Vampire: A Theorem Prover for First- Order Classical Logic Jason Gruber COSC Tuesday May 3, 2005

2 What is Vampire? developed by Andrei Voronkov and Alexandre Riazanov –Computer Science Department, University of Manchester The main goals of the project are: –creating an experimental environment for developing and evaluating novel efficient implementation techniques for automated theorem proving in first- order logic –developing software whose efficiency can meet practical needs in automatic theorem proving arising in a number of areas such as formal verification and development of software and hardware

3 Vampire At the moment our system Vampire is –a relatively fast resolution- and superposition- based experimental system –a developing system; many ideas are waiting to be implemented Vampire is not –an example of good design and coding style, though the situation is improving

4 Vampire It solves problems in the Thousands of Problems for Theorem Prover syntax in both CNF (conjunctive normal form) and full first-order logic syntax.Thousands of Problems for Theorem Prover Vampire supports the following inference rules: –ordered binary resolution with negative selection –superposition –special form of splitting Vampire exploits a number of redundancy control and simplification techniques, such as –forward and backward subsumption –forward and backward demodulation –forward subsumption resolution –cheap variant of basicness restrictions

5 Vampire Its kernel implements the calculi of ordered binary resolution and Quick Facts about: superposition (geometry) the placement of one object ideally in the position of another one in order to show that the two coincidesuperposition for handling equality. The splitting rule and negative equality splitting are simulated by the introduction of new predicate definitions and dynamic folding of such definitions. A number of standard redundancy criteria and simplification techniques are used for pruning the search space: subsumption, tautology deletion (optionally modulo commutativity), subsumption resolution, rewriting by ordered unit equalities, basicness restrictions and irreducibility of substitution terms. The reduction orderings used are the standard Knuth-Bendix ordering and a special non-recursive version of the Knuth-Bendix ordering.superposition

6 Vampire A number of efficient indexing techniques are used to implement all major operations on sets of terms and clauses. Run-time algorithm specialization is used to accelerate some costly operations, e.g., checks of ordering constraints. Although the kernel of the system works only with clausal normal forms, the preprocessor component accepts a problem in the full first-order logic syntax, clausifies it and performs a number of useful transformations before passing the result to the kernel. When a theorem is proven, the system produces a verifiable proof, which validates both the classification phase and the refutation of the CNF.

7 Vampire The efficiency of Vampire is due to a combination of compilation and indexing techniques used to implement all costly operations code trees are used for forward subsumption and search for generalizations, combination of path indexing and database style joins for backward subsumption and demodulation, and a number of specialized versions of discrimination trees for resolution and superposition.

8 Vampire Another interesting feature, which makes the system competitive, is the so-called limited resource strategy (LRS)*. Its purpose is to provide the best performance when time is limited by minimizing the amount of work that has been wasted when the time limit is reached.

9 Trophies CASC-16 (1999) : 1st place in MIX CASC-17 (2000) : 1st place in FOF (combined with Flotter) CASC JC (2001) : 1st place in MIX Proof Class and joint 1st place with E-SETHEO in MIX Assurance Class CASC-18 (2002) : 1st place in MIX (both Proof Class and Assurance Class), 1st place in FOF CASC-19 (2003) : 1st place in MIX (v6.0, both Proof Class and Assurance Class), 1st place in FOF (v5.0) CASC-J2 (2004) : 1st place in MIX (both Proof Class and Assurance Class), 1st place in FOFCASC-16 (1999) CASC-17 (2000) CASC JC (2001) CASC-18 (2002) CASC-19 (2003) CASC-J2 (2004)

10 Related work (incomplete list) The kernel of Vampire is used in a new temporal prover TeMP Vampire is integrated into the Sigma knowledge engineering environment. Vampire is used in the Hoolet reasoner for OWL-DL Vampire is being integrated into Isabelle

First Order Logic An overview

12 Syntax of First Order Logic

13 Semantics for First Order Logic

14 First Order Logic First order logic is used to model the world in terms of: –Objects, which are things with individual identities –Properties of objects that distinguish them from other objects –Relations that hold among objects –Functions which are a subset of the relations in which there is only one ‘value’ for any given ‘input’ Examples: –Objects: Students, lectures, companies, cars… –Relations: Brother of, bigger than, outside, part of, has color, occurs after, owns, visits, precedes… –Properties: blue, oval, even, large –Functions: father of, best friend, second half, one more than…

15 User Provides Constant symbols which represent individuals in the world –Mary –3 –green Function symbols which map individuals to individuals –Father-of(Mary)=John –Color-of(Sky)=Blue Predicate symbols mapping from individuals to truth values –Greater(5,3) –Green(Grass) –Color(Grass, Green)

16 FOL Provides Variable symbols –E.g., x, y, foo Connectives –Same as in PL; not (¬), and (  ), or (  ), implies (  ), if and only if (  ) Quantifiers –Universal  x –Existential  x

17 Quantifiers Universal quantification –(  x)P(x) means that P holds for all values of x in the domain associated with that variable E.g., (  x) dolphin(x)  mammal(x) Existential quantification –(  x)P(x) means that P holds for some value of x in the domain associated with that variable –E.g., (  x)mammal(x)  lays-eggs(x) –Make a statement about some object without naming it

18 Sentences are built from terms and atoms A term (denoting a real-world individual) is a constant symbol, a variable symbol, or an n-place function of n terms. –X and f(x1,…,xn) are terms, where each xi is a term. A term with no variables is a Ground Term An atom (which has value true or false) is either –An n-place predicate of n terms, or, –¬P, P  Q, P  Q, P  Q, P  Q where P and Q are atoms A sentence is an atom, or, if P is a sentence and x is a variable, then (  x)P and (  x)P are sentences. A well-formed formula (wff) is a sentence containing no “free” variables, IE, all variables are ‘bound’ by universal or existential quantifiers. –(  x)P(x,y) has x bound as a universally quantified variable, but y is free

19 Translating English to FOL Every gardener likes the sun –(  x) gardener(x)  likes(x,sun) You can fool some of the people all of the time. –(  x)(  t)(person(x)  time(t))  can-fool(x,t) You can fool all of the people some of the time. –(  x)(  t)(person(x)  time(t))  can-fool(x,t) All purple mushrooms are poisonous. –(  x)(mushroom(x)  purple(x))  poisonous(x) No purple mushroom is poisonous. –¬(  x)purple(x)  mushroom(x)  poisonous(x) –(  x)(mushroom(x)  purple(x))  ¬poisonous(x) There are exactly two purple mushrooms. –(  x)(  y)mushroom(x)  purple(x)  mushroom(y)  purple(y)  ¬(x=y)  (Az)(m ushroom(z)  purple(z))  ((x=z)v)y=z)) Clinton is not tall. –¬tall(Clinton) X is above Y if X is directly on top of Y or there is a pile of one or more other objects directly on top of another starting with X and ending with Y. –(  x)(  y)above(x,y)  (on(x,y)v(  z)(on(x.z)  above(z,y)))

20 Quantifiers Universal quantifiers often used with “implies” to form “rules.” –(  x) student(x)  smart(x) means “all students are smart.” You rarely use universal quantification to make blanket statements about every individual in the world. –(  x)student(x)  smart(x) means everyone in the world is a student and is smart. Existential quantifiers usually used with “and” to specify a list of properties about an individual. –(  x)student(x)  smart(x) means “there is a student who is smart.” A common mistake is to represent this English sentence as the FOL sentence: –(  x)student(x)  smart(x) But what happens when there is a person who is NOT a student?

21 Connections between All & Exist We can relate sentences involving for all and exists with DeMorgan’s Laws: –(  x) ¬P  ¬(  x) P –¬(  x) P  (  x) ¬ P –(  x) P  ¬(  x) ¬ P –(  x) P  ¬(  x) ¬ P

22 Axioms, definitions and theorems Axioms are facts and rules which attempt to capture all of the (important) facts and concepts about a domain that can be used to prove theorems. –Mathematicians don’t want any unnecessary (dependent) axioms—ones that can be derived from other axioms. –Dependent axioms can make reasoning faster, however. –Choosing a good set of axioms for a domain is a kind of design problem. A definition of a predicate is of the form “p(X)  …” and can be decomposed into two parts –Necessary description: “p(x)  …” –Sufficient description: “p(x)  …” –Some concepts do not have complete definitions (e.g. person(x))

23 Conversion to Normal Form STEP 1 - Eliminate implications: –The first step is to convert all implications with corresponding disjunctions. –ie. p  q becomes ¬ p  q STEP 2 - Move  inwards: –Use de Morgan’s laws, quantifier equivalences, and double negations –¬(p  q) becomes ¬ p  ¬ q ¬(p  q) becomes ¬ p  ¬ q ¬  x, p becomes  x ¬ p ¬  x, p becomes  x ¬ p ¬ ¬ p becomes p

24 Steps 3 & 4 STEP 3 - Standardize variables: –Change all duplicate variable names to separate names to Avoid confusion later when we drop quantifiers (  x P(x))  (  x Q(x)) becomes (  x P(x))  (  y Q(y)) STEP 4 - Move quantifiers left: –This can be done because we removed all duplicate variables p   x q becomes  x p  q

25 Steps 5-8 STEP 5 - Skolemize: –Remove all existential quantifiers Replace variables with names not existing elsewhere in KB Simple case:  x Q(x) becomes Q(A) where A is unique STEP 6 - Distribute  over  : –(a  b)  c becomes (a  c)  (b  c) STEP 7 - Flatten nested conjunctions and disjunctions: –(a  b)  c becomes (a  b  c) (a  b)  c becomes (a  b  c) STEP 8 - Convert disjunctions to implications: – Optional step for converting to INF –Gather all negative literals to left, and positive literals to right (  a   b  c  d) becomes (a  b  c  d)

26 English & FOL example Russell and Norvig Example

27 Now we apply the conversion procedure

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29 References ampire_theorem_prover.htmhttp:// ampire_theorem_prover.htm htmlhttp:// html tions/L2_9B_Kolackowsky_Sit/ tions/L2_9B_Kolackowsky_Sit/ A.Riazanov,A.Voronkov, Limited Resource Strategy in Resolution Theorem Proving, Preprint CSPP-7, University of Manchester, 2000CSPP-7