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| Original author(s) | Microsoft Research |
|---|---|
| Developer(s) | Microsoft |
| Initial release | 2012 |
| Stable release | |
| Repository | |
| Written in | C++ |
| Operating system | Windows, FreeBSD, Linux (Debian, Ubuntu), macOS |
| Platform | IA-32, x86-64, WebAssembly, arm64 |
| Type | Theorem prover |
| License | MIT License |
| Website | github |
Z3, also known as the Z3 Theorem Prover, is a satisfiability modulo theories (SMT) solver developed by Microsoft. [2]
Z3 was developed in the Research in Software Engineering (RiSE) group at Microsoft Research Redmond and is targeted at solving problems that arise in software verification and program analysis. Z3 supports arithmetic, fixed-size bit-vectors, extensional arrays, datatypes, uninterpreted functions, and quantifiers. Its main applications are extended static checking, test case generation, and predicate abstraction.[ citation needed ]
Z3 was open sourced in the beginning of 2015. [3] The source code is licensed under MIT License and hosted on GitHub. [4] The solver can be built using Visual Studio, a makefile or using CMake and runs on Windows, FreeBSD, Linux, and macOS.
The default input format for Z3 is SMTLIB2. It also has officially supported bindings for several programming languages, including C, C++, Python, .NET, Java, and OCaml. [5]
In this example propositional logic assertions are checked using functions to represent the propositions a and b. The following Z3 script checks to see if :
(declare-fun a () Bool) (declare-fun b () Bool) (assert (not (= (not (and a b)) (or (not a)(not b))))) (check-sat)
Result:
unsat
Note that the script asserts the negation of the proposition of interest. The unsat result means that the negated proposition is not satisfiable, thus proving the desired result (De Morgan's law).
The following script solves the two given equations, finding suitable values for the variables a and b:
(declare-const a Int) (declare-const b Int) (assert (= (+ a b) 20)) (assert (= (+ a (* 2 b)) 10)) (check-sat) (get-model)
Result:
sat (model (define-fun b () Int -10) (define-fun a () Int 30) )
In 2015, Z3 received the Programming Languages Software Award from ACM SIGPLAN. [6] [7] In 2018, Z3 received the Test of Time Award from the European Joint Conferences on Theory and Practice of Software (ETAPS). [8] Microsoft researchers Nikolaj Bjørner and Leonardo de Moura received the 2019 Herbrand Award for Distinguished Contributions to Automated Reasoning in recognition of their work in advancing theorem proving with Z3. [9] [10]
Automated theorem proving is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a major impetus for the development of computer science.
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
In classical deductive logic, a consistent theory is one that does not lead to a logical contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if it has a model, i.e., there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states a theory is consistent if there is no formula such that both and its negation are elements of the set of consequences of . Let be a set of closed sentences and the set of closed sentences provable from under some formal deductive system. The set of axioms is consistent when there is no formula such that and .
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In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality. SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing.
The Bernays–Schönfinkel class of formulas, named after Paul Bernays, Moses Schönfinkel and Frank P. Ramsey, is a fragment of first-order logic formulas where satisfiability is decidable.
In computer science and formal methods, a SAT solver is a computer program which aims to solve the Boolean satisfiability problem. On input a formula over Boolean variables, such as "(x or y) and (x or not y)", a SAT solver outputs whether the formula is satisfiable, meaning that there are possible values of x and y which make the formula true, or unsatisfiable, meaning that there are no such values of x and y. In this case, the formula is satisfiable when x is true, so the solver should return "satisfiable". Since the introduction of algorithms for SAT in the 1960s, modern SAT solvers have grown into complex software artifacts involving a large number of heuristics and program optimizations to work efficiently.
Z3 may refer to:
The CADE ATP System Competition (CASC) is an annual competition of fully automated theorem provers for classical logic
In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and n-ary form. Function symbols are used, together with constants and variables, to form terms.
Concolic testing is a hybrid software verification technique that performs symbolic execution, a classical technique that treats program variables as symbolic variables, along a concrete execution path. Symbolic execution is used in conjunction with an automated theorem prover or constraint solver based on constraint logic programming to generate new concrete inputs with the aim of maximizing code coverage. Its main focus is finding bugs in real-world software, rather than demonstrating program correctness.
Verve is a research operating system developed by Microsoft Research. Verve is verified end-to-end for type safety and memory safety.
F* is a functional programming language inspired by ML and aimed at program verification. Its type system includes dependent types, monadic effects, and refinement types. This allows expressing precise specifications for programs, including functional correctness and security properties. The F* type-checker aims to prove that programs meet their specifications using a combination of SMT solving and manual proofs. Programs written in F* can be translated to OCaml, F#, and C for execution. Previous versions of F* could also be translated to JavaScript.
Alt-Ergo, an automatic solver for mathematical formulas, is mainly used in formal program verification. It operates on the principle of satisfiability modulo theories (SMT). Development was undertaken by researchers at the Paris-Sud University, Laboratoire de Recherche en Informatique, Inria Saclay Ile-de-France, and CNRS. Since 2013, project management and oversight has been conducted by OCamlPro company. It is released under the free and open-source software CeCILL-C license.
Dafny is an imperative and functional compiled language that compiles to other programming languages, such as C#, Java, JavaScript, Go and Python. It supports formal specification through preconditions, postconditions, loop invariants, loop variants, termination specifications and read/write framing specifications. The language combines ideas from the functional and imperative paradigms; it includes support for object-oriented programming. Features include generic classes, dynamic allocation, inductive datatypes and a variation of separation logic known as implicit dynamic frames for reasoning about side effects. Dafny was created by Rustan Leino at Microsoft Research after his previous work on developing ESC/Modula-3, ESC/Java, and Spec#.
Liquid Haskell is a program verifier for the programming language Haskell which allows specifying correctness properties by using refinement types. Properties are verified using a satisfiability modulo theories (SMT) solver which is SMTLIB2-compliant, such as the Z3 Theorem Prover.
In computer science, DPLL(T) is a framework for determining the satisfiability of SMT problems. The algorithm extends the original SAT-solving DPLL algorithm with the ability to reason about an arbitrary theory T. At a high level, the algorithm works by transforming an SMT problem into a SAT formula where atoms are replaced with Boolean variables. The algorithm repeatedly finds a satisfying valuation for the SAT problem, consults a theory solver to check consistency under the domain-specific theory, and then (if a contradiction is found) refines the SAT formula with this information.
In computer science, an e-graph is a data structure that stores an equivalence relation over terms of some language.
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