A brief Introduction to Automated Theorem Proving
Theoretical Foundations, History and the Resolution Calculus for classical First-order Logic
Uwe Keller
based on material by B. Beckert, R. Hähnle, A. Voronkov, A. Leitsch and T. Tammet
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
Modelling(automated)
Deduction
Logic and Theorem Proving
Real-world descriptionin natural language.Mathematical ProblemsProgram + Specification
Syntax (formal language).First-order Logic, Dynamic Logic, …
Valid Formulae
Provable Formulae
Formalization
Semantics(truth function)
Calculus(derivation / proof)
Correctness
Completeness
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!
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 …
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 :- )
Theorem Proving (II) Research areas
Interactive / tactic TP vs. Automated TP Classical Logic vs. Non-classical logics
Calculi for … ATP - General principle: Refutation approach
Resolution, Tableau, Inverse Method, Instance-based Methods ITP – General principle: Show Proof situation/context
Sequent Calculi others – General principle: Generation of complex
formulae based on very simple axioms Hilbert-style Calculi
Central difference: What are the elements in a proof & what is a proof?
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 …
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
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) … Formalization
Compact (only LEQ) Redundancy-free Specific definitions
Continous functions Main idea of proof
is 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
Part II:Foundations
FOL, Normalforms & Preprocessing, Metaresults
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
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!
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!
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
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)))
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
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
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 -> 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
Metaresults Metaresult = Property of a Logic L
Most famous example: Gödels Incompleteness Theorems! 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 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
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
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,…,t
) = f(t
,…,t
) for all n-ary function symbols f 2 and
ground terms t,…,t
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!
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
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!)
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 respective instance of the orginial formula set is determined as being inconsistent (unsatisfiable)
The theorem immediately gives an algorithm for ATP! Problem: How to construct / find T in the theorem in a clever
way?
Herband‘s Theorem:From Clause Logic to Propositional Logic
Clause Logic
Propositional Logic
(Ground) Substitutions
Clauses
Ground clauses
Incons-istent set
Part III:The Resolution Calculus
Pre-resolution phase: Gilmore‘s Methods, Davis-Putnam Procedure
Unification Basic Resolution Calculus Refinements, Redundancy
Pre-Resolution period: Gilmore‘s Method
First ATP procedure for First-order logic Directly based on Herbrands Theorem
Reduction of FOL entailment to satisfiability in Prop. Logic How to generate candidates C‘ for propositional satisiability checking
from a FOL clause set C Saturation by ground instances from Hn(C) (= set of ground terms of
depth · n) More precisely: Successively generate the sets C‘n of ground clauses :=
{c : c 2 C and rg() µ Hn(C) } Since H_n( C) grow exponentially it is very important to have a good
algorithm for checking satisfiability
Pre-Resolution period: Gilmore‘s Method
„Easy“ test of satisfiability of the generated C‘ set of ground clauses: Transform C‘ into Disjunctive Normal Form D = DNF(C‘) is unsatisfiable iff every consitutent of D contains a
contradiction L Æ ¬L for some literal L Can be done in deterministic time O(n log(n)) Problem: Convertion from CNF into DNF (almost always) exponential
(inherently complex, since otherwise P = NP), (not known at that time!) Pseudocode
begin contr := false while not contr do D‘ := DNF(C‘_n) contr := all constitutents of D‘ contain complementary literals n:=n+1 end whileend
Pre-Resolution period: Gilmore‘s Method
Weak points of Gilmore‘s approach … The generation of the candidate ground clause sets C‘n to be checked the discjunctive normal form transfomation
First weakness is inherent to all procedures directly applying Herbrands theorem
The second problem concerns propositional logic only
Gilmore‘s pioneering implementation did not yield actual proofs for quite simple predicate logic formulas
A possible improvement Avoid transformation to DNF and try to find „good“ decision methods for
satisfiability on CNFs This is basically what was achieved by Davis and Putnam [DP,1960] shortly after
Gilmore‘s implementation
Pre-Resolution period:Davis-Putnam Procedure
Like Gilmore‘s method based on successive production of ground caluse sets C‘N and testing of their unsatisfiability
(Still) very efficient decision method for satisfiability. Requires CNF for ground clauses.
Invented originally for FOL, it became the most powerful SAT decision procedure for Propositional Logic. Many very powerful SAT solvers still are refining DPP today.
Davis-Logemann-Loveland Rules [DLL, 1962] Preliminary step: Reduce all clauses in C
Eliminate multiple occurrences of the same literal (leave only one). Generates a clause set C‘
Then apply the follwing rules non-deterministically to C‘ Tautology-Rule One-Literal-Rule Pure-Literal-Rule Splitting-Rule
Pre-Resolution period:Davis-Putnam Procedure
Davis-Logemann-Loveland Rules [DLL, 1962] Tautology-Rule: Delete all clauses in C‘ containing complementary literals One-Literal-Rule: If there is a clauses c = {l} with only one literal l, remove all
clauses d from C‘ which contain l, and remove the dual literal ld from all other clauses
Pure-Literal-Rule: Let D‘ µ C‘ with the following property: There exists a literal l appearing in all clauses of D‘, but ld does not appear in C‘. Then delete D‘ from C‘
Splitting-Rule: Let C‘ = {A1,…,An,B1,…,Bm} [ R such that R contains l nor ld, all Ai contain l but not ld and all Bj contain ld but not l. Let A‘i = Ai after deletion of l and let B‘j = Bj after deletion of ld.Then split C‘ into C‘1 = {A‘1,…,A‘n} [ R and C‘2={B‘1,…,B‘m} [ R
Properties of the DLL procedure The rules are essentially reductive (atoms are in each step deleted) The rules are correct (rules preserve satisfiability; in case of split only for one of
the new introduced clauses sets The procedure generates sets that contain the empty clause for all cases (of the
applied splits) iff C‘ is unsatisfiable (decision criteria: correctness and completeness, termination)
Example: C = {P Ç Q, R Ç S Ç S, ¬R Ç S, R Ç ¬S, ¬R Ç ¬S, P Ç ¬Q Ç ¬P}
Pre-Resolution period:Davis-Putnam Procedure
Pseudocode of the First-order ATP procedure by Davis & Putnam
begin {C finite set of clauses}
if C does not contain („real“) function symbols
then apply DP1 – DP3 to C‘_0; check the DP decision tree for unsatisfiability
else begin
n:= 0; contr := false
while not contr do
perform DP1 – DP3 on C‘_n
if the DP-decision tree proves unsatisfiability
then contr := true
else contr := false
n:=n+1
end while
end
end
DP1: Reduce all clausesDP2: Delete all tautologiesDP3: Construct a DP decision tree according to the given rules
• Nondeterministic (DP3)• If C does not contain function symbols (with arity > 0) then the procedure always terminates (== decision procedure for FOL clause set)•If C is satisfiable and C contains function symbols then the algorithm does not terminate•Yields a decision procedure for validity of the Bernays-Schönfinkel class in FOL (8* 9*)
Interlude:Inferences & Inference systems An inference I has the form
where n ¸ 0, F1,…,Fn, G are formulae
An inference rule R is a set of inferences more precisely a decidable (usually efficiently computable) n+1-ary relation over
formuale Usually one uses schematic variables for representing formulae in inference rules and
attach some (most often syntactic) conditions to these variables Every instance I 2 R is called an instance of R
An inference system § is a (finite) set of inference rules A proof of G from P in § is a finite sequence of formulae F1, … Fn such that
Fn = F and for all Fi (i · n) it holds that either Fi 2 N or there is an inference I such that Fi is the
conclusion of I and all the premisses P1, … Pj of I are contained in the prefix F1, …, F(i-1)
Here we mainly consider inference systems on clauses, for instance Resolution
F1 F2 … Fn
G
Premisses
Conclusion
A Revolution in ATP: Robinson‘s Resolution Principle In some sense the simplest possible calculus for FOL (without equality)
In principle only a single inference rule which combines substution and atomic cut
Possible since it requires set of input formulae in CNF (very simple and uniform syntactic form)
Binary substitution rule computing a „minimal“ substitution which makes two atoms equal
A quote from Robinsons landmarking paper [Robinson, 1965] …
Theorem-proving on the computer, using procedures based on the fundamental theorem of Herbrand concerning the FOL Predicate Calculus, is examined with a view towards improving the efficiency and widening the range of practical applicability of these procedures. A close analysis of the process of substitution (of terms for variables) and the process of truth-functional analysis of the results of such substitutions reveals that both processes can be combined into a single new iterating process (called resolution) which is vastly more efficient than the older cylcic procedures consisting of substitution stages alternating with truth-functional analysis stages.
A Revolution in ATP: Robinson‘s Resolution Principle
The basic Resolution Calculus (BRC) Ground case
General case
Fundamental aspects: Iterative grounding of the clause set „Guided“ guessing of interesting instances (Unification) built into the calculus Resolving upon an atom L does not require L to be ground (unnecessary
grounding avoided)
L Ç C ¬L Ç DC Ç D
C Ç L Ç L C Ç L
BinaryResolution Factoring
L Ç C ¬L‘ Ç D(C Ç D)
C Ç L Ç L‘ (C Ç L)
BinaryResolution Factoring
where is the most general unifier of L and L‘
Basic Resolution Calculus:Properties
Properties of the basic Resolution Calculus Given any two clauses, there are only finitely many resolvents using the Resolution
Inference Rule. The Resolution Calculus is sound
If c is provable from C in BRC then C ² c This means in particular: If we can derive the emtpy clause then C is unsatisfiable
The Resolution Calculus is refutationally complete A set C of clauses is unsatisfiable then the empty clause can be proven (derived) from
C Altogether
A set C of clauses is unsatisfiable iff. there is a proof for the empty clause from C in BRC
Remark: Soundness of the inference system can be relaxed to satisfiability- preserving!
How to find a contradiction (empty clause) starting with an initial (unsatisfiable) formula set? Saturation approach (wrt. the inference system BRC)
Resolution:Proof search by Saturation Saturated sets
A set of clauses C is called saturated (wrt. inference system ) if every inference in with premises in C gives a clause in C
Completness reformulated (in terms of saturated sets) A set C of clauses is unsatisfiable iff every saturated set S of clauses with C
µ S also contains the empty clause
That means: Simply construct a(ny) saturated set S of clauses (wrt. BRC) S (saturation algorithm)
Simple algorithm
S:= set of input clauseswhile not finished do(1) Repeatedly apply all inferences to clauses in S, adding to S conclusions of these inferences(2) If the empty clause is proved, terminate with success.
If no inference rule is applicable, terminate with failure
Resolution:Proof search by Saturation
given clausecandidate clause
Conclusions
Search spaceSearch space
Goal clause
Resolution:Proof search by Saturation
Most likely scenario ….
Search space
Resolution:Proof search by Saturation Possible theoretical scenarios
At some moment the empty clause is generated, in this case the input set of clauses is unsatisfiable
Saturation will terminate without ever generating the empty clause, in this case the input set of clauses is satisfiable
Saturation will run forever, but without generating the empty clause. In this case the input set of clauses is satisfiable
Possible practical scenarios At some moment the empty clause is generated, in this case the input set of
clauses is unsatisfiable Saturation will terminate without ever generating the empty clause, in this
case the input set of clauses is satisfiable Saturation will run until we run out of resources, but without generating
the empty clause. In this case it is unknown whether the input set of clauses is (un)satisfiable
Resolution:How to saturate in clever way ? The simple saturation algorithm is highly inefficient Apply inferences not in an arbitrary way, but within some senseful /
useful order. Generate the empty clause as early as possible in the saturation
process
„Prefer“ some inferences over others (in a sense), for instance goal directedness
Actually what we need to ensure then to have completness guaranteed is fairness: A saturation algorithm is fair iff every possible inference is eventually
selected Completness Theorem reformulated (for Saturation Algorithms)
Let A be a fair saturation algorithm. A set C of clauses is unsatisfiable iff A eventually produces the empty clause
Central problem: How to find „good“ saturation algorithms!
How to guess suitable instances?Unification
Example:Basic Resolution Calculus
Enhancing Efficiency: Refinements of Resolution
Resolution Refinements:Hyperresolution
Resolution Refinements:Ordered Resolution
Enhancing Efficiency: Redundancy Criteria in Resolution
Part IV:Chain Resolution
A Variant of Resolution for the Semantic Web
Part IV:Demo
assisted by a Resolution-based ATP System: VAMPIRE
And there is a lot we have not talked about yet … Different Calculi
Tableaux Methods, Instance-based Methods, Inverse Method
Decision Procedures Theory Reasoning, in particular equality ATP in other logics
Modal, temporal logics, description logics Logics for non-monotonic reasoning Paraconsistent logics
Reasoning tasks other than logical entailment / unsatisfiability Query answering
References & further Reading …
