By Michael Huth
ISBN-10: 052154310X
ISBN-13: 9780521543101
Contemporary years have visible the improvement of strong instruments for verifying and software program structures, as businesses around the globe fully grasp the necessity for more suitable technique of validating their items. there's expanding call for for education in simple equipment in formal reasoning in order that scholars can achieve skillability in logic-based verification tools. the second one variation of this winning textbook addresses either these requisites, via carrying on with to supply a transparent advent to formal reasoning that's either suitable to the wishes of recent computing device technological know-how and rigorous sufficient for functional software. advancements to the 1st variation were made all through, with additional and elevated sections on SAT solvers, existential/universal second-order good judgment, micro-models, programming through agreement and overall correctness. The assurance of model-checking has been considerably up-to-date. extra workouts were additional. net aid for the ebook contains labored options for all workouts for lecturers, and version ideas to a few workouts for college kids.
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Additional resources for Logic in computer science: modelling and reasoning about systems
Example text
We denote this by p∧q →r p → (q → r). Since there can be only one formula to the right of , we observe that each instance of can only relate two formulas to each other. 15 Here is an example of a proof that uses introduction and elimination rules for conjunction; it shows the validity of the sequent p → q p ∧ r → q ∧ r: 1 p→q premise 2 p∧r assumption 3 p ∧e1 2 4 r ∧e2 2 5 q →e 1, 3 6 q∧r ∧i 5, 4 7 p∧r →q∧r →i 2−6 The rules for disjunction The rules for disjunction are different in spirit from those for conjunction.
The first one has the Latin name reductio ad absurdum. It means ‘reduction to absurdity’ and we will simply call it proof by contradiction (PBC for short). The rule says: if from ¬φ we obtain a contradiction, then we are entitled to deduce φ: ¬φ .. ⊥ φ PBC. 2 Natural deduction 25 This rule looks rather similar to ¬i, except that the negation is in a different place. This is the clue to how to derive PBC from our basic proof rules. Suppose we have a proof of ⊥ from ¬φ. By →i, we can transform this into a proof of ¬φ → ⊥ and proceed as follows: 1 ¬φ → ⊥ given 2 ¬φ assumption 3 ⊥ →e 1, 2 4 ¬¬φ ¬i 2−3 5 φ ¬¬e 4 This shows that PBC can be derived from →i, ¬i, →e and ¬¬e.
Naturally, the constructed disjunction depends upon the assumptions needed in establishing its respective disjunct p or q. Now let’s consider or-elimination. How can we use a formula of the form φ ∨ ψ in a proof? Again, our guiding principle is to disassemble assumptions into their basic constituents so that the latter may be used in our argumentation such that they render our desired conclusion. Let us imagine that we want to show some proposition χ by assuming φ ∨ ψ. Since we don’t know which of φ and ψ is true, we have to give two separate proofs which we need to combine into one argument: 1.
Logic in computer science: modelling and reasoning about systems by Michael Huth
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