By Roland Backhouse, Wei Chen, João F. Ferreira (auth.), Claude Bolduc, Jules Desharnais, Béchir Ktari (eds.)

This publication constitutes the refereed court cases of the tenth overseas convention on arithmetic of software building, MPC 2010, held in Québec urban, Canada in June 2010. the nineteen revised complete papers provided including 1 invited speak and the abstracts of two invited talks have been rigorously reviewed and chosen from 37 submissions. the focal point is on thoughts that mix precision with conciseness, allowing courses to be built by way of formal calculation. inside of this subject, the scope of the sequence is particularly varied, together with programming method, software specification and transformation, application research, programming paradigms, programming calculi, programming language semantics, protection and application logics.

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**Extra resources for Mathematics of Program Construction: 10th International Conference, MPC 2010, Québec City, Canada, June 21-23, 2010. Proceedings**

**Example text**

0 do {inv: v<=w & w<=rtc(y^);v & u=w’*y^;w} w:=w+u u:=w’*y^;u od {post: w=rtc(y^);v} We now assume that our imaginary tool translates these assertions into a Kleene algebra with range (with dual domain axioms). The range operation is used behind the scene for typing sets and for computing sets of successor states: The set of states that are reachable (in one step) from a set v with respect to a relation x is given by the range of v;x. Now a(x) denotes the antirange of x and r(x) the range of x.

We will automatically analyse the ﬁrst three proof obligations in our case studies, hence prove partial program correctness. We do not formally consider termination (total correctness) since this requires a diﬀerent kind of analysis. For program veriﬁcation, the precondition, the postcondition and the invariant are added as assertions to the given program code, and the proof obligations can then be generated and analysed automatically. This approach is based on the seminal work of Floyd and Hoare [10,12].

0;x=0. An idempotent semiring expanded by the reﬂexive transitive closure operation axiomatised above is a Kleene algebra. Conversion can now be axiomatised as (x^)^=x & (x+y)^=x^+y^ & (x;y)^=y^;x^ & x<=x;(x^;x). The universal relation can be axiomatised as x<=U. Sets can again be modelled as vectors or as subidentities, but, in contrast to relation algebras, the subalgebras of all subidentities in idempotent semirings or Kleene algebras are not necessarily Boolean algebras. The simplest approach — and best suited for ATP — uses domain semirings and Kleene algebras with domain [8].