Download Logic

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
 Resolution
 Basic calculus, Unification
 Refinements, Redundancy
 Decision procedures
 Chain Resolution
 A Variant of Resolution for the Semantic Web
 Demo
Part I:
Introduction
 Motivation & History
 Theorem Proving, ATP and Calculi
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)
Valid
Formulae
Modelling
Calculus
(derivation / proof)
Correctness
Completeness
Provable
Formulae
(automated)
Deduction
How did it start …
 Results from first-half of the 20th century in
mathematical logic showed …
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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
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First: Computers where needed :- )
AI as a prominent field: Reasoning as a basic skill!
Mid 1950‘s first attempts to implement an ATP
 Today
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(A)TP is no longer only a part of main stream AI
Central shared problem: How to represent and search
extremely large search spaces!
A rough timeline in ATP …
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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
…
Theorem Proving
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Given
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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
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Can we construct a proof for S (from A) in calculus C?
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Logic = Syntax + Semantics + Calculus
TP = Proof-search in C (Huge search problem)
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Correctness and completeness of Calculi essential properties
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Calculus = Non-deterministic Algorithm
Central problem in ATP: How to implement a non-deterministic
algorithm „efficiently“ on a deterministic machine :- )
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?
Main TP Applications
 Main Applications
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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 …
Why is FOL of special interest in
the ATP community ?
 There are less & more expressive logics than FOL
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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
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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
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
Example (II) …
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Formalization
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Compact (only LEQ)
Redundancy-free
Specific definitions
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Main idea of proof
is already encoded
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Continous functions
Use Supremum
Can be done by an
ATP system
… but without proper
Formalization ?!?
ATP better than human
prover? Robbins Problem
in Algebra
Intelligent Proving vs.
Combinatorical proving
Part II:
Foundations
 FOL, Normalforms & Preprocessing,
Metaresults
Classical First-order Logic (FOL)
 Syntax
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Signature §
 Function Symbols, Predicate Symbols, Arity, logical
Connectives, Quantors
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Terms (over §), Atomic Formulae (over §), Formluae (over §)
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Definition relative to the signature § of the predicate logic
 Semantics
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First-order structure / interpretation S = (U,I)
 Universe U + Signature-Interpretation I
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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
Classical FOL (II)
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Model of a statement
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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?
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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!
Normal Forms
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What is a normal form?
Why are they interesting?
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Relation to ATP?
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Normalforms in FOL
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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
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
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Certain calculi then can not be applied in these logics!
Negation Normal Form
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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)
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How to achieve this NF ?
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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
Standard Form
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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
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How to generate this NF?
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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)))
Prenex Normal Form
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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
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How to construct this NF?
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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
Clause Normal Form
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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?
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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
Clause Normal Form (II)
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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)
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Since every formula can be transformed into CNF, the CNF can be seen as „logic
free“ representation of a formulae
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All quantors are universal, no free variables are allowed -> drop quantors
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Matrix is in CNF = Conjunction of Disjunction of Literals -> Model as a Set of Sets of
Literals
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The sketched transformation to CNF is not optimal
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Example
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!
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Skolemization before PNF transformation, Definitorial CNF for Matrix, Reuse of Skolem
functions
Metaresults
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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)
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Deduction Theorem
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If M [ s ² s‘ then M ² s‘ ! s
Logical entailment can be reduced to validity
Proof by contradiction
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If M is a set of closed formulae then
M 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
Metaresults (II)
 Complexity of logical entailment, validity and
satisfiability
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Propositional Logic
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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
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Logical entailment / validity / satisfiability is undecidable
Set of valid formulae is semi-decidable (recursively
enumerable)
Set of satisfiable formulae is not recursively enumerable
Metaresults (III)
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Term Interpretations and Herbrand Theorem
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S = (U,I) is term-interpretation if U = Term0
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Let Term0 be non-empty. An interpretation S = (U,I) is called
Herbrand-Interpretation if
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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
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Herbrand-Modell of s is Herbrand-Intp. I with I ² s
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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)
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Term Interpretations and Herbrand Theorem
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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
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Let Term0 be non-empty and M a set of formulae in Prenex-NF
without existential quantors.
Then the following statements are equivalent
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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
Metaresults (V)
 Compactness of FOL
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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
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Let Term0 be non-empty and M a set of formulae in Prenex-NF
without existential quantors.
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Then M is unsatisfiable iff. T(M) is unsatisfiable for a finite
set of ground terms T ½ Term0
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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)
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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
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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
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The theorem immediately gives an algorithm for ATP!
 Problem: How to construct / find T in the theorem in
a clever way?
Part III:
The Resolution Calculus
 Pre-resolution phase:
 Gilmore‘s Methods, Davis-Putnam Procedure
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Unification
Basic Resolution Calculus
Refinements, Redundancy
Decision procedures
Pre-Resolution period:
Gilmore‘s Method
Pre-Resolution period:
Davis-Putnam Procedure
A Revolution in ATP:
Robinson‘s Resolution Principle
Part IV:
Chain Resolution
 A Variant of Resolution for the
Semantic Web
Part IV:
Demo
 assisted by a Resolution-based ATP
System: VAMPIRE