Waldmann, Uwe - Max-Planck-Institut für Informatik

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• I N F O R M A T I K Superposition and Chaining
• Theorem (Model construction): Let N be a set of clauses that is sat urated up to redundancy and does not contain the empty clause. Then we
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• Recapitulation: (\precedence")
• Theorem (Model construction): Let N be a set of clauses that is sat-urated up to redundancy and does not contain the empty clause. Then we
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• An Extension of the Knuth-Bendix Ordering with LPO-like Properties
• Part 2: FirstOrder Logic #Firstorder logic
• Praxis des Programmierens Vorlesung 13.07.1999
• Literature: FirstOrder
• Superposition and Chaining for Totally Ordered Divisible Abelian Groups
• 2 Events, characters, codes, glyphs As computer users, we tend to identify
• L A T E X 2 " -Kurzbeschreibung J org Knappen
• Uwe Waldmann Saarbr ucken, 31.5.1999 Ubungen zur Vorlesung
• 3.8 Inference Systems and Proofs Inference systems # (proof calculi) are sets of tuples
• Assignment 1 (DPLL) (10 points) Let N be the following set of propositional clauses
• Einfuhrung in die Informatik fur Horer aller Fakultaten II
• Automatic Verification of Hybrid Systems with Large Discrete State Space
• Problem 1 (Algebras and semantics) (8 points) ; ) be a signature, let p(t 1 ; : : : ; t n ) be a ground -atom, and let
• Problem 1 (OBDDs) (10 points) Let = fP; Qg be a set of propositional variables; let P < Q be an ordering
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• Typen und Klassen Variable bzw. Ausdruck
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• 3 File system The material of this section is taken from E. Nemeth, G. Snyder, S. Seebass, T. R.
• Problem 1 (Algebras and semantics) (8 points) Let = (, ) be a signature, let p(t1, . . . , tn) be a ground -atom, and let
• 5 Termination Termination
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• Fenster in Java AWT: Abstract Window Toolkit
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• Termination Termination
• 2.11 General Resolution Propositional resolution
• Automated Reasoning Harald Ganzinger
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• Comparing Instance Generation Methods for Automated Reasoning #
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• 6 Shell programming There is no script for this section. Have a look at the commented shell script(s) on the
• 2 Propositional Logic Propositional logic
• Klassenhierarchie (I) class Point {
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• Reconsidered terminating.
• 3 FirstOrder Logic #Firstorder logic
• Problem 1 (Algebras and semantics) (4 + 4 = 8 points) Prove the following statement: If F and G are rst-order formulas and F ! G
• Simpli cation Simpli cation
• Problem 1 (Semantics) (14 points) Show that the following inference rule is sound
• Wiederholung Klassenhierarchie
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• Draft of \Cancellative Abelian Monoids and Related Structures in Refutational Theorem Proving, Part I and II", to appear in Journal of Symbolic Computation
• Hillenbrand
• 3.12 Ordered Resolution with Selection Motivation: Search space for Res very large.
• Knuth-Bendix Completion
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• Introductory Introductory
• Problem 1 (Uni cation) (6 points) For each of the following uni cation problems, compute either an mgu or show
• WellFounded Literature
• procedures. 21(2):211{243,
• MAXPLANCKINSTITUT \Omega \Psi
• Cancellative Abelian Monoids in Refutational Theorem Proving
• Uwe Waldmann Saarbr ucken, 20.4.1999 Ubungen zur Vorlesung
• Automated Reasoning Harald Ganzinger
• Fehlerbehandlung Fehlerbehandlung
• 3.3 Models, Validity, and Satisfiability F is valid in A under assignment #
• FirstOrder mathematics
• Theorem Proving in Cancellative Abelian Monoids (Extended Abstract) ?
• Implementation Refutational
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• 3.13 A Resolution Prover So far: static view on completeness of resolution
• 2.5 The DPLL Procedure Given a propositional formula in CNF (or alternatively, a finite set N of clauses), check
• Objektorientierte Programmierung
• Objektorientierte Programmierung Geschichte
• 8 Introduction to Perl There is no script for this section. Have a look at the commented Perl transcript on the
• Aufgabe 1 (Algebren und Semantik) (8 Punkte) ; ) eine Signatur, sei p(t 1 ; : : : ; t n ) ein -Grundatom, und sei N
• Wiederholung: Datentypen).
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• Automated Reasoning Uwe Waldmann
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• 3.13 A Resolution Prover So far: static view on completeness of resolution
• 3 File system The material of this section is taken from E. Nemeth, G. Snyder, S. Seebass, T. R.
• What is Automated Deduction? Automated deduction
• inconsistent, inconsistent
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• Problem 1 (Semantics) (14 points) Show that the following inference rule is sound
• I N F O R M A T I K Cancellative Superposition
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• Automated Reasoning # Uwe Waldmann
• Superposition completeness
• Problem 1 (Algebras and semantics) (3 + 4 + 4 = 11 points) ; ) be a signature such
• EnglishGerman Dictionary of Deductionrelated Terms
• Knuth-Bendix Completion: Outlook Classical completion
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• Rewrite Relations Let E be a set of equations.
• 3.4 Critical Pairs Showing local confluence (Sketch)
• Wiederholung Nachrichten,
• Problem 1 (DPLL) (10 points) Prove the (un-)satis ability of the following set of propositional clauses us-
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• Unsatis ability Unsatis ability
• Problem 1 (Abstract Reduction Systems) (5 points) Prove: There is no abstract reduction system (A; !) with A 6= ; such that
• 6 Shell programming There is no script for this section. Have a look at the commented shell script(s) on the
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• Theorem (Model construction): Let N be a set of clauses that is sat-urated up to redundancy and does not contain the empty clause. Then we
• Problem 1 (File System) (4 + 5 + 5 = 14 points) What are the outputs of the following command sequences? If several outputs
• 4 Unification and Critical Pairs Unification
• Wiederholung: Zeiger in C In C (und C++)
• Verkettete Datenstrukturen: Listen Liste = endliche Folge von Elementen [a1, a2, . . . , an].
• 2.7 Example: Sudoku 1 2 3 4 5 6 7 8 9
• 2 Events, characters, codes, glyphs As computer users, we tend to identify
• page 2, beginning of Sect. 1.1: replace ``A is a set'' by ``A is a nonempty set''
• Superposition: Refutational
• Java = C + Klassen (?) (Fast) wie in C (I)
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• Modular Proof Systems for Partial Functions with Weak Equality
• composition substitutions
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• Propositional resolution
• 3.4 Critical Pairs Showing local confluence (Sketch)
• 1.4 Ordered Binary Decision Diagrams see Chapter 6.1/6.2 of Michael Huth and Mark Ryan: Logic in
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• Modular Proof Systems for Partial Functions with Evans Equality
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• Checking Unsatisfiability Unsatisfiability of finite sets of first-order formulas (or clauses)
• page 2, beginning of Sect. 1.1: replace "A is a set" by "A is a non-empty set"
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• 2.15 Semantic Tableaux Literature
• Motivation: improvement
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• WALDMEISTER Hillenbrand
• Praxis des Programmierens Vorlesung: Dienstags, 14:15{16:00 Uhr,
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• 1.6 The DPLL Procedure Given a propositional formula in CNF (or alternatively, a finite
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• 3.10 Refutational Completeness of Resolution How to show refutational completeness of propositional resolution
• 2.7 Example: Sudoku 1 2 3 4 5 6 7 8 9
• 2.10 Refutational Completeness of Resolution How to show refutational completeness of propositional
• Recapitulation: AC1-uni cation
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• Part 4: Implementation Issues Refutational completeness is nice in theory, but . . .
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• Uwe Waldmann Saarbr ucken, 27.4.1999 Ubungen zur Vorlesung
• If N is inconsistent, then N | # | # # # N # # {#} | | . Does this imply that every derivation starting from an
• 3.16 Other Inference Systems . Instantiationbased methods
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• Operational Semantics of Constraint Logic Programs
• Organisatorisches Donnerstags,
• Datenstrukturen: Spezialfall
• Refutational Completeness
• 2. Objektorientierung Dahl, Nygaard: Simula 67
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• 4 FirstOrder Logic with Equality Equality is the most important relation in mathematics and functional programming.
• graphischer Dateimanager
• Uwe Waldmann Saarbr ucken, 22.6.1999 Ubungen zur Vorlesung
• Superposition: Refutational Completeness A -interpretation A is called term-generated, if for every b UA
• Problem 1 (OBDDs) (10 points) Let = {P, Q} be a set of propositional variables; let P < Q be an ordering
• Unification in Order-Sorted Signatures Uwe Waldmann
• Simplification Simplification
• expression). Contracting
• Termination: well-founded,
• Uwe Waldmann Saarbr ucken, 20.7.1999 Ubungen zur Vorlesung
• 2 Propositional Logic Propositional logic
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• 3.6 Getting Small Skolem Functions A clause set that is better suited for automated theorem proving can be obtained using
• 2.5 The DPLL Procedure Given a propositional formula in CNF (or alternatively, a finite set N of clauses), check
• Assignment 1 (DPLL) (10 points) Let N be the following set of propositional clauses
• Wiederholung Klassen, Objekte, Nachrichten, Methoden
• Superposition and Model Evolution Combined Peter Baumgartner
• Virgile Prevosto1 and Uwe Waldmann2
• A New Input Technique for Accented Letters in Alphabetical Scripts
• Feedback der Veranstaltungsevaluation -SS 2011 Automated Reasoning
• Proving Termination: Monotone Mappings Let (A, >A) and (B, >B ) be partial orderings.
• The Superposition Calculus One problem
• Problem 1 (Unification) (6 points) For each of the following unification problems, compute either an mgu or show
• Problem 1 (Abstract Reduction Systems) (5 points) Prove: There is no abstract reduction system (A, ) with A = such that
• Java Fehlerbehandlung Fehlerbehandlung
• 3 First-Order Logic First-order logic
• 3.3 Models, Validity, and Satisfiability F is valid in A under assignment
• 3.6 Getting Small Skolem Functions A clause set that is better suited for automated theorem proving can be obtained using
• 3.8 Inference Systems and Proofs Inference systems (proof calculi) are sets of tuples
• 3.10 Refutational Completeness of Resolution How to show refutational completeness of propositional resolution
• 3.12 Ordered Resolution with Selection Motivation: Search space for Res very large.
• 3.16 Other Inference Systems Instantiation-based methods
• 4 First-Order Logic with Equality Equality is the most important relation in mathematics and functional programming.
• Unix for Advanced Users Uwe Waldmann
• 8 Introduction to Perl There is no script for this section. Have a look at the commented Perl transcript on the
• 1.4 Ordered Binary Decision Diagrams see Chapter 6.1/6.2 of Michael Huth and Mark Ryan: Logic in
• 1.6 The DPLL Procedure Given a propositional formula in CNF (or alternatively, a finite
• 2.9 Well-Founded Orderings Literature: Franz Baader and Tobias Nipkow: Term rewriting
• If N is inconsistent, then N | | N {} | | .
• 2.16 Other Inference Systems Instantiation-based methods for FOL
• Simplification Orderings The proper subterm ordering is defined by s t if and only if
• 3.7 Superposition Combine the ideas of ordered resolution (overlap maximal
• Theorem (Model construction): Let N be a set of clauses that is sat-urated up to redundancy and does not contain the empty clause. Then we
• Problem 1 (DPLL) (10 points) Prove the (un-)satisfiability of the following set of propositional clauses us-
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• Problem 1 (Algebras and semantics) (3 + 4 + 4 = 11 points) Let = (, ) be a signature such that contains at least one constant
• Aufgabe 1 (Algebren und Semantik) (8 Punkte) Sei = (, ) eine Signatur, sei p(t1, . . . , tn) ein -Grundatom, und sei N
• Problem 1 (Algebras and semantics) (4 + 4 = 8 points) Prove the following statement: If F and G are first-order formulas and F G
• English-German Dictionary of Deduction-related Terms
• Recursive Path Orderings Recapitulation
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• Simplification Orderings The proper subterm ordering is defined by s t if and only if
• Problem 1 (File System) (4 + 5 + 5 = 14 points) What are the outputs of the following command sequences? If several outputs
• I N F O R M A T I K Universit at
• I N F O R M A T I K Universit at
• A New Input Technique for Accented Letters in Alphabetical Scripts
• Algorithmic Satisfiability(F
• Unix for Advanced Users # Uwe Waldmann
• Wiederholung Klassenhierarchie
• Instantiationbased instantiation;
• 3.7 Superposition Combine the ideas of ordered resolution (overlap maximal
• Automated Deduction for Equational Uwe Waldmann
• 6 Termination Revisited So far: Termination as a subordinate task for entailment checking.
• 4.7 Unfailing Completion Classical completion
• 5 Implementing Saturation Procedures Refutational completeness is nice in theory, but . . .
• 4.5 Termination Termination problems
• Further topics in automated reasoning. 7.1 Satisfiability Modulo Theories (SMT)
• Assignment 1 (CNF) (10 points) Transform the formula
• 6.3 Reduction Pairs and Argument Filterings Goal: Show the non-existence of K-minimal infinite rewrite sequences
• 4.6 Knuth-Bendix Completion Completion
• 4.5 Termination Termination problems
• Assignment 1 (CNF) (10 points) Transform the formula
• 4.6 KnuthBendix Completion Completion
• 6.3 Reduction Pairs and Argument Filterings Goal: Show the nonexistence of Kminimal infinite rewrite sequences
• 5 Implementing Saturation Procedures Refutational completeness is nice in theory, but . . .
• Further topics in automated reasoning. 7.1 Satisfiability Modulo Theories (SMT)
• 6 Termination Revisited So far: Termination as a subordinate task for entailment checking.
• 4.7 Unfailing Completion Classical completion
• Assignment 1 (Propositional Logic) (10 points) Let F[G H] be a propositional formula that contains G H as a subformula
• Universitat FR Informatik
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• 1.8 Combining Decision Procedures Let T1 and T2 be first-order theories over the signatures 1 and 2.
• Automated Reasoning II Uwe Waldmann
• 2 Satisfiability Modulo Theories (SMT) decision procedures for satisfiability for various fragments of firstorder theories;
• Automated Reasoning II # Uwe Waldmann
• 1.8 Combining Decision Procedures Let T 1 and T 2 be firstorder theories over the signatures # 1 and # 2 .
• The Nelson--Oppen Algorithm (Deterministic Version for Convex Theories) if ##x F i is unsatisfiable w. r. t. T i for some i.
• Assignment 1 (Propositional Logic) (10 points) Let F [G #H] be a propositional formula that contains G#H as a subformula
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• 1.3 Linear Rational Arithmetic There are several ways to define linear rational arithmetic.
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• The NelsonOppen Algorithm (Deterministic Version for Convex Theories) if x Fi is unsatisfiable w. r. t. Ti for some i.
• 2 Satisfiability Modulo Theories (SMT) decision procedures for satisfiability for various fragments of first-order theories;
• 1.3 Linear Rational Arithmetic There are several ways to define linear rational arithmetic.
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• 3.6 Constraint Superposition Refutational completeness proof for superposition is based on the analysis of inferences
• 3.8 Integrating Theories I: EUnification Dealing with mathematical theories naively in a superposition prover is di#cult
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• Existentiallyquantified LRA So far, we have considered formulas that may contain free, existentially quantified, and
• 3.5 Improvements and Refinements The superposition calculus as described so far can be improved and refined in several
• 2.2 Heuristic Instantiation DPLL(T) is limited to ground (or existentially quantified) formulas. Even if we have
• 3 Superposition Firstorder calculi considered so far
• 3.6 Constraint Superposition Refutational completeness proof for superposition is based on the analysis of inferences
• 2.2 Heuristic Instantiation DPLL(T) is limited to ground (or existentially quantified) formulas. Even if we have
• 3.8 Integrating Theories I: E-Unification Dealing with mathematical theories naively in a superposition prover is difficult
• Existentially-quantified LRA So far, we have considered formulas that may contain free, existentially quantified, and
• 3 Superposition First-order calculi considered so far
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• 3.5 Improvements and Refinements The superposition calculus as described so far can be improved and refined in several