A brief Introduction to Automated Theorem Proving

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A brief Introduction to Automated Theorem Proving

• Theoretical Foundations, History and the
  Resolution Calculus for classical First-order
  Logic
• Uwe Keller

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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

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Part IIntroduction

• Motivation History
• Theorem Proving, ATP and Calculi

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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
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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 1950s 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!

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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

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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 - )

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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?

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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

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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

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Example

• Theorem in (elementary) Calculus
• Nullstellensatz Every function which is
  continous over a closed interval Ia,b must
  take the value 0 somewhere in I if f(a) lt 0 and
  f(b) gt 0
• Proof idea Consider the Supremum l of set M
  x f(x) lt 0, altxltb and show that f(l) 0

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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

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Part IIFoundations

• FOL, Normalforms Preprocessing, Metaresults

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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

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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!

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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!

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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

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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) Ç 9 z (P(z) Æ
  Q(z)))

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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

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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 ( (L1,1 Ç Ç
  L1,m1) Æ Æ (Lk,1 Ç 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

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Clause Normal Form (II)

• A formula A is in Clause NF can be written as A
  8 x1 8 xn ( (L1,1 Ç Ç L1,m1) Æ Æ (Lk,1 Ç
  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 -gt drop quantors
• Matrix is in CNF Conjunction of Disjunction of
  Literals -gt 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

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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

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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

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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!

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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

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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 Herbrands 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!)

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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?

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Part IIIThe Resolution Calculus

• Pre-resolution phase
• Gilmores Methods, Davis-Putnam Procedure
• Unification
• Basic Resolution Calculus
• Refinements, Redundancy
• Decision procedures

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Pre-Resolution period Gilmores Method
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Pre-Resolution periodDavis-Putnam Procedure
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A Revolution in ATP Robinsons Resolution
Principle
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Part IVChain Resolution

• A Variant of Resolution for the Semantic Web

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Part IVDemo

• assisted by a Resolution-based ATP System
  VAMPIRE