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Software Verification 1 Deductive Verification. Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität und Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik. Plan for today. While-Programs Syntax Semantics denotational: Scott Domains operational: SOS
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Software Verification 1Deductive Verification Prof. Dr. Holger Schlingloff Institut für Informatik der Humboldt Universität und Fraunhofer Institut für Rechnerarchitektur und Softwaretechnik
Plan for today • While-Programs • Syntax • Semantics • denotational: Scott Domains • operational: SOS • axiomatic: Hoare Logic
Syntax of while-Programs • Given a (typed) signature =(D, F, R) and a (denumerable) set V of program variables. • (each program variable has a type) • (T is the set of terms in the signature) • for simplicity, assume always R contains equality == • A while-program is defined as follows whileProg ::= skip | V=T| {whileProg; whileProg} | if (FOL-) whileProg else whileProg | while (FOL-) whileProg whereFOL- is a quantifier-free first-order formula over (,V)
Examples • =({int}, {0,%}, {==}), V=(a, b, c) • 1 = while (a==0) {{c = a; a = b%a}; b = c} • 2 = if (0==(a%0)%a) skip else {skip;skip} • =({int}, {0,1,48,+,-,**}, {<,isprim}), V=(n,k) • 3 = if (isprim(n)) n=k • Mersenne = {n=0; k=0; while (k<48) {n++; if (isprim((2**n)-1)) k++}} • Note: in C, “skip” and “else skip” is omitted, and n++ denotes n=n+1
Semantics • What is the “meaning” of such a program? • e.g., 3 = if (isprim(n)) k=n • need a first-order model M: (U,I,V) for (,V) • e.g., U=({zero,one,two,three,...}), I(0)=zero, I(1)=one, ..., I(isprim)={two, three, five,...},V(n)=two, V(k)=zero • Program modifies states (valuations) • V’(n)=two, V’(k)=two • semantics = function from initial to final valuations? • [[3]] = {(two,zero)(two,two), (one,two)(one,two),..., (two,three)(two,two), (one,three)(one,three), ...}
Nonterminating Programs • What is the meaning of the following? • e.g., 5 = if (isprim(n)) while(n==n) skip; • 5: zerozero, oneone, two? • Theory of Scott-Domains • extend every domain with an element # “undefined” • intuitively, # denotes nontermination • 1< 2 if 2 is “more defined” than 1 • 5 < if (n>9isprim(n)) while(n==n) skip;
Denotational Semantics • Given a universe U#=U{#} and interpretation Ifor =(D, F, R), the semantics of a program is a function mapping a program variable valuation into a program variable valuation: • [[]]: VV • [[skip]]=Id, where x(Id(x)==x)) (identity function) • [[v=t]]=Upd(v,t), where Upd(v,t)(V)(v)=tM and Upd(v,t)(V)(w)=wM
Denotational Semantics • [[{1; 2}]]=2(1) (function application) • [[if (b) 1 else 2]](V)=#, if b contains any v s.t. V(v)=#,[[if (b) 1 else 2]](V)= 1, if (U#,I,V)⊨ b[[if (b) 1 else 2]](V)= 2, if (U#,I,V)⊭ b • Define {while (b) }k as follows: • {while (b) }0=skip • {while (b) }k+1={if (b) ; {while(b) }k } • [[while(b) ]]=[[{while(b) }k]], where k is the smallest number for which (U#,I, [[{while(b) }k]](V))⊭ b(or else, [[while(b) ]](V)=#)
Examples • [[if (isprim(n)) k=n]](n=x, k=y) = (x, y+(x-y)*|isprim(x)|) • [[(while (a!=0) {c = a; a = b%a; b = c}]](x,y,z) = (0, gcd(x,y), gcd(x,y))
Structured Operational Semantics • Denotational semantics can be made mathematically sound, but is not “intuitive” • Operations of a “real” machine? • transitions from valuation to valuation • program counter is increased with the program • Abstract representation: • state=(program, valuation) • program means the part which is still to be executed • transition=(state1, state2) • “Meaning” of a program is a (possibly infinite) set of such transitions
SOS-Rules • (v=t, V)(skip, V[v:=t]); • ({skip; },V) (,V) • if (1, V1) (2,V2), then ({1; }, V1) ({2; },V2) • if (U,I,V) ⊨ b, then (if (b) 1 else 2, V) (1,V) • if (U,I,V) ⊭ b, then (if (b) 1 else 2, V) (2,V) • (while (b) , V) ({if (b) {; while (b) }}, V)
