Title: A brief Introduction to Automated Theorem Proving
1A brief Introduction to Automated Theorem Proving
- Theoretical Foundations, History and the
Resolution Calculus for classical First-order
Logic - Uwe Keller
2Content
- 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
3Part IIntroduction
- Motivation History
- Theorem Proving, ATP and Calculi
4Logic 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
5How 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!
6A 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 -
7Theorem 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 - )
8Theorem 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?
9Main 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
10Why 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
11Example
- 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
12Example (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
13Part IIFoundations
- FOL, Normalforms Preprocessing, Metaresults
14Classical 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
15Classical 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!
16Normal 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!
17Negation 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
-
18Standard 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)))
19Prenex 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
20Clause 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
21Clause 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
22Metaresults
- 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
23Metaresults (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
24Metaresults (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!
25Metaresults (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
26Metaresults (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!)
27Metaresults (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?
28Part IIIThe Resolution Calculus
- Pre-resolution phase
- Gilmores Methods, Davis-Putnam Procedure
- Unification
- Basic Resolution Calculus
- Refinements, Redundancy
- Decision procedures
29Pre-Resolution period Gilmores Method
30Pre-Resolution periodDavis-Putnam Procedure
31A Revolution in ATP Robinsons Resolution
Principle
32Part IVChain Resolution
- A Variant of Resolution for the Semantic Web
33Part IVDemo
- assisted by a Resolution-based ATP System
VAMPIRE