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A brief Introduction to Automated Theorem Proving

A brief Introduction to Automated Theorem Proving. Theoretical Foundations, History and the Resolution Calculus for classical First-order Logic Uwe Keller. Content. Intoduction Motivation & History Theorem Proving, ATP and Calculi Foundations FOL, Normalforms & Preprocessing, Metaresults

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A brief Introduction to Automated Theorem Proving

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  1. A brief Introduction to Automated Theorem Proving Theoretical Foundations, History and the Resolution Calculus for classical First-order Logic Uwe Keller

  2. Content • Intoduction • Motivation & History • Theorem Proving, ATP and Calculi • Foundations • FOL, Normalforms & Preprocessing, Metaresults • Resolution • Basic calculus, Unification • Refinements, Redundancy • Decision procedures • Chain Resolution • A Variant of Resolution for the Semantic Web • Demo

  3. Part I:Introduction Motivation & History Theorem Proving, ATP and Calculi

  4. (automated) Deduction Modelling Logic and Theorem Proving Real-world description in natural language. Mathematical Problems Program + Specification Formalization Syntax (formal language). First-order Logic, Dynamic Logic, … Semantics (truth function) Calculus (derivation / proof) Correctness Valid Formulae Provable Formulae Completeness

  5. How did it start … • Results from first-half of the 20th century in mathematical logic showed … • we can do logical reasoning with a limited set of simple (computable) rules in restricted formal languages like First-order Logic (FOL) • That means computers can do reasoning! • Implementation of ATP • First: Computers where needed :- ) • AI as a prominent field: Reasoning as a basic skill! • Mid 1950‘s first attempts to implement an ATP • Today • (A)TP is no longer only a part of main stream AI • Central shared problem: How to represent and search extremely large search spaces!

  6. A rough timeline in ATP … • before 1950: Proof-theoretic Work by Skolem, Herbrand, Gentzen and Schütte • 1954: First machine-generated Proof (Davis) • 1955ff: Semantic Tableaus (Beth, Hinitkka) • 1957: First machine-generated Proof in Logic Calculus (Newell & Simon) • 1957: Lazy substitution by free (dummy) Vars (Kanger, Prawitz) • 1958: First prover for Predicate Logic (Prawitz) • 1959: More provers (Gilmore, Wang) • 1960: Davis-Putnam Procedure (Davis, Putnam, Longman) • 1963: Unification (J.A. Robinson) • 1963ff: Resolution (J.A. Robinson); Inverse Method (Maslov) • 1963ff: Modern Tableau Method (Smullyan, Lis) without Unification • 1968: Modelelimination (Loveland), with Unification • 1970ff: PROLOG (Colmerauer, Kowalski), Refinements of Resolution • 1971: Connection Method (Bibel), Matings (Andrews) with Unification • 1985: ATP in non-classical logics, Renaissance of Tableaux Methods • 1987: Tableaus with Unification • 1993ff: Renewed interest in Instance-based Methods: DPLL, Modelevolution • …

  7. Theorem Proving • Given • a formal language (or logic) L • a calculus C for this language (= set of rules) • a conjecture S and a set of assumptions or axioms A in the language L • Determine • Can we construct a proof for S (from A) in calculus C? • Logic = Syntax + Semantics + Calculus • TP = Proof-search in C (Huge search problem) • Correctness and completeness of Calculi essential properties • Calculus = Non-deterministic Algorithm • Central problem in ATP: How to implement a non-deterministic algorithm „efficiently“ on a deterministic machine :- )

  8. Theorem Proving (II) • Research areas • Interactive / tactic TP vs. Automated TP • Classical Logic vs. Non-classical logics • Calculi for … • ATP - General principle: Refutation • Resolution, Tableau, Inverse Method, Instance-based Methods • ITP – General principle: Proof situation / context • Sequent Calculi • others – General principle: Generation from Axioms • Hilbert-style Calculi • Central difference: • What are the elements in a proof & what is a proof?

  9. Main TP Applications • Main Applications • Software & Hardware Verification • Theorem proving in Mathematics • Query answering in rich knowledge bases (Ontologies) • Verification of cryptographic protocols • Retrieval of Software Components • Reasoning in non-classical Logics • Program synthesis … • … many systems implemented • ATP: Vampire, Otter, Spass, E-SETHEO, Darwin, Epilog, SNARK, Gandalf … • ITP: Isabelle/HOL, Coq, Theorema, KeY-Prover …

  10. Why is FOL of special interest in the ATP community ? • There are less & more expressive logics than FOL • Classical Propositional Logic, Modal Propositional Logic, Description Logics, Temporal Propositional Logic • Higher-order Predicate Logics, Dynamic Predicate Logics, Type Theory • Research in ATP mainly focused on FOL • FOL is very expressive, many real-world problems can be formalized in FOL • FOL turned out to be the most expressive logic that one can adequately approach with ATP techniques

  11. Example … • Theorem in (elementary) Calculus • Nullstellensatz: Every function which is continous over a closed interval I=[a,b] must take the value 0 somewhere in I if f(a) <= 0 and f(b) >= 0 • Proof idea: Consider the Supremum l of set M = {x : f(x) <= 0, a<=x<=b} and show that f(l) = 0

  12. Example (II) … • Formalization • Compact (only LEQ) • Redundancy-free • Specific definitions • Continous functions • Main idea of proofis already encoded • Use Supremum • Can be done by anATP system • … but without properFormalization ?!? • ATP better than humanprover? Robbins Problem in Algebra • Intelligent Proving vs.Combinatorical proving

  13. Part II:Foundations FOL, Normalforms & Preprocessing, Metaresults

  14. Classical First-order Logic (FOL) • Syntax • Signature § • Function Symbols, Predicate Symbols, Arity, logical Connectives, Quantors • Terms (over §), Atomic Formulae (over §), Formluae (over §) • Definition relative to the signature § of the predicate logic • Semantics • First-order structure / interpretation S = (U,I) • Universe U + Signature-Interpretation I • Constants I(c) = element of U • Functionsymbols I(f) = total functions on U • Relationsymbols I(R) = relation on U • Logical connectives and quantors in the usual way • Definition relative to the signature § of the predicate logic

  15. Classical FOL (II) • Model of a statement • An interpretation S = (U,I) is called a model of a statement s iff valS(s) = t • What does it mean to infer a statement from given premisses? • Informally: Whenever our premisses P hold it is the case that the statement holds as well • Formally: Logical Entailment • For every interpretation S which is a model of P it holds that S is a model of S as well • Special case: Validity – Set of premisses is empty • Logical entailment in a logic L is the (semantic) relation that a calculus C aims at formalizing syntactically (by means of a derivability relation)! • Logical entailment considers semantics (Interpretations) relative to a set of premisses or axioms!

  16. Normal Forms • What is a normal form? • Why are they interesting? • Relation to ATP? • Conversion of input to a specifc NF my be required by a calculus (e.g. Resolution) )Preprocessing step • ATP in a sense can be seen as a conversion in a NF itself, borderline is fuzzy in a sense • Normalforms in FOL • Negation Normal Form • Standard Form • Prenex Normal Form • Clause Normal Form (in a sense a „logic free“ form) • There are logics where certain NF do not exist, like CNF in a Dynamic First-order Logic • Certain calculi then can not be applied in these logics!

  17. Negation Normal Form • A formula is in Negation NF (NNF) iff. it contains no implication and no bi-implication symbols and all negation symbols occur only as part of a literal (directly in front of atomic formulae) • How to achieve this NF ? • Replace implication and bi-implication by their definition (in terms of Æ and Ç) • Move negation symbols inside to atomic formulae • De Morgan laws • Dualize quantifiers when moving negation symbols over a quantor • Eliminate multiple negations • All these syntactical transformations generate semantically equivalent formulae • Example

  18. Standard Form • A formula A is in Standard Form if no variable x in A occurs both bound and free and no bound variable is used as a quantor variable for multiple subformulae • How to generate this NF? • Bounded renaming of quantor variables and the respective occurrences • Transformed formulae is semantically equivalent to original one • Example (8 x P(x) Æ Q(z)) ! (9 x R(x) Ç9 z (P(z) Æ Q(z)))

  19. Prenex Normal Form • A formula A is in Prenex NF iff. it is of the form A = Q1x1 … Qnxn B where Qk is a universal or existential quantor and B contains no quantors. B is called the Matrix of A • How to construct this NF? • Transform A in NNF and Standard Form • Move iteratively outermost quantor to the outside until it reaches another quantor. Quantors may not cross quantors of different sort (in-scope relation between quantor occurrences may not be changed) • This transformation generates a formulae which is logically equivalent to the original one. • Example

  20. Clause Normal Form • A formula A is in Clause NF iff. it is in PNF, closed, the prefix only contains universal quantors and the Matrix is on conjunctive normal form. • In other words: A = 8 x1 … 8 xn ( (L1,1Ç … Ç L1,m1) Æ … Æ (Lk,1Ç … Ç Lk,mk)) where Li,j is a literal (negated or positive atomic formula) • How to construct this NF? • Transform A in NNF and Standard Form • Transform result in PNF • Remove existential quantors by Skolemization (Function terms) • Apply Distributivity laws to convert Matrix of the result in conjuntive normal form (conjunction of discjunction of literals) • This transformation results in a formula which is not logically equivalent, but it is satisfiability-preserving (which is enough for the ATP methods later) • Example

  21. Clause Normal Form (II) • A formula A is in Clause NF can be written as A = 8 x1 … 8 xn ( (L1,1Ç … Ç L1,m1) Æ … Æ (Lk,1Ç … Ç Lk,mk)) where Li,j is a literal (negated or positive atomic formula) • Since every formula can be transformed into CNF, the CNF can be seen as „logic free“ representation of a formulae • All quantors are universal, no free variables are allowed -> drop quantors • Matrix is in CNF = Conjunction of Disjunction of Literals -> Model as a Set of Sets of Literals • Example • The sketched transformation to CNF is not optimal • Exponential blowup possible (already for NNF) • Syntactical structure of the original formula gets lost • Skolemsymbols have unnecessarily many parameters • Unnecessarily many new skolem systems are introduced • One can improve all these aspects of a transformation to CNF! • Skolemization before PNF transformation, Definitorial CNF for Matrix, Reuse of Skolem functions

  22. Metaresults • Metaresult = Property of a Logic L • Here some metaresults for FOL which form the theoretical foundation of ATP (carry over to many other logics as well) • Deduction Theorem • If M [ s ² s‘ then M ² s‘ ! s • Logical entailment can be reduced to validity • Proof by contradiction • If M is a set of closed formulae thenM ² s iff. M [ {¬s} is unsatisfiable (i.e. has no model) • Logical entailment can be reduced to unsatisfiability checking • Refutation can be used as a universal principle for inference in FOL

  23. Metaresults (II) • Complexity of logical entailment, validity and satisfiability • Propositional Logic • Logical entailment (²-relation) is decidable, Satisfiability too • Set of valid formulae is co-NP-complete • Set of satisfiable formulae is NP-complete • First-order Predicate Logic • Logical entailment / validity / satisfiability is undecidable • Set of valid formulae is semi-decidable (recursively enumerable) • Set of satisfiable formulae is not recursively enumerable

  24. Metaresults (III) • Term Interpretations and Herbrand Theorem • S = (U,I) is term-interpretation if U = Term0 • Let Term0 be non-empty. An interpretation S = (U,I) is called Herbrand-Interpretation if • S is term-interpretation and • I(f)(t1,…,tn) = f(t1,…,tn) for all n-ary function symbols f 2 and ground terms t1,…,tn • Herbrand-Modell of s is Herbrand-Intp. I with I ² s • Herbrand-Interpretations are special because they have a simple universe (syntactical) and Terms are basically uninterpreted. Quantifiers then have ground terms as their range! • Computers can deal with such special (syntactical) interpretations, but not with interpretations in general!

  25. Metaresults (IV) • Term Interpretations and Herbrand Theorem • Let M be a set of closed formulae s in Prenex-Normalform that contain no existential quantors (for instance s in CNF) • Let T be a set of terms (over signature ) • T(M) := set of T-instances of M, i.e. replace every occurence of a (universal) variable in any formulae in M with any term in T • Herbrand Theorem • Let Term0 be non-empty and M a set of formulae in Prenex-NF without existential quantors. • Then the following statements are equivalent • M has a model • M has a Herbrand-model • Term0(M) has a model • The last set is a set of formulae in propositional logic

  26. Metaresults (V) • Compactness of FOL • A (possibly infinite) set M of formulae has a model iff every finite subset M‘ ½ M has a model (i.e. is satisfiable) • Combining Compactness with Herbrand‘s Theorem • Let Term0 be non-empty and M a set of formulae in Prenex-NF without existential quantors. • Then M is unsatisfiable iff. T(M) is unsatisfiable for a finite set of ground terms T ½ Term0 • Note that T is a finite set of ground terms over the signature  of the formula set M • No „external“ functions symbols have to be considered! • Allows for using guided substitutions (Unification!)

  27. Metaresults (VI) • That means: logical entailment / validity can be checked • by reduction to unsatisfiabiliy of a set of formulae M‘ • which can done by finding suitable finite (counter)-examples for the quantfied variables such that a contradiction arises • One can only use the Signature  of the given set M‘ to find the counterexamples • Basically this is what all ATP procedures do: Find a finite set of counterexamples (objects) such that a the instance of the orginial set is determined as being • The theorem immediately gives an algorithm for ATP! • Problem: How to construct / find T in the theorem in a clever way?

  28. Part III:The Resolution Calculus Pre-resolution phase: Gilmore‘s Methods, Davis-Putnam Procedure Unification Basic Resolution Calculus Refinements, Redundancy Decision procedures

  29. Pre-Resolution period: Gilmore‘s Method

  30. Pre-Resolution period:Davis-Putnam Procedure

  31. A Revolution in ATP: Robinson‘s Resolution Principle

  32. Part IV:Chain Resolution A Variant of Resolution for the Semantic Web

  33. Part IV:Demo assisted by a Resolution-based ATP System: VAMPIRE

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