By Daniel Kroening, Ofer Strichman
A choice technique is an set of rules that, given a call challenge, terminates with an accurate yes/no resolution. right here, the authors specialise in theories which are expressive sufficient to version genuine difficulties, yet are nonetheless decidable. particularly, the ebook concentrates on determination methods for first-order theories which are common in automatic verification and reasoning, theorem-proving, compiler optimization and operations learn. The suggestions defined within the booklet draw from fields similar to graph conception and common sense, and are sometimes utilized in industry.
The authors introduce the elemental terminology of SAT, Satisfiability Modulo Theories (SMT) and the DPLL(T) framework. Then, in separate chapters, they learn determination methods for propositional good judgment; equalities and uninterpreted services; linear mathematics; bit vectors; arrays; pointer common sense; and quantified formulation. additionally they examine the matter of figuring out mixed theories in response to the Nelson-Oppen procedure.
The first variation of this ebook was once followed as a textbook in classes all over the world. It was once released in 2008 and the sphere now known as SMT used to be then in its infancy, with out the normal terminology and canonic algorithms it has now; this moment variation displays those alterations. It brings ahead the DPLL(T) framework. It additionally expands the SAT bankruptcy with glossy SAT heuristics, and features a new part approximately incremental satisfiability, and the similar Constraints pride challenge (CSP). The bankruptcy approximately quantifiers was once accelerated with a brand new part approximately common quantification utilizing E-matching and a piece approximately successfully Propositional Reasoning (EPR). The ebook additionally incorporates a new bankruptcy at the software of SMT in commercial software program engineering and in computational biology, coauthored via Nikolaj Bjørner and Leonardo de Moura, and Hillel Kugler, respectively.
Each bankruptcy contains a targeted bibliography and routines. academics’ slides and a C++ library for speedy prototyping of determination approaches can be found from the authors’ website.
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Additional resources for Decision Procedures: An Algorithmic Point of View
For each (i, j) ∈ E, k (xit =⇒ ¬xjt ) . 3) t=1 Note that the input of this problem can be represented by a graph, where the stations are the graph’s nodes and E corresponds to the graph’s edges. Checking whether the allocation problem is solvable corresponds to solving what is known in graph theory as the k-colorability problem: can all nodes be assigned one of k colors such that two adjacent nodes are assigned different colors? Indeed, one way to solve k-colorability is by reducing it to propositional logic.
The first step in building the verification condition is to model the variables a and b and the procedures that are called using the Boolean variables a, b, f , g, and h, as can be seen in Fig. 2. The if-then-else construct can be replaced by an equivalent propositional logic expression as follows: (if x then y else z) ≡ (x ∧ y) ∨ (¬x ∧ z) . 2 SAT Solvers 29 if ¬a ∧ ¬b then h else if ¬a then g else f if a then f else if b then g else h Fig. 2. In the process of building a formula—the verification condition—we replace the program variables and the function symbols with new Boolean variables ⇐⇒ (¬a ∧ ¬b) ∧ h ∨ ¬(¬a ∧ ¬b) ∧ (¬a ∧ g ∨ a ∧ f ) a ∧ f ∨ ¬a ∧ (b ∧ g ∨ ¬b ∧ h) .
3 and 10. We must assume in this subsection that the reader is familiar to some extent with first-order logic—a reasonable exposition of this subject is beyond the scope of this book. See [43, 138] for a more organized study of these matters. Let us recall some of the terms that are directly relevant to our topic. First-order logic (also called predicate logic) is based on the following elements: 1. Variables: a set of variables. 4 The Theoretical Point of View 15 2. , “∧”, “¬”, and “∨”), quantifiers (“∃” and “∀”), and parentheses.
Decision Procedures: An Algorithmic Point of View by Daniel Kroening, Ofer Strichman