Software Verification 1 Deductive Verification

10.11.2011

Software Verification 1Deductive Verification

Prof. Dr. Holger SchlingloffInstitut für Informatik der Humboldt Universität

und

Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik

http://www.hu-berlin.de/index.html

Folie 2H. Schlingloff, Software-Verifikation I

Predicate Logic

• used to formalize mathematical reasoning dates back to Frege (1879) „Begriffsschrift“

- „Eine der arithmetischen nachgebildete Formelsprache des reinen Denkens“

individuals, predicates (sets of individuals), relations (sets of pairs), ...

quantification of statements (quantum = how much)- all, none, at least one, at most one, some, most, many, ...

- need for variables to denote “arbitrary” objects In contrast to propositional logic, first-order logic adds

- structure to basic propositions- quantification on (infinite) domains


FOL: Syntax

• New syntactic elements R is a set of relation symbols,

where each pR has an arity nN0

V is a denumerable set of (first-order or individual) variables

An atomic formula is p(x1,…,xn), where pR is n-ary and (x1,…,xn)Vn.

• Syntax of first-order logicFOL ::= R (Vn) | | (FOL FOL) | V FOL


FOL: Syntax

• Abbreviations and parenthesis as in PL Of course, x = ¬x ¬

• Propositions = 0-ary relationsPredicates = 1-ary relations if all predicates are propositions, then FOL = PL

• Examples xxx (p() x(q() p())) xxy ¬p(x) xy (p(x,y) p(y,x)) (xy p(x,y) yx p(x,y))


Typed FOL

• Often, types/sorts are used to differentiate domains

• Signature =(D, F, R), where D is a (finite) set of domain names F is a set of function symbols, where each fF has an

arity nN0 and a type DDn+1

- 0-ary functions are called constants R is a set of relation symbols, where each pR has an

arity nN0 and a type DDn

- unary relations are called predicates- propositions can be seen as 0-ary relations

• Remark: domains and types are for ease of use only (can be simulated in an untyped setting by additional predicates)


Terms and Formulas

• Let again V be a (denumerable) set of (first-order) variables, where each variable has a type DD (written as x:D)(for any type, there is an unlimited supply of variables of that type)

• The notions Term and Atomic Formula AtF are defined recursively: each variable of type D is a term of type D if f is an n-ary function symbol of type (D1,…Dn,Dn+1) and t1, …, tn

are terms of type D1, …, Dn, then f(t1,…,tn) is a term of type Dn+1 if p is an n-ary relation symbol of type (D1,…Dn) and t1, …, tn are

terms of type D1, …, Dn, then p(t1,…,tn) is an atomic formula

• Revised syntax of first-order logicFOL ::= AtF | | (FOL FOL) | V:D FOL


Examples

x:Boy y:Girl loves(x,y)x:Human y:Human (needs(x,y) loves(y,x))x,y:Int equals(plus(x,y), plus(y,x))x:Int ¬equals(zero(), succ(x))• …


FOL: Models

• (We give the typed semantics only)

•First-Order Model Let a universe U be some nonempty set, and

let DU U for every DD be the domain of D

Interpretation I: assignment F ↦ Un+1

R ↦ Un

Valuation V: assignment V ↦ Uinterpretations and valuations must respect typing

Model M: (U,I,V)


FOL: Semantics

• Given a model M: (U,I,V), the value tM of term t (of type D) can be defined inductively if t=xV, then tM=V(x) if t=f(t1,…,tn) , then tM=I(f)(t1

M,…,tnM)

• Likewise, the validation relation ⊨ between model M and formula M ⊨ p(t1,…,tn) if (t1

M,…,tnM)I(p)

M ⊭ ; M ⊨ () if M ⊨ implies M ⊨ M ⊨x if M‘ ⊨ for some M‘ which differs at most

in V(x) from M

• Validity and satisfiability is defined as in the propositional case


Examples

• ⊨ x x • ⊨ x x x ( )

• ⊨ x x x ( )

• ⊨ x y y x • ⊨ x (x:=t)

• If ⊨ , then ⊨ x


FOL: Calculus

• A sound and complete axiom system for FOL: all substitution instances of axioms of PL modus ponens: , () ⊢ ⊢((x:=t)x) instantiation

() ⊢(x) if x doesn‘t occur in particularization

• Relaxation: particularization may be applied if there is no free occurrence of x in ; i.e., x may occur in inside the scope of a quantification


FOL: Completeness

• As in the propositional case, correctness is easy (⊢ ⊨, “every derivable formula is valid”)

• Completeness (⊨ ⊢, “every valid formula is derivable”) follows with a similar proof as previously:given a consistent formula, construct a model satisfying it ~⊢¬ ~⊨¬

• Extension lemma: If Φ is a finite consistent set of formulæ and is any formula, then Φ{} or Φ{¬} is consistent

• Needs additionally: If Φ is any consistent set of formulæ and x is a formula in Φ, then Φ{(t)} is consistent for any term t

• From this, a canonical model can be constructed as before


Example

• Consider the formula xyz ((p(x, y) ∧ p(y, z)) → p(x, z))

∧ x ¬p(x, x) ∧ x p(x, f(x) )This formula is satifiable only in infinite models


FOL: Undecidability

• Completeness means the set of valid formulæ can be recursively enumerated

• Turing showed that the invalid formulæ are not r.e., i.e., there is no algorithm deciding whether a formula is valid or not strictly speaking, FOL= with at least one binary

relation certain sublanguages of FOL are still decidable


FOL=

• Equality is not definable in FOL• First order logic with equality contains an

additional (binary) relation == which is always interpreted as equality of domain elements Written in infix notation, i.e. (x==y) for ==(x,y)

• Axioms (x==x) reflexivity

(x==y (y==z x==z)) transitivity

(x==y y==x) symmetry

(x==y ( (y:=x))) substitution


Presburger arithmetic

• Given a signature (N, 0,´,+) of FOL=, define n (n´==0) m n (m´==n´ m==n) p(0) n(p(n) p(n´)) n p(n)

• If the third axiom holds for all p, then this uniquely characterizes the natural numbers (“monomorphic”) n (n+0==n) mn ((m+n)+1 == m+(n+1))

• This theory is decidable!

25.4.2009


Peano arithmetic

• Given the signature (N, 0,´,+,*) and above axioms, plus n (n*0==0) mn (m*n´ == (m*n)+m)

• This theory is undecidable

25.4.2009


Formalizing C in FOL

• Consider the following C programint gcd (int a, int b){ int c; while ( a != 0 ) { c = a; a = b%a; b = c; } return b;}

• Consider the following FOL formula : t:N (a(t)==0 c(t+1)==a(t) a(t+1)==b(t)%a(t) b(t+1)=c(t)

a(t)==0 a(t+1)==a(t) b(t+1)==b(t) c(t+1)==c(t) )

• In which way are these equivalent?


Correctness

From this formalization, we expect that ⊨ t (a(t)==0 → b(t)==gcd(a(0),b(0)))

(partial correctness) ⊨ t (a(t)==0 b(t)==gcd(a(0),b(0)))

(total correctness)

Can we prove these statements?


First order theorem proving

• Despite the undecidability of first order logic, provers have reached a remarkable proficiency SPASS Vampire Otter, Prover9

• Need (some) arithmetic solver

Software Verification 1 Deductive Verification

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