By Marcel Vinícius Medeiros Oliveira, Jim Woodcock

This publication constitutes the completely refereed post-conference court cases of the twelfth Brazilian Symposium on Formal equipment, SBMF 2009, held in Gramado, Brazil, in August 2009 -- co-located with SAST 2009, the Brazilian Workshop on Systematic and automatic software program Testing.

The 20 revised complete papers awarded including three invited papers have been conscientiously chosen from a variety of submissions in the course of rounds of reviewing and development. The papers are dedicated to the dissemination of the advance and use of formal equipment for the layout and verification of computational platforms. The symposium supplied a chance for researchers with a large diversity of pursuits in formal equipment for constructing computing platforms and software program to debate contemporary advancements during this field.

**Read Online or Download Formal Methods: Foundations and Applications: 12th Brazilian Symposium on Formal Methods, SBMF 2009 Gramado, Brazil, August 19-21, 2009 Revised Selected ... / Programming and Software Engineering) PDF**

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**Extra info for Formal Methods: Foundations and Applications: 12th Brazilian Symposium on Formal Methods, SBMF 2009 Gramado, Brazil, August 19-21, 2009 Revised Selected ... / Programming and Software Engineering)**

**Sample text**

The free theory over a signature Σ is the ﬁrst-order theory with an empty set of sentences. 1. It is also known as the theory of uninterpreted functions. , arrays) can be reduced to this one. Given a conjunction of equalities between terms using free functions, a congruence closure can be used for representing the smallest set of implied equalities. This representation can be used to check if a mixture of equalities and disequalities are satisﬁable. Simply check that the terms on both sides of each disequality are in diﬀerent equivalence classes.

Given a formula ϕ and a structure M , satisfaction M |= ϕ is deﬁned as: M M M M M M |= p(t1 , . . , tn ) ⇐⇒ M [[t1 ]], . . , M [[tn ]] ∈ M (p) |= ¬ϕ ⇐⇒ M |= ϕ ⇐⇒ M |= ϕ0 or M |= ϕ1 |= ϕ0 ∨ ϕ1 |= ϕ0 ∧ ϕ1 ⇐⇒ M |= ϕ0 and M |= ϕ1 |= (∃x : σ. ϕ) ⇐⇒ M {x → ν} |= ϕ for some ν ∈ |M |σ |= (∀x : σ. ϕ) ⇐⇒ M {x → ν} |= ϕ for all ν ∈ |M |σ Note that an implication ϕ0 ⇒ ϕ1 is equivalent to ¬ϕ0 ∨ϕ1 , and a bi-implication ϕ0 ⇔ ϕ1 is equivalent to (¬ϕ0 ∨ ϕ1 ) ∧ (ϕ0 ∨ ¬ϕ1 ). t. M |= ϕ, and is valid if for all structures M , M |= ϕ.

Congruence closure example: a = b, b = c, f (a, g(a)) = f (b, g(c)). (a) A DAG for all terms in the example. (b) Equivalences a = b and b = c are shown by dashed lines. (c) Nodes g(a) and g(c) are congruent because a = c is implied by ﬁrst two equalities. (d) Nodes f (a, g(a)) and f (b, g(c)) are also congruent, hence the example is unsatisﬁable because f (a, g(a)) = f (b, g(c)). A more suitable domain for machine arithmetic is to represent every number as a ﬁxed-size sequences of bits. On a 64 bit CPU, for instance, an integer is represented as a bit-vector with 64 bits.