WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008
|
|
- Lambert Harrington
- 6 years ago
- Views:
Transcription
1 WHAT IS AN SMT SOLVER? Jaeheon Yi - April 17, 2008
2 WHAT I LL TALK ABOUT Propositional Logic Terminology, Satisfiability, Decision Procedure First-Order Logic Terminology, Background Theories Satisfiability Modulo Theories Combining Theories, SMT Solvers
3 TERMINOLOGY: PROPOSITIONAL LOGIC Propositional variables: Propositional formulae: p, q, r p, φ 1 φ 2, φ 1 φ 2, φ Literals: Clauses: p, p p q r s Conjunctive Normal Form: (p q) r (s q) Truth Assignment: Let p be false, q be true, etc.
4 SATISFIABILITY Given a propositional formula φ, does it have a truth assignment M, such that M = φ? Decidable: there exists a decision procedure for this Complexity: NP-complete
5 DECISION PROCEDURE DPLL Method Sound Complete Backtracking Conjunctive Normal Form Davis, Putnam, Logemann, Loveland: 1960, 1962
6 DECISION PROCEDURE Function DPLL Input: A formula, Output: A boolean φ If φ is consistent Return true If φ contains an empty clause Return false l := choose_literal( φ ) Return DPLL( φ l ) Or DPLL( φ l )
7 DECISION PROCEDURE Performance: Depends on choice of branching literal Satisfying assignment may be found quickly Unsatisfiability requires exhaustive search Modern Heuristics
8 TERMINOLOGY: FIRST-ORDER LOGIC Variables: X = {x 1,x 2,x 3,... } Function Symbols: Predicate Symbols: F = {f 1,f 2,f 3,... } P = {p 1,p 2,p 3,... } Equality, quantification, propositional connectives: =,,,,,
9 TERMINOLOGY: FIRST-ORDER LOGIC Term τ ::= x f(τ 1,..., τ n ) Formula ϕ ::= τ 1 = τ 2 p(τ 1,..., τ n )... Sentence: fully quantified formula Structure: a triple (δ, σ, ι) Domain δ : set of values Signature σ : set of function and predicate symbols, with arities ι Interpretation : what X, F, P do in δ
10 TERMINOLOGY: EXAMPLES Term: x 1, f 1 (x 3 ), f 2 (x 2,f 1 (x 3 )) Formula: Sentence: p 1 (x 1 ), x 1 = f 1 (x 3 ), p 1 (x 1 ), x 1. p 1 (x 1 ) p 2 (x 1,x 2 ) x 1.p 1 (x 1 ), x 3. x 1.x 1 = f 1 (x 3 ) Structure: a triple (δ, σ, ι) δ Domain : N σ Signature : c n 0, for all, ι n N is zero 1 Interpretation : for is_zero: if argument is 0 then true, else false
11 BACKGROUND THEORIES Theory : A set of sentences in signature σ. T σ Some theories: Theory of Equality with Uninterpreted Functions Theory of Integers Theory of Reals Theory of Lists
12 BACKGROUND THEORIES Two ways to define a theory: Enumerate a set of axioms, and close the set under logical consequence Define a structure in which all sentences are satisfied
13 BACKGROUND THEORIES T E Theory of Equality with Uninterpreted Functions ( Empty Theory ) {} T Z (Z, +,, ) Theory of Integers T R Theory of Reals (Q, +,, ) T L Theory of Lists cons(car(x),cdr(x)) = x, car(cons(x,y)) = x, cdr(cons(x,y)) = y,...
14 BACKGROUND THEORIES T E Theory of Equality with Uninterpreted Functions ( Empty Theory ) Undecidable. Quantifier-free fragment is decidable T Z Theory of Integers Decidable. Quantifier-free fragment is NP-complete T R Theory of Reals Decidable. Quantifier-free fragment is solvable in polynomial time T L Theory of Lists Decidable. Quantifier-free fragment is solvable in linear time
15 SATISFIABILITY IN A THEORY Given a signature σ and a theory T σ, a formula ϕ, constructed in σ, is satisfiable in T σ, if ϕ evaluates to true under some interpretation of T σ
16 SATISFIABILITY MODULO THEORIES Assume we have signatures σ 1,..., σ n, and theories T,..., T σ1 σn, so that we have a combined signature σ = σ 1 σ n and combined theory of T σ = T T σ1 σn Is a formula ϕ, constructed in σ, satisfiable in T σ?
17 SMT EXAMPLE Let T = T E T Z. Let s assume we have a formula with the following terms conjuncted: 1 x x 2 Γ = f(x) f(1) f(x) f(2) Is Γsatisfiable in T?
18 DECISION PROCEDURE Nelson-Oppen method Combines decision procedures for first-order theories into a single decision procedure for the union theory - equality symbol is shared ϕ must be quantifier-free Signatures σ 1,..., σ n must be disjoint Theories T σ1,..., T σn must be stably infinite
19 STABLY INFINITE THEORIES For signature σ, a theory T σ is stably infinite, if for every quantifier-free formula ϕ in σ that is satisfiable in T σ, there exists an interpretation satisfying ϕ, whose domain is infinite. T E,T Z,T R,T L are all stably infinite theories. For signature σ = {a, b}, the following theory is not stably infinite: T = { x. x = a x = b}
20 NELSON-OPPEN METHOD Phase 1: Purify Phase 2: Equality Propagation Contradiction Rule Equality Propagation Rule Case Split Rule
21 NELSON-OPPEN METHOD Purify phase: Assume formula from σ 1 σ 2 Γ, a conjunction of literals Convert gamma to conjunction Γ 1 Γ 2, satisfying two properties: Each literal in Γ i is a literal in σ i, for all i Γ 1 Γ 2 is satisfiable in T 1 T 2, iff Γ is
22 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each term and inequality purified
23 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each { term and inequality purified Γ = f(g(x)) g(f(x)) }
24 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each term and inequality purified { } f(w1 ) g(f(x)) Γ = w 1 = g(x)
25 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each term and inequality purified f(w 1 ) g(w 2 ) Γ = w 1 = g(x) w 2 = f(x)
26 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each term and inequality purified w 3 w 4 w 1 = g(x) Γ = w 2 = f(x) w 3 = f(w 1 ) w 4 = g(w 2 )
27 NELSON-OPPEN METHOD Purify phase example Assume Formula σ 1 = {f 1 }, σ 2 = {g 1 } Γ = {f(g(x)) g(f(x))} Each term and inequality purified { } w2 = f(x) Γ 1 = w 3 = f(w 1 ) Γ 2 = w 1 = g(x) w 4 = g(w 2 ) w 3 w 4
28 NELSON-OPPEN METHOD Equality Propagation Phase Tree with states Γ 1, Γ 2,E as nodes Initial derivation s 0 : Γ 1, Γ 2, Apply inference rules If all leaves are labeled false, then unsat Else sat
29 NELSON-OPPEN METHOD Equality Propagation Phase Contradiction If unsatisfiable Equality Propagation If equality is satisfiable Γ 1, Γ 2,E false Γ 1, Γ 2,E Γ 1, Γ 2,E {x = y} Case Split Γ 1, Γ 2,E Γ 1, Γ 2,E {x 1 = y 1 } Γ 1, Γ 2,E {x n = y n }
30 NELSON-OPPEN METHOD Equality Propagation Phase example Satisfiability of following formula in T Z T E 1 x 1 x x 2 x 2 Γ = Γ f(x) f(1) Z = w 1 =1 f(x) f(2) w 2 =2 After purify: { f(x) f(w1 ) Γ E = f(x) f(w 2 ) }
31 NELSON-OPPEN METHOD Equality Propagation Phase example Satisfiability of following formula in T Z T E s 0 : Γ Z, Γ E, s 1 : Γ Z, Γ E, {x = w 1 } s 2 : Γ Z, Γ E, {x = w 2 } s 3 : false s 4 : false
32 THEORY CONVEXITY Case-Split inference rule creates many, many subtrees Can be avoided altogether, if the combined theories are convex T R,T E,T L are convex. T Z is not.
33 THEORY CONVEXITY A theory T σ is convex, if for every conjunction Γ of literals in σ, and for every disjunction x 1 = y 1 x n = y n n T Γ = some j i=1 x i = y i iff T Γ = x j = y j, for
34 THEORY CONVEXITY How is T Z not convex? We have a formula: the conjunction Γ = We have a disjunction, x = z y = z x =1 y =2 1 z z 2 While T Z Γ logically implies the disjunction, both x = z and y = z do not logically follow.
35 IMPLEMENTATION Theory of Equality with Uninterpreted Functions has decision procedure: Congruence Closure algorithm E-Graphs: a set of terms, and an equivalence relation on those terms
36 SMT SOLVERS Simplify (HP Research) CVC3 (Stanford) Yices (SRI International) Z3 (MS Research) Paradox (Chalmers)
37 ISSUES Modularity of theories More theories Extending to solve formulas with quantification
38 CONCLUSION Thank you for coming Questions welcome Wanna hack one up?
39 BIBLIOGRAPHY Zohar Manna and Calogero Zarba. Combining Decision Procedures. In Formal Methods at the Crossroads: from Panacea to Foundational Support, Lecture Notes in Computer Science, Volume 2787, Springer-Verlag, November 2003, pp David Detlefs, Greg Nelson, James B. Saxe. Simplify: A Theorem Prover for Program Checking. In Journal of the ACM, Vol 52, No 3, May 2005, pp Greg Nelson and Derek C. Oppen. Simplification by Cooperating Decision Procedures. In ACM Transactions on Programming Languages and Systems, Vol 1, No 2. October 1979, pp Greg Nelson and Derek C. Oppen. Fast Decision Procedures Based on Congruence Closure. In Journal of the ACM, Vol 27, No 2, April 1980, pp Stanley N. Burris. Logic for Mathematics and Computer Science. Prentice Hall, August 1997.
Tutorial 1: Modern SMT Solvers and Verification
University of Illinois at Urbana-Champaign Tutorial 1: Modern SMT Solvers and Verification Sayan Mitra Electrical & Computer Engineering Coordinated Science Laboratory University of Illinois at Urbana
More informationCS156: The Calculus of Computation
CS156: The Calculus of Computation Zohar Manna Winter 2010 It is reasonable to hope that the relationship between computation and mathematical logic will be as fruitful in the next century as that between
More informationThe Calculus of Computation: Decision Procedures with Applications to Verification. Part I: FOUNDATIONS. by Aaron Bradley Zohar Manna
The Calculus of Computation: Decision Procedures with Applications to Verification Part I: FOUNDATIONS by Aaron Bradley Zohar Manna 1. Propositional Logic(PL) Springer 2007 1-1 1-2 Propositional Logic(PL)
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT
More informationFirst-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
More informationSatisfiability Modulo Theories (SMT)
Satisfiability Modulo Theories (SMT) Sylvain Conchon Cours 7 / 9 avril 2014 1 Road map The SMT problem Modern efficient SAT solvers CDCL(T) Examples of decision procedures: equality (CC) and difference
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 31. Propositional Logic: DPLL Algorithm Malte Helmert and Gabriele Röger University of Basel April 24, 2017 Propositional Logic: Overview Chapter overview: propositional
More informationCS156: The Calculus of Computation Zohar Manna Autumn 2008
Page 3 of 52 Page 4 of 52 CS156: The Calculus of Computation Zohar Manna Autumn 2008 Lecturer: Zohar Manna (manna@cs.stanford.edu) Office Hours: MW 12:30-1:00 at Gates 481 TAs: Boyu Wang (wangboyu@stanford.edu)
More informationComputational Logic. Davide Martinenghi. Spring Free University of Bozen-Bolzano. Computational Logic Davide Martinenghi (1/30)
Computational Logic Davide Martinenghi Free University of Bozen-Bolzano Spring 2010 Computational Logic Davide Martinenghi (1/30) Propositional Logic - sequent calculus To overcome the problems of natural
More informationTopics in Model-Based Reasoning
Towards Integration of Proving and Solving Dipartimento di Informatica Università degli Studi di Verona Verona, Italy March, 2014 Automated reasoning Artificial Intelligence Automated Reasoning Computational
More informationCombining Decision Procedures
Combining Decision Procedures Ashish Tiwari tiwari@csl.sri.com http://www.csl.sri.com/. Computer Science Laboratory SRI International 333 Ravenswood Menlo Park, CA 94025 Combining Decision Procedures (p.1
More informationLecture 1: Logical Foundations
Lecture 1: Logical Foundations Zak Kincaid January 13, 2016 Logics have two components: syntax and semantics Syntax: defines the well-formed phrases of the language. given by a formal grammar. Typically
More informationRewriting for Satisfiability Modulo Theories
1 Dipartimento di Informatica Università degli Studi di Verona Verona, Italy July 10, 2010 1 Joint work with Chris Lynch (Department of Mathematics and Computer Science, Clarkson University, NY, USA) and
More informationIntroduction 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
More informationThe Simplify Theorem Prover
The Simplify Theorem Prover Class Notes for Lecture No.8 by Mooly Sagiv Notes prepared by Daniel Deutch Introduction This lecture will present key aspects in the leading theorem proving systems existing
More informationSolvers for the Problem of Boolean Satisfiability (SAT) Will Klieber Aug 31, 2011
Solvers for the Problem of Boolean Satisfiability (SAT) Will Klieber 15-414 Aug 31, 2011 Why study SAT solvers? Many problems reduce to SAT. Formal verification CAD, VLSI Optimization AI, planning, automated
More informationFoundations of Lazy SMT and DPLL(T)
Foundations of Lazy SMT and DPLL(T) Cesare Tinelli The University of Iowa Foundations of Lazy SMT and DPLL(T) p.1/86 Acknowledgments: Many thanks to Albert Oliveras for contributing some of the material
More informationPropositional and First Order Reasoning
Propositional and First Order Reasoning Terminology Propositional variable: boolean variable (p) Literal: propositional variable or its negation p p Clause: disjunction of literals q \/ p \/ r given by
More informationCombined Satisfiability Modulo Parametric Theories
Intel 07 p.1/39 Combined Satisfiability Modulo Parametric Theories Sava Krstić*, Amit Goel*, Jim Grundy*, and Cesare Tinelli** *Strategic CAD Labs, Intel **The University of Iowa Intel 07 p.2/39 This Talk
More informationCOMP219: Artificial Intelligence. Lecture 20: Propositional Reasoning
COMP219: Artificial Intelligence Lecture 20: Propositional Reasoning 1 Overview Last time Logic for KR in general; Propositional Logic; Natural Deduction Today Entailment, satisfiability and validity Normal
More informationAn Introduction to Satisfiability Modulo Theories
ICCAD 2009 Tutorial p. 1/78 An Introduction to Satisfiability Modulo Theories Clark Barrett and Sanjit Seshia ICCAD 2009 Tutorial p. 2/78 Roadmap Theory Solvers Examples of Theory Solvers Combining Theory
More informationClassical Propositional Logic
Classical Propositional Logic Peter Baumgartner http://users.cecs.anu.edu.au/~baumgart/ Ph: 02 6218 3717 Data61/CSIRO and ANU July 2017 1 / 71 Classical Logic and Reasoning Problems A 1 : Socrates is a
More informationChapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau CMSC 722, AI Planning University of Maryland, Spring 2008 1 Motivation Propositional
More informationMotivation. 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
More informationComp487/587 - 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
More informationLogic and Inferences
Artificial Intelligence Logic and Inferences Readings: Chapter 7 of Russell & Norvig. Artificial Intelligence p.1/34 Components of Propositional Logic Logic constants: True (1), and False (0) Propositional
More informationLecture 9: The Splitting Method for SAT
Lecture 9: The Splitting Method for SAT 1 Importance of SAT Cook-Levin Theorem: SAT is NP-complete. The reason why SAT is an important problem can be summarized as below: 1. A natural NP-Complete problem.
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 7: Procedures for First-Order Theories, Part 1
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 7: Procedures for First-Order Theories, Part 1 Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson Theory Procedures
More informationSAT Solvers: Theory and Practice
Summer School on Verification Technology, Systems & Applications, September 17, 2008 p. 1/98 SAT Solvers: Theory and Practice Clark Barrett barrett@cs.nyu.edu New York University Summer School on Verification
More informationPart 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
More informationLecture Notes on SAT Solvers & DPLL
15-414: Bug Catching: Automated Program Verification Lecture Notes on SAT Solvers & DPLL Matt Fredrikson André Platzer Carnegie Mellon University Lecture 10 1 Introduction In this lecture we will switch
More informationSatisfiability Modulo Theories (SMT)
CS510 Software Engineering Satisfiability Modulo Theories (SMT) Slides modified from those by Aarti Gupta Textbook: The Calculus of Computation by A. Bradley and Z. Manna 1 Satisfiability Modulo Theory
More informationSMT: Satisfiability Modulo Theories
SMT: Satisfiability Modulo Theories Ranjit Jhala, UC San Diego April 9, 2013 Decision Procedures Last Time Propositional Logic Today 1. Combining SAT and Theory Solvers 2. Theory Solvers Theory of Equality
More informationSMT BASICS WS 2017/2018 ( ) LOGIC SATISFIABILITY MODULO THEORIES. Institute for Formal Models and Verification Johannes Kepler Universität Linz
LOGIC SATISFIABILITY MODULO THEORIES SMT BASICS 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
More informationSolving SAT Modulo Theories
Solving SAT Modulo Theories R. Nieuwenhuis, A. Oliveras, and C.Tinelli. Solving SAT and SAT Modulo Theories: from an Abstract Davis-Putnam-Logemann-Loveland Procedure to DPLL(T) Mooly Sagiv Motivation
More informationCSE507. Introduction. Computer-Aided Reasoning for Software. Emina Torlak courses.cs.washington.edu/courses/cse507/17wi/
Computer-Aided Reasoning for Software CSE507 courses.cs.washington.edu/courses/cse507/17wi/ Introduction Emina Torlak emina@cs.washington.edu Today What is this course about? Course logistics Review of
More informationLecture 2 Propositional Logic & SAT
CS 5110/6110 Rigorous System Design Spring 2017 Jan-17 Lecture 2 Propositional Logic & SAT Zvonimir Rakamarić University of Utah Announcements Homework 1 will be posted soon Propositional logic: Chapter
More informationLeonardo de Moura Microsoft Research
Leonardo de Moura Microsoft Research Logic is The Calculus of Computer Science (Z. Manna). High computational complexity Naïve solutions will not scale Is formula F satisfiable modulo theory T? SMT solvers
More informationQuantifiers. Leonardo de Moura Microsoft Research
Quantifiers Leonardo de Moura Microsoft Research Satisfiability a > b + 2, a = 2c + 10, c + b 1000 SAT a = 0, b = 3, c = 5 Model 0 > 3 + 2, 0 = 2 5 + 10, 5 + ( 3) 1000 Quantifiers x y x > 0 f x, y = 0
More informationPropositional Reasoning
Propositional Reasoning CS 440 / ECE 448 Introduction to Artificial Intelligence Instructor: Eyal Amir Grad TAs: Wen Pu, Yonatan Bisk Undergrad TAs: Sam Johnson, Nikhil Johri Spring 2010 Intro to AI (CS
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Bruno Dutertre SRI International Leonardo de Moura Microsoft Research Satisfiability a > b + 2, a = 2c + 10, c + b 1000 SAT a = 0, b = 3, c = 5 Model 0 > 3 + 2, 0 = 2 5 +
More informationFoundations of Artificial Intelligence
Foundations of Artificial Intelligence 8. Satisfiability and Model Construction Davis-Putnam-Logemann-Loveland Procedure, Phase Transitions, GSAT Joschka Boedecker and Wolfram Burgard and Bernhard Nebel
More informationModel Based Theory Combination
Model Based Theory Combination SMT 2007 Leonardo de Moura and Nikolaj Bjørner {leonardo, nbjorner}@microsoft.com. Microsoft Research Model Based Theory Combination p.1/20 Combination of Theories In practice,
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationChapter 7 Propositional Satisfiability Techniques
Lecture slides for Automated Planning: Theory and Practice Chapter 7 Propositional Satisfiability Techniques Dana S. Nau University of Maryland 12:58 PM February 15, 2012 1 Motivation Propositional satisfiability:
More informationFormal Verification Methods 1: Propositional Logic
Formal Verification Methods 1: Propositional Logic John Harrison Intel Corporation Course overview Propositional logic A resurgence of interest Logic and circuits Normal forms The Davis-Putnam procedure
More informationPropositional Logic. Methods & Tools for Software Engineering (MTSE) Fall Prof. Arie Gurfinkel
Propositional Logic Methods & Tools for Software Engineering (MTSE) Fall 2017 Prof. Arie Gurfinkel References Chpater 1 of Logic for Computer Scientists http://www.springerlink.com/content/978-0-8176-4762-9/
More informationSolving Quantified Verification Conditions using Satisfiability Modulo Theories
Solving Quantified Verification Conditions using Satisfiability Modulo Theories Yeting Ge, Clark Barrett, Cesare Tinelli Solving Quantified Verification Conditions using Satisfiability Modulo Theories
More informationCombinations of Theories for Decidable Fragments of First-order Logic
Combinations of Theories for Decidable Fragments of First-order Logic Pascal Fontaine Loria, INRIA, Université de Nancy (France) Montreal August 2, 2009 Montreal, August 2, 2009 1 / 15 Context / Motivation
More informationAn Introduction to SAT Solving
An Introduction to SAT Solving Applied Logic for Computer Science UWO December 3, 2017 Applied Logic for Computer Science An Introduction to SAT Solving UWO December 3, 2017 1 / 46 Plan 1 The Boolean satisfiability
More informationLearning 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
More informationTheory Combination. Clark Barrett. New York University. CS357, Stanford University, Nov 2, p. 1/24
CS357, Stanford University, Nov 2, 2015. p. 1/24 Theory Combination Clark Barrett barrett@cs.nyu.edu New York University CS357, Stanford University, Nov 2, 2015. p. 2/24 Combining Theory Solvers Given
More informationLOGIC 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
More informationSums of Products. Pasi Rastas November 15, 2005
Sums of Products Pasi Rastas November 15, 2005 1 Introduction This presentation is mainly based on 1. Bacchus, Dalmao and Pitassi : Algorithms and Complexity results for #SAT and Bayesian inference 2.
More informationA two-tier technique for supporting quantifiers in a lazily proof-explicating theorem prover
A two-tier technique for supporting quantifiers in a lazily proof-explicating theorem prover K. Rustan M. Leino 0, Madan Musuvathi 0, and Xinming Ou 1 0 Microsoft Research, Redmond, WA, USA {leino,madanm@microsoft.com
More informationDecision Procedures for Satisfiability and Validity in Propositional Logic
Decision Procedures for Satisfiability and Validity in Propositional Logic Meghdad Ghari Institute for Research in Fundamental Sciences (IPM) School of Mathematics-Isfahan Branch Logic Group http://math.ipm.ac.ir/isfahan/logic-group.htm
More informationConstraint Solving for Finite Model Finding in SMT Solvers
myjournal manuscript No. (will be inserted by the editor) Constraint Solving for Finite Model Finding in SMT Solvers Andrew Reynolds Cesare Tinelli Clark Barrett Received: date / Accepted: date Abstract
More informationOverview, cont. Overview, cont. Logistics. Optional Reference #1. Optional Reference #2. Workload and Grading
Course staff CS389L: Automated Logical Reasoning Lecture 1: ntroduction and Review of Basics şıl Dillig nstructor: şil Dillig E-mail: isil@cs.utexas.edu Office hours: Thursday after class until 6:30 pm
More informationSymbolic Analysis. Xiangyu Zhang
Symbolic Analysis Xiangyu Zhang What is Symbolic Analysis CS510 S o f t w a r e E n g i n e e r i n g Static analysis considers all paths are feasible Dynamic considers one path or a number of paths Symbolic
More informationConstraint Logic Programming and Integrating Simplex with DPLL(T )
Constraint Logic Programming and Integrating Simplex with DPLL(T ) Ali Sinan Köksal December 3, 2010 Constraint Logic Programming Underlying concepts The CLP(X ) framework Comparison of CLP with LP Integrating
More informationSAT/SMT/AR Introduction and Applications
SAT/SMT/AR Introduction and Applications Ákos Hajdu Budapest University of Technology and Economics Department of Measurement and Information Systems 1 Ákos Hajdu About me o PhD student at BME MIT (2016
More informationPrice: $25 (incl. T-Shirt, morning tea and lunch) Visit:
Three days of interesting talks & workshops from industry experts across Australia Explore new computing topics Network with students & employers in Brisbane Price: $25 (incl. T-Shirt, morning tea and
More informationPropositional Logic: Models and Proofs
Propositional Logic: Models and Proofs C. R. Ramakrishnan CSE 505 1 Syntax 2 Model Theory 3 Proof Theory and Resolution Compiled at 11:51 on 2016/11/02 Computing with Logic Propositional Logic CSE 505
More informationPropositional 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/
More informationPropositional and Predicate Logic. jean/gbooks/logic.html
CMSC 630 February 10, 2009 1 Propositional and Predicate Logic Sources J. Gallier. Logic for Computer Science, John Wiley and Sons, Hoboken NJ, 1986. 2003 revised edition available on line at http://www.cis.upenn.edu/
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 3: Practical SAT Solving Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson SAT Solving 1 / 36 Review: Propositional
More informationKnowledge 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
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT
More informationThe Impact of Craig s Interpolation Theorem. in Computer Science
The Impact of Craig s Interpolation Theorem in Computer Science Cesare Tinelli tinelli@cs.uiowa.edu The University of Iowa Berkeley, May 2007 p.1/28 The Role of Logic in Computer Science Mathematical logic
More informationFirst-Order Theorem Proving and Vampire. Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester)
First-Order Theorem Proving and Vampire Laura Kovács (Chalmers University of Technology) Andrei Voronkov (The University of Manchester) Outline Introduction First-Order Logic and TPTP Inference Systems
More informationFrom SAT To SMT: Part 1. Vijay Ganesh MIT
From SAT To SMT: Part 1 Vijay Ganesh MIT Software Engineering & SMT Solvers An Indispensable Tactic for Any Strategy Formal Methods Program Analysis SE Goal: Reliable/Secure Software Automatic Testing
More informationPropositional 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
More informationFinite model finding in satisfiability modulo theories
University of Iowa Iowa Research Online Theses and Dissertations Fall 2013 Finite model finding in satisfiability modulo theories Andrew Joseph Reynolds University of Iowa Copyright 2013 Andrew J. Reynolds
More informationSatisfiability Modulo Theories
Satisfiability Modulo Theories Clark Barrett and Cesare Tinelli Abstract Satisfiability Modulo Theories (SMT) refers to the problem of determining whether a first-order formula is satisfiable with respect
More informationPROPOSITIONAL LOGIC. VL Logik: WS 2018/19
PROPOSITIONAL LOGIC VL Logik: WS 2018/19 (Version 2018.2) Martina Seidl (martina.seidl@jku.at), Armin Biere (biere@jku.at) Institut für Formale Modelle und Verifikation BOX Game: Rules 1. The game board
More informationInteger vs. constraint programming. IP vs. CP: Language
Discrete Math for Bioinformatics WS 0/, by A. Bockmayr/K. Reinert,. Januar 0, 0:6 00 Integer vs. constraint programming Practical Problem Solving Model building: Language Model solving: Algorithms IP vs.
More information1 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
More informationPropositional Calculus
Propositional Calculus Dr. Neil T. Dantam CSCI-498/598 RPM, Colorado School of Mines Spring 2018 Dantam (Mines CSCI, RPM) Propositional Calculus Spring 2018 1 / 64 Calculus? Definition: Calculus A well
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationSatisability Modulo Structures as Constraint Satisfaction: An Introduction
janvier 2007 Journées Francophones des Langages Applicatifs JFLA07 Satisability Modulo Structures as Constraint Satisfaction: An Introduction Hassan Aït-Kaci 1 & Bruno Berstel 2 & Ulrich Junker 2 & Michel
More informationNotes. 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
More informationIntroduction to SAT (constraint) solving. Justyna Petke
Introduction to SAT (constraint) solving Justyna Petke SAT, SMT and CSP solvers are used for solving problems involving constraints. The term constraint solver, however, usually refers to a CSP solver.
More informationAutomated Program Verification and Testing 15414/15614 Fall 2016 Lecture 8: Procedures for First-Order Theories, Part 2
Automated Program Verification and Testing 15414/15614 Fall 2016 Lecture 8: Procedures for First-Order Theories, Part 2 Matt Fredrikson mfredrik@cs.cmu.edu October 17, 2016 Matt Fredrikson Theory Procedures
More informationPolite Theories Revisited
Polite Theories Revisited Dejan Jovanović and Clark Barrett New York University dejan@cs.nyu.edu, barrett@cs.nyu.edu c Springer-Verlag Abstract. The classic method of Nelson and Oppen for combining decision
More informationPropositional 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
More informationNP-Complete Reductions 2
x 1 x 1 x 2 x 2 x 3 x 3 x 4 x 4 12 22 32 CS 447 11 13 21 23 31 33 Algorithms NP-Complete Reductions 2 Prof. Gregory Provan Department of Computer Science University College Cork 1 Lecture Outline NP-Complete
More informationUsing E-Unification to Handle Equality in Universal Formula Semantic Tableaux Extended Abstract
Using E-Unification to Handle Equality in Universal Formula Semantic Tableaux Extended Abstract Bernhard Beckert University of Karlsruhe Institute for Logic, Complexity und Deduction Systems 76128 Karlsruhe,
More informationDeliberative Agents Knowledge Representation I. Deliberative Agents
Deliberative Agents Knowledge Representation I Vasant Honavar Bioinformatics and Computational Biology Program Center for Computational Intelligence, Learning, & Discovery honavar@cs.iastate.edu www.cs.iastate.edu/~honavar/
More information1 Algebraic Methods. 1.1 Gröbner Bases Applied to SAT
1 Algebraic Methods In an algebraic system Boolean constraints are expressed as a system of algebraic equations or inequalities which has a solution if and only if the constraints are satisfiable. Equations
More informationOn the Complexity of the Reflected Logic of Proofs
On the Complexity of the Reflected Logic of Proofs Nikolai V. Krupski Department of Math. Logic and the Theory of Algorithms, Faculty of Mechanics and Mathematics, Moscow State University, Moscow 119899,
More informationMathematical Logic Part Three
Mathematical Logic Part hree riday our Square! oday at 4:15PM, Outside Gates Announcements Problem Set 3 due right now. Problem Set 4 goes out today. Checkpoint due Monday, October 22. Remainder due riday,
More informationPlanning as Satisfiability
Planning as Satisfiability Alan Fern * Review of propositional logic (see chapter 7) Planning as propositional satisfiability Satisfiability techniques (see chapter 7) Combining satisfiability techniques
More informationWorst-Case Upper Bound for (1, 2)-QSAT
Worst-Case Upper Bound for (1, 2)-QSAT Minghao Yin Department of Computer, Northeast Normal University, Changchun, China, 130117 ymh@nenu.edu.cn Abstract. The rigorous theoretical analysis of the algorithm
More informationCombining Non-Stably Infinite Theories
Combining Non-Stably Infinite Theories Cesare Tinelli Calogero G. Zarba 1 tinelli@cs.uiowa.edu zarba@theory.stanford.edu Department of Computer Science The University of Iowa Report No. 03-01 April 2003
More informationThe Wumpus Game. Stench Gold. Start. Cao Hoang Tru CSE Faculty - HCMUT
The Wumpus Game Stench Stench Gold Stench Start 1 The Wumpus Game Stench in the square containing the wumpus and in the directly adjacent squares in the squares directly adjacent to a pit Glitter in the
More informationChapter 7 R&N ICS 271 Fall 2017 Kalev Kask
Set 6: Knowledge Representation: The Propositional Calculus Chapter 7 R&N ICS 271 Fall 2017 Kalev Kask Outline Representing knowledge using logic Agent that reason logically A knowledge based agent Representing
More informationTitle: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5)
B.Y. Choueiry 1 Instructor s notes #12 Title: Logical Agents AIMA: Chapter 7 (Sections 7.4 and 7.5) Introduction to Artificial Intelligence CSCE 476-876, Fall 2018 URL: www.cse.unl.edu/ choueiry/f18-476-876
More informationVerification using Satisfiability Checking, Predicate Abstraction, and Craig Interpolation. Himanshu Jain THESIS ORAL TALK
Verification using Satisfiability Checking, Predicate Abstraction, and Craig Interpolation Himanshu Jain THESIS ORAL TALK 1 Computer Systems are Pervasive Computer Systems = Software + Hardware Software/Hardware
More informationCS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati
CS 514, Mathematics for Computer Science Mid-semester Exam, Autumn 2017 Department of Computer Science and Engineering IIT Guwahati Important 1. No questions about the paper will be entertained during
More informationCS:4420 Artificial Intelligence
CS:4420 Artificial Intelligence Spring 2018 Propositional Logic Cesare Tinelli The University of Iowa Copyright 2004 18, Cesare Tinelli and Stuart Russell a a These notes were originally developed by Stuart
More information