Cylindrical Algebraic Decomposition in Coq

Cylindrical Algebraic Decomposition in Coq

Cylindrical Algebraic Decomposition in Coq MAP 2010 - Logroño 13-16 November 2010 Assia Mahboubi INRIA Microsoft Research Joint Centre (France) INRIA Saclay Île-de-France École Polytechnique, Palaiseau

Cylindrical Algebraic Decomposition in Coq

Cylindrical Algebraic Decomposition in Coq MAP 2010 - Logroño 13-16 November 2010 Assia Mahboubi INRIA Microsoft Research Joint Centre (France) INRIA Saclay Île-de-France École Polytechnique, Palaiseau

Cylindrical Algebraic Decomposition in Coq

Cylindrical Algebraic Decomposition in Coq MAP 2010 - Logroño 13-16 November 2010 Assia Mahboubi INRIA Microsoft Research Joint Centre (France) INRIA Saclay Île-de-France École Polytechnique, Palaiseau

Construction of Real Algebraic Numbers in Coq

Construction of Real Algebraic Numbers in Coq INRIA Saclay Île-de-France LIX École Polytechnique INRIA Microsoft Research Joint Centre cohen@crans.org August 13, 2012 Why algebraic numbers? Field strictly

FORMAL PROOFS IN REAL ALGEBRAIC GEOMETRY: FROM ORDERED FIELDS TO QUANTIFIER ELIMINATION

Logical Methods in Computer Science Vol. 8 (1:02) 2012, pp. 1 40 www.lmcs-online.org Submitted May 16, 2011 Published Feb. 16, 2012 FORMAL PROOFS IN REAL ALGEBRAIC GEOMETRY: FROM ORDERED FIELDS TO QUANTIFIER

Constructive Formalization of Classical Modal Logic

Constructive Formalization of Classical Modal Logic Christian Doczkal and Gert Smolka Saarland University June 30, 2011 This paper reports about the formalization of classical modal logic in the constructive

More Model Theory Notes

More Model Theory Notes Miscellaneous information, loosely organized. 1. Kinds of Models A countable homogeneous model M is one such that, for any partial elementary map f : A M with A M finite, and any

1 FUNDAMENTALS OF LOGIC NO.10 HERBRAND THEOREM Tatsuya Hagino hagino@sfc.keio.ac.jp lecture URL https://vu5.sfc.keio.ac.jp/slide/ 2 So Far Propositional Logic Logical connectives (,,, ) Truth table Tautology

Outline. Formale Methoden der Informatik First-Order Logic for Forgetters. Why PL1? Why PL1? Cont d. Motivation

Outline Formale Methoden der Informatik First-Order Logic for Forgetters Uwe Egly Vienna University of Technology Institute of Information Systems Knowledge-Based Systems Group Motivation Syntax of PL1

FIRST-ORDER QUERY EVALUATION ON STRUCTURES OF BOUNDED DEGREE

FIRST-ORDER QUERY EVALUATION ON STRUCTURES OF BOUNDED DEGREE INRIA and ENS Cachan e-mail address: kazana@lsv.ens-cachan.fr WOJCIECH KAZANA AND LUC SEGOUFIN INRIA and ENS Cachan e-mail address: see http://pages.saclay.inria.fr/luc.segoufin/

First Order Logic (FOL) 1 znj/dm2017

First Order Logic (FOL) 1 http://lcs.ios.ac.cn/ znj/dm2017 Naijun Zhan March 19, 2017 1 Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course.

Notes. Corneliu Popeea. May 3, 2013

Notes Corneliu Popeea May 3, 2013 1 Propositional logic Syntax We rely on a set of atomic propositions, AP, containing atoms like p, q. A propositional logic formula φ Formula is then defined by the following

A Dichotomy. in in Probabilistic Databases. Joint work with Robert Fink. for Non-Repeating Queries with Negation Queries with Negation

Dichotomy for Non-Repeating Queries with Negation Queries with Negation in in Probabilistic Databases Robert Dan Olteanu Fink and Dan Olteanu Joint work with Robert Fink Uncertainty in Computation Simons

Reasoning with Quantified Boolean Formulas

Reasoning with Quantified Boolean Formulas Martina Seidl Institute for Formal Models and Verification Johannes Kepler University Linz 1 What are QBF? Quantified Boolean formulas (QBF) are formulas of propositional

Model theory and algebraic geometry in groups, non-standard actions and algorithms

Model theory and algebraic geometry in groups, non-standard actions and algorithms Olga Kharlampovich and Alexei Miasnikov ICM 2014 1 / 32 Outline Tarski s problems Malcev s problems Open problems 2 /

Syntax. Notation Throughout, and when not otherwise said, we assume a vocabulary V = C F P.

First-Order Logic Syntax The alphabet of a first-order language is organised into the following categories. Logical connectives:,,,,, and. Auxiliary symbols:.,,, ( and ). Variables: we assume a countable

Quantified Boolean Formulas Part 1

Quantified Boolean Formulas Part 1 Uwe Egly Knowledge-Based Systems Group Institute of Information Systems Vienna University of Technology Results of the SAT 2009 application benchmarks for leading solvers

SKETCHY NOTES FOR WEEKS 7 AND 8

SKETCHY NOTES FOR WEEKS 7 AND 8 We are now ready to start work on the proof of the Completeness Theorem for first order logic. Before we start a couple of remarks are in order (1) When we studied propositional

Beyond First-Order Logic

Beyond First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) Beyond First-Order Logic MFES 2008/09 1 / 37 FOL

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

What is the decimal (base 10) representation of the binary number ? Show your work and place your final answer in the box.

Question 1. [10 marks] Part (a) [2 marks] What is the decimal (base 10) representation of the binary number 110101? Show your work and place your final answer in the box. 2 0 + 2 2 + 2 4 + 2 5 = 1 + 4

Applied Logic. Lecture 1 - Propositional logic. Marcin Szczuka. Institute of Informatics, The University of Warsaw

Applied Logic Lecture 1 - Propositional logic Marcin Szczuka Institute of Informatics, The University of Warsaw Monographic lecture, Spring semester 2017/2018 Marcin Szczuka (MIMUW) Applied Logic 2018

Constructive Algebra in Type Theory

Constructive Algebra in Type Theory Anders Mörtberg mortberg@chalmers.se September 4, 2012 ( 0 2 1 0 ) T ( x1 x 2 ) = ( 2 0 ) ( 0 2 1 0 ) T ( x1 x 2 ) = ( 2 0 ) ( x1 x 2 ) = ( 0 2 ) Matlab 2009b A=[ 0

Logic Part I: Classical Logic and Its Semantics

Logic Part I: Classical Logic and Its Semantics Max Schäfer Formosan Summer School on Logic, Language, and Computation 2007 July 2, 2007 1 / 51 Principles of Classical Logic classical logic seeks to model

Normal Forms for First-Order Logic

Logic and Proof Hilary 2016 James Worrell Normal Forms for First-Order Logic In this lecture we show how to transform an arbitrary formula of first-order logic to an equisatisfiable formula in Skolem form.

Propositional Logic. Testing, Quality Assurance, and Maintenance Winter Prof. Arie Gurfinkel

Propositional Logic Testing, Quality Assurance, and Maintenance Winter 2018 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/

The Zariski Spectrum of a ring

Thierry Coquand September 2010 Use of prime ideals Let R be a ring. We say that a 0,..., a n is unimodular iff a 0,..., a n = 1 We say that Σa i X i is primitive iff a 0,..., a n is unimodular Theorem:

Model theory, stability, applications

Model theory, stability, applications Anand Pillay University of Leeds June 6, 2013 Logic I Modern mathematical logic developed at the end of the 19th and beginning of 20th centuries with the so-called

Section 2.3: Statements Containing Multiple Quantifiers

Section 2.3: Statements Containing Multiple Quantifiers In this section, we consider statements such as there is a person in this company who is in charge of all the paperwork where more than one quantifier

Applied Logic for Computer Scientists. Answers to Some Exercises

Applied Logic for Computer Scientists Computational Deduction and Formal Proofs Springer, 2017 doi: http://link.springer.com/book/10.1007%2f978-3-319-51653-0 Answers to Some Exercises Mauricio Ayala-Rincón

Decidability of Sub-theories of Polynomials over a Finite Field

Decidability of Sub-theories of Polynomials over a Finite Field Alla Sirokofskich Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 45, D-53115, Bonn, Germany Department of Mathematics University

Knowledge base (KB) = set of sentences in a formal language Declarative approach to building an agent (or other system):

Logic Knowledge-based agents Inference engine Knowledge base Domain-independent algorithms Domain-specific content Knowledge base (KB) = set of sentences in a formal language Declarative approach to building

Model theory and constructive mathematics. Thierry Coquand

Thierry Coquand Model theory A challenge to constructive mathematics Important results: conjecture Ax-Kochen, differential algebraic closure, Mordell-Lang 1 Model theory The starting point of model theory,

First-Order Logic (FOL)

First-Order Logic (FOL) Also called Predicate Logic or Predicate Calculus 2. First-Order Logic (FOL) FOL Syntax variables x, y, z, constants a, b, c, functions f, g, h, terms variables, constants or n-ary

Finite Model Theory: First-Order Logic on the Class of Finite Models

1 Finite Model Theory: First-Order Logic on the Class of Finite Models Anuj Dawar University of Cambridge Modnet Tutorial, La Roche, 21 April 2008 2 Finite Model Theory In the 1980s, the term finite model

MaanavaN.Com MA1256 DISCRETE MATHEMATICS. DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : MA1256 DISCRETE MATHEMATICS

DEPARTMENT OF MATHEMATICS QUESTION BANK Subject & Code : UNIT I PROPOSITIONAL CALCULUS Part A ( Marks) Year / Sem : III / V. Write the negation of the following proposition. To enter into the country you

A Constructive Formalisation of Semi-algebraic Sets and Functions

A Constructive Formalisation of Semi-algebraic Sets and Functions To cite this version:. A Constructive Formalisation of Semi-algebraic Sets and Functions. June Andronick and Amy Felty. Certified Programs

Lecture 7: Polynomial rings

Lecture 7: Polynomial rings Rajat Mittal IIT Kanpur You have seen polynomials many a times till now. The purpose of this lecture is to give a formal treatment to constructing polynomials and the rules

Skills Practice Skills Practice for Lesson 10.1

Skills Practice Skills Practice for Lesson.1 Name Date Higher Order Polynomials and Factoring Roots of Polynomial Equations Problem Set Solve each polynomial equation using factoring. Then check your solution(s).

The Potential and Challenges of CAD with Equational Constraints for SC-Square

The Potential and Challenges of with Equational Constraints for SC-Square Matthew England (Coventry University) Joint work with: James H. Davenport (University of Bath) 7th International Conference on

Using the Prover I: Lee Pike. June 3, NASA Langley Formal Methods Group Using the Prover I:

Basic Basic NASA Langley Formal Methods Group lee.s.pike@nasa.gov June 3, 2005 Basic Sequents Basic Sequent semantics: The conjunction of the antecedents above the turnstile implies the disjunction of

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

Part 2: First-Order Logic

Part 2: First-Order Logic First-order logic formalizes fundamental mathematical concepts is expressive (Turing-complete) is not too expressive (e. g. not axiomatizable: natural numbers, uncountable sets)

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,

First-Order Logic First-Order Theories. Roopsha Samanta. Partly based on slides by Aaron Bradley and Isil Dillig

First-Order Logic First-Order Theories Roopsha Samanta Partly based on slides by Aaron Bradley and Isil Dillig Roadmap Review: propositional logic Syntax and semantics of first-order logic (FOL) Semantic

Truth-Functional Logic

Truth-Functional Logic Syntax Every atomic sentence (A, B, C, ) is a sentence and are sentences With ϕ a sentence, the negation ϕ is a sentence With ϕ and ψ sentences, the conjunction ϕ ψ is a sentence

Herbrand Theorem, Equality, and Compactness

CSC 438F/2404F Notes (S. Cook and T. Pitassi) Fall, 2014 Herbrand Theorem, Equality, and Compactness The Herbrand Theorem We now consider a complete method for proving the unsatisfiability of sets of first-order

Lecture Notes 1 Basic Concepts of Mathematics MATH 352

Lecture Notes 1 Basic Concepts of Mathematics MATH 352 Ivan Avramidi New Mexico Institute of Mining and Technology Socorro, NM 87801 June 3, 2004 Author: Ivan Avramidi; File: absmath.tex; Date: June 11,

Arithmetic Decision Procedures: a simple introduction

Arithmetic Decision Procedures: a simple introduction Michael Norrish Abstract Fourier-Motzkin variable elimination is introduced as a complete method for deciding linear arithmetic inequalities over R.

Exercises 1 - Solutions

Exercises 1 - Solutions SAV 2013 1 PL validity For each of the following propositional logic formulae determine whether it is valid or not. If it is valid prove it, otherwise give a counterexample. Note

Sequent calculus for predicate logic

CHAPTER 13 Sequent calculus for predicate logic 1. Classical sequent calculus The axioms and rules of the classical sequent calculus are: Axioms { Γ, ϕ, ϕ for atomic ϕ Γ, Left Γ,α 1,α 2 Γ,α 1 α 2 Γ,β 1

6-1 The Positivstellensatz P. Parrilo and S. Lall, ECC

6-1 The Positivstellensatz P. Parrilo and S. Lall, ECC 2003 2003.09.02.10 6. The Positivstellensatz Basic semialgebraic sets Semialgebraic sets Tarski-Seidenberg and quantifier elimination Feasibility

15. Polynomial rings Definition-Lemma Let R be a ring and let x be an indeterminate.

15. Polynomial rings Definition-Lemma 15.1. Let R be a ring and let x be an indeterminate. The polynomial ring R[x] is defined to be the set of all formal sums a n x n + a n 1 x n +... a 1 x + a 0 = a

Comp487/587 - Boolean Formulas

Comp487/587 - Boolean Formulas 1 Logic and SAT 1.1 What is a Boolean Formula Logic is a way through which we can analyze and reason about simple or complicated events. In particular, we are interested

Constructive algebra. Thierry Coquand. May 2018

Constructive algebra Thierry Coquand May 2018 Constructive algebra is algebra done in the context of intuitionistic logic 1 H. Lombardi, C. Quitté Commutative Algebra: Constructive Methods, 2016 I. Yengui

Concept Learning Mitchell, Chapter 2. CptS 570 Machine Learning School of EECS Washington State University

Concept Learning Mitchell, Chapter 2 CptS 570 Machine Learning School of EECS Washington State University Outline Definition General-to-specific ordering over hypotheses Version spaces and the candidate

Datalog and Constraint Satisfaction with Infinite Templates

Datalog and Constraint Satisfaction with Infinite Templates Manuel Bodirsky 1 and Víctor Dalmau 2 1 CNRS/LIX, École Polytechnique, bodirsky@lix.polytechnique.fr 2 Universitat Pompeu Fabra, victor.dalmau@upf.edu

Löwenheim-Skolem Theorems, Countable Approximations, and L ω. David W. Kueker (Lecture Notes, Fall 2007)

Löwenheim-Skolem Theorems, Countable Approximations, and L ω 0. Introduction David W. Kueker (Lecture Notes, Fall 2007) In its simplest form the Löwenheim-Skolem Theorem for L ω1 ω states that if σ L ω1

University of Oxford, Michaelis November 16, Categorical Semantics and Topos Theory Homotopy type theor

Categorical Semantics and Topos Theory Homotopy type theory Seminar University of Oxford, Michaelis 2011 November 16, 2011 References Johnstone, P.T.: Sketches of an Elephant. A Topos-Theory Compendium.

MATH 770 : Foundations of Mathematics. Fall Itay Ben-Yaacov

MATH 770 : Foundations of Mathematics Fall 2005 Itay Ben-Yaacov Itay Ben-Yaacov, University of Wisconsin Madison, Department of Mathematics, 480 Lincoln Drive, Madison, WI 53706-1388, USA URL: http://www.math.wisc.edu/~pezz

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In

Motivation. CS389L: Automated Logical Reasoning. Lecture 10: Overview of First-Order Theories. Signature and Axioms of First-Order Theory

Motivation CS389L: Automated Logical Reasoning Lecture 10: Overview of First-Order Theories Işıl Dillig Last few lectures: Full first-order logic In FOL, functions/predicates are uninterpreted (i.e., structure

The model companion of the class of pseudo-complemented semilattices is finitely axiomatizable

The model companion of the class of pseudo-complemented semilattices is finitely axiomatizable (joint work with Regula Rupp and Jürg Schmid) Joel Adler joel.adler@phbern.ch Institut S1 Pädagogische Hochschule

Negation introduction

Negation introduction How do we prove a negation? P = P F -introduction {Assume} P (l-1) F { -intro on and (l-1)} (l) P -intro Negation elimination -elimination How do we use a negation in a proof? P (l)

Some consequences of compactness in Lukasiewicz Predicate Logic

Some consequences of compactness in Lukasiewicz Predicate Logic Luca Spada Department of Mathematics and Computer Science University of Salerno www.logica.dmi.unisa.it/lucaspada 7 th Panhellenic Logic

Basic Algebraic Logic

ELTE 2013. September Today Past 1 Universal Algebra 1 Algebra 2 Transforming Algebras... Past 1 Homomorphism 2 Subalgebras 3 Direct products 3 Varieties 1 Algebraic Model Theory 1 Term Algebras 2 Meanings

Conjunction: p q is true if both p, q are true, and false if at least one of p, q is false. The truth table for conjunction is as follows.

Chapter 1 Logic 1.1 Introduction and Definitions Definitions. A sentence (statement, proposition) is an utterance (that is, a string of characters) which is either true (T) or false (F). A predicate is

Introduction to Artificial Intelligence Propositional Logic & SAT Solving. UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010

Introduction to Artificial Intelligence Propositional Logic & SAT Solving UIUC CS 440 / ECE 448 Professor: Eyal Amir Spring Semester 2010 Today Representation in Propositional Logic Semantics & Deduction

Proofs of transcendance

Proofs of transcendance Sophie Bernard, Laurence Rideau, Pierre-Yves Strub Yves Bertot November 2015 1 / 19 Objectives Study the gap between practical mathematics and computer-verified reasoning Explore

Computing Rational Points in Convex Semi-algebraic Sets and Sums-of-Squares Decompositions

Computing Rational Points in Convex Semi-algebraic Sets and Sums-of-Squares Decompositions Mohab Safey El Din 1 Lihong Zhi 2 1 University Pierre et Marie Curie, Paris 6, France INRIA Paris-Rocquencourt,

A brief introduction to Logic. (slides from

A brief introduction to Logic (slides from http://www.decision-procedures.org/) 1 A Brief Introduction to Logic - Outline Propositional Logic :Syntax Propositional Logic :Semantics Satisfiability and validity

Classical First-Order Logic

Classical First-Order Logic Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) First-Order Logic (Classical) MFES 2008/09

Chapter 1 : The language of mathematics.

MAT 200, Logic, Language and Proof, Fall 2015 Summary Chapter 1 : The language of mathematics. Definition. A proposition is a sentence which is either true or false. Truth table for the connective or :

MODEL THEORY FOR ALGEBRAIC GEOMETRY

MODEL THEORY FOR ALGEBRAIC GEOMETRY VICTOR ZHANG Abstract. We demonstrate how several problems of algebraic geometry, i.e. Ax-Grothendieck, Hilbert s Nullstellensatz, Noether- Ostrowski, and Hilbert s

Logic and Proofs. (A brief summary)

Logic and Proofs (A brief summary) Why Study Logic: To learn to prove claims/statements rigorously To be able to judge better the soundness and consistency of (others ) arguments To gain the foundations

On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic

On the satisfiability problem for a 4-level quantified syllogistic and some applications to modal logic Domenico Cantone and Marianna Nicolosi Asmundo Dipartimento di Matematica e Informatica Università

CSE 1400 Applied Discrete Mathematics Definitions

CSE 1400 Applied Discrete Mathematics Definitions Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Arithmetic 1 Alphabets, Strings, Languages, & Words 2 Number Systems 3 Machine

tp(c/a) tp(c/ab) T h(m M ) is assumed in the background.

Model Theory II. 80824 22.10.2006-22.01-2007 (not: 17.12) Time: The first meeting will be on SUNDAY, OCT. 22, 10-12, room 209. We will try to make this time change permanent. Please write ehud@math.huji.ac.il

LOGIC. Mathematics. Computer Science. Stanley N. Burris

LOGIC for Mathematics and Computer Science Stanley N. Burris Department of Pure Mathematics University of Waterloo Prentice Hall Upper Saddle River, New Jersey 07458 Contents Preface The Flow of Topics

COMP 409: Logic Homework 5

COMP 409: Logic Homework 5 Note: The pages below refer to the text from the book by Enderton. 1. Exercises 1-6 on p. 78. 1. Translate into this language the English sentences listed below. If the English

2-4: The Use of Quantifiers

2-4: The Use of Quantifiers The number x + 2 is an even integer is not a statement. When x is replaced by 1, 3 or 5 the resulting statement is false. However, when x is replaced by 2, 4 or 6 the resulting

CMPS 217 Logic in Computer Science. Lecture #17

CMPS 217 Logic in Computer Science https://courses.soe.ucsc.edu/courses/cmps217/spring13/01 Lecture #17 1 The Complexity of FO-Truth on a Structure Structure A Complexity of Th(A) Structure of the natural

INTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4

INTRODUCTION TO PREDICATE LOGIC HUTH AND RYAN 2.1, 2.2, 2.4 Neil D. Jones DIKU 2005 Some slides today new, some based on logic 2004 (Nils Andersen), some based on kernebegreber (NJ 2005) PREDICATE LOGIC:

586 Index. vertex, 369 disjoint, 236 pairwise, 272, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 465 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

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

cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018

cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics

7 RC Simulates RA. Lemma: For every RA expression E(A 1... A k ) there exists a DRC formula F with F V (F ) = {A 1,..., A k } and

7 RC Simulates RA. We now show that DRC (and hence TRC) is at least as expressive as RA. That is, given an RA expression E that mentions at most C, there is an equivalent DRC expression E that mentions

Predicate Calculus. Formal Methods in Verification of Computer Systems Jeremy Johnson

Predicate Calculus Formal Methods in Verification of Computer Systems Jeremy Johnson Outline 1. Motivation 1. Variables, quantifiers and predicates 2. Syntax 1. Terms and formulas 2. Quantifiers, scope

First-Order Theorem Proving and Vampire

First-Order Theorem Proving and Vampire Laura Kovács 1,2 and Martin Suda 2 1 TU Wien 2 Chalmers Outline Introduction First-Order Logic and TPTP Inference Systems Saturation Algorithms Redundancy Elimination

Decision Procedures 1: Survey of decision procedures

Decision Procedures 1: Survey of decision procedures John Harrison Intel Corporation TYPES summer school 2005, Göteborg Fri 19th August 2005 (09:00 09:45) 0 Summary Interesting and uninteresting proofs

Basing Decisions on Sentences in Decision Diagrams

Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence Basing Decisions on Sentences in Decision Diagrams Yexiang Xue Department of Computer Science Cornell University yexiang@cs.cornell.edu

Constructive Logic. Thierry Coquand. August 2008

Thierry Coquand August 2008 This course To present constructive mathematics using logic Introduction to recent work in constructive algebra (H. Lombardi, P. Schuster, I. Yengui,... ) Connection with computer

CHAPTER 11. Introduction to Intuitionistic Logic

CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated

AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH Jimmy T. Arnold

AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH 3034 Jimmy T. Arnold i TABLE OF CONTENTS CHAPTER 1: The Structure of Mathematical Statements.............................1 1.1. Statements..................................................................

Propositional and Predicate Logic - VII

Propositional and Predicate Logic - VII Petr Gregor KTIML MFF UK WS 2015/2016 Petr Gregor (KTIML MFF UK) Propositional and Predicate Logic - VII WS 2015/2016 1 / 11 Theory Validity in a theory A theory

Foundations of Artificial Intelligence

Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Wolfram Burgard, Maren Bennewitz, and Marco Ragni Albert-Ludwigs-Universität Freiburg Contents 1 Agents

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes

Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements

Foundations of Artificial Intelligence

Foundations of Artificial Intelligence 7. Propositional Logic Rational Thinking, Logic, Resolution Joschka Boedecker and Wolfram Burgard and Bernhard Nebel Albert-Ludwigs-Universität Freiburg May 17, 2016

Model Checking for Modal Intuitionistic Dependence Logic

1/71 Model Checking for Modal Intuitionistic Dependence Logic Fan Yang Department of Mathematics and Statistics University of Helsinki Logical Approaches to Barriers in Complexity II Cambridge, 26-30 March,

Outline. Logic. Definition. Theorem (Gödel s Completeness Theorem) Summary of Previous Week. Undecidability. Unification

Logic Aart Middeldorp Vincent van Oostrom Franziska Rapp Christian Sternagel Department of Computer Science University of Innsbruck WS 2017/2018 AM (DCS @ UIBK) week 11 2/38 Definitions elimination x φ