0703106v1 [math.lo] 5 Mar 2007 March 6, 2008 Abstract It is well-known that the basic modal logic of all topological spaces is S4. However,
On the set of intermediate logics between the truth and degree preserving Lukasiewicz logics
On the set of intermediate logics between the truth and degree preserving Lukasiewicz logics Marcelo Coniglio 1 Francesc Esteva 2 Lluís Godo 2 1 CLE and Department of Philosophy State University of Campinas
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter p. 1/33
CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer Science (Arkoudas and Musser) Chapter 4.1-4.8 p. 1/33 CSCI.6962/4962 Software Verification Fundamental Proof Methods in Computer
The Countable Henkin Principle
The Countable Henkin Principle Robert Goldblatt Abstract. This is a revised and extended version of an article which encapsulates a key aspect of the Henkin method in a general result about the existence
Introduction to Metalogic
Philosophy 135 Spring 2008 Tony Martin Introduction to Metalogic 1 The semantics of sentential logic. The language L of sentential logic. Symbols of L: Remarks: (i) sentence letters p 0, p 1, p 2,... (ii)
0.1 Tangent Spaces and Lagrange Multipliers
01 TANGENT SPACES AND LAGRANGE MULTIPLIERS 1 01 Tangent Spaces and Lagrange Multipliers If a differentiable function G = (G 1,, G k ) : E n+k E k then the surface S defined by S = { x G( x) = v} is called
The Many Faces of Modal Logic Day 4: Structural Proof Theory
The Many Faces of Modal Logic Day 4: Structural Proof Theory Dirk Pattinson Australian National University, Canberra (Slides based on a NASSLLI 2014 Tutorial and are joint work with Lutz Schröder) LAC
LOGIC PROPOSITIONAL REASONING
LOGIC PROPOSITIONAL REASONING WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität Linz Version 2018.1
Model Theory of Modal Logic Lecture 5. Valentin Goranko Technical University of Denmark
Model Theory of Modal Logic Lecture 5 Valentin Goranko Technical University of Denmark Third Indian School on Logic and its Applications Hyderabad, January 29, 2010 Model Theory of Modal Logic Lecture
Teaching Natural Deduction as a Subversive Activity
Teaching Natural Deduction as a Subversive Activity James Caldwell Department of Computer Science University of Wyoming Laramie, WY Third International Congress on Tools for Teaching Logic 3 June 2011
Intelligent Systems. Propositional Logic. Dieter Fensel and Dumitru Roman. Copyright 2008 STI INNSBRUCK
Intelligent Systems Propositional Logic Dieter Fensel and Dumitru Roman www.sti-innsbruck.at Copyright 2008 STI INNSBRUCK www.sti-innsbruck.at Where are we? # Title 1 Introduction 2 Propositional Logic
Deep Sequent Systems for Modal Logic
Deep Sequent Systems for Modal Logic Kai Brünnler abstract. We see a systematic set of cut-free axiomatisations for all the basic normal modal logics formed from the axioms t, b,4, 5. They employ a form
Propositional Logic: Evaluating the Formulas
Institute for Formal Models and Verification Johannes Kepler University Linz VL Logik (LVA-Nr. 342208) Winter Semester 2015/2016 Propositional Logic: Evaluating the Formulas Version 2015.2 Armin Biere
Substitution Theorem. Equivalences. Proposition 2.8 The following equivalences are valid for all formulas φ, ψ, χ: (φ φ) φ Idempotency
Substitution Theorem Proposition 2.7 Let φ 1 and φ 2 be equivalent formulas, and ψ[φ 1 ] p be a formula in which φ 1 occurs as a subformula at position p. Then ψ[φ 1 ] p is equivalent to ψ[φ 2 ] p. Proof.
Syntactic Characterisations in Model Theory
Department of Mathematics Bachelor Thesis (7.5 ECTS) Syntactic Characterisations in Model Theory Author: Dionijs van Tuijl Supervisor: Dr. Jaap van Oosten June 15, 2016 Contents 1 Introduction 2 2 Preliminaries