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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
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- 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
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- 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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- 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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- Klassenhierarchie (I) class Point {
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- Reconsidered terminating.
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- 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
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- 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
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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
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- 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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- What is Automated Deduction? Automated deduction
- inconsistent, inconsistent
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- Problem 1 (Algebras and semantics) (3 + 4 + 4 = 11 points) ; ) be a signature such
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- 3.4 Critical Pairs Showing local confluence (Sketch)
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- Problem 1 (DPLL) (10 points) Prove the (un-)satis ability of the following set of propositional clauses us-
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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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- composition substitutions
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- 1.4 Ordered Binary Decision Diagrams see Chapter 6.1/6.2 of Michael Huth and Mark Ryan: Logic in
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- Checking Unsatisfiability Unsatisfiability of finite sets of first-order formulas (or clauses)
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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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- 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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- 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
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- 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
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- Simplification Simplification
- expression). Contracting
- Termination: well-founded,
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- 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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- Problem 1 (File System) (4 + 5 + 5 = 14 points) What are the outputs of the following command sequences? If several outputs
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- Algorithmic Satisfiability(F
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- 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
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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