Satisfiability Modulo Theories

# Satisfiability Modulo Theories

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In computer science
Computer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...

, the Satisfiability Modulo Theories (SMT) problem is a decision problem
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...

for logical formulas with respect to combinations of background theories
Theory (mathematical logic)
In mathematical logic, a theory is a set of sentences in a formal language. Usually a deductive system is understood from context. An element \phi\in T of a theory T is then called an axiom of the theory, and any sentence that follows from the axioms is called a theorem of the theory. Every axiom...

expressed in classical first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

with equality. Examples of theories typically used in computer science are the theory of real numbers, the theory of integers, and the theories of various data structure
Data structure
In computer science, a data structure is a particular way of storing and organizing data in a computer so that it can be used efficiently.Different kinds of data structures are suited to different kinds of applications, and some are highly specialized to specific tasks...

s such as lists, arrays, bit vectors and so on. SMT can be thought of as a form of the constraint satisfaction problem
Constraint satisfaction problem
Constraint satisfaction problems s are mathematical problems defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constraints over variables, which is solved by constraint...

and thus a certain formalized approach to constraint programming
Constraint programming
Constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints. Constraints differ from the common primitives of imperative programming languages in that they do not specify a step or sequence of steps to execute, but rather the properties...

.

## Basic terminology

Formally speaking, an SMT instance is a formula
Formula
In mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....

in first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

, where some function and predicate symbols have additional interpretations, and SMT is the problem of determining whether such a formula is satisfiable. In other words, imagine an instance of the Boolean satisfiability problem
Boolean satisfiability problem
In computer science, satisfiability is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE...

(SAT) in which some of the binary variables are replaced by predicates over a suitable set of non-binary variables. A predicate is basically a binary-valued function of non-binary variables. Example predicates include linear inequalities (e.g., ) or equalities involving uninterpreted terms and function symbols (e.g., where is some unspecified function of two unspecified arguments.) These predicates are classified according to the theory they belong to. For instance, linear inequalities over real variables are evaluated using the rules of the theory of linear real arithmetic
Arithmetic
Arithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...

, whereas predicates involving uninterpreted terms and function symbols are evaluated using the rules of the theory of uninterpreted function
Uninterpreted function
In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and arity. Function symbols are used, together with constants and variables, to form terms....

s with equality (sometimes referred to as the empty theory). Other theories include the theories of arrays and list structures (useful for modeling and verifying software programs), and the theory of bit vectors (useful in modeling and verifying hardware designs). Subtheories are also possible: for example, difference logic is a sub-theory of linear arithmetic in which each inequality is restricted to have the form for variables and and constant .

Most SMT solvers support only quantifier free fragments of their logics.

## Expressive power of SMT

An SMT instance is a generalization of a Boolean SAT
Boolean satisfiability problem
In computer science, satisfiability is the problem of determining if the variables of a given Boolean formula can be assigned in such a way as to make the formula evaluate to TRUE...

instance in which various sets of variables are replaced by predicates from a variety of underlying theories. Obviously, SMT formulas provide a much richer modeling language
Language
Language may refer either to the specifically human capacity for acquiring and using complex systems of communication, or to a specific instance of such a system of complex communication...

than is possible with Boolean SAT formulas. For example, an SMT formula allows us to model the datapath
Datapath
A datapath is a collection of functional units, such as arithmetic logic units or multipliers, that perform data processing operations. Most central processing units consist of a datapath and a control unit, with a large part of the control unit dedicated to regulating the interaction between the...

operations of a microprocessor
Microprocessor
A microprocessor incorporates the functions of a computer's central processing unit on a single integrated circuit, or at most a few integrated circuits. It is a multipurpose, programmable device that accepts digital data as input, processes it according to instructions stored in its memory, and...

at the word rather than the bit level.

Answer set programming is a form of declarative programming oriented towards difficult search problems. It is based on the stable model semantics of logic programming. In ASP, search problems are reduced to computing stable models, and answer set solvers -- programs for generating stable...

is also based on predicates (more precisely, on atomic sentence
Atomic sentence
In logic, an atomic sentence is a type of declarative sentence which is either true or false and which cannot be broken down into other simpler sentences...

s created from atomic formula
Atomic formula
In mathematical logic, an atomic formula is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic...

). Unlike SMT, answer-set programs do not have quantifiers, and cannot easily express constraints such as linear arithmetic or difference logic -- ASP is at best suitable for boolean problems that reduce to the free theory of uninterpreted functions. Implementing 32-bit integers as bitvectors in ASP suffers from most of the same problems that early SMT solvers faced: "obvious" identities such as x+y=y+x are difficult to deduce.

Constraint logic programming does provide support for linear arithmetic constraints, but within a completely different theoretical framework.

## SMT solvers

Early attempts for solving SMT instances involved translating them to Boolean SAT instances (e.g., a 32-bit integer variable would be encoded by 32 bit variables with appropriate weights and word-level operations such as 'plus' would be replaced by lower-level logic operations on the bits) and passing this formula to a Boolean SAT solver. This approach, which is referred to as the eager approach, has its merits: by pre-processing the SMT formula into an equivalent Boolean SAT formula we can use existing Boolean SAT solvers "as-is" and leverage their performance and capacity improvements over time. On the other hand, the loss of the high-level semantics of the underlying theories means that the Boolean SAT solver has to work a lot harder than necessary to discover "obvious" facts (such as for integer addition.) This observation led to the development of a number of SMT solvers that tightly integrate the Boolean reasoning of a DPLL
DPLL algorithm
The Davis–Putnam–Logemann–Loveland algorithm is a complete, backtracking-based algorithm for deciding the satisfiability of propositional logic formulae in conjunctive normal form, i.e. for solving the CNF-SAT problem....

-style search with theory-specific solvers (T-solvers) that handle conjunctions
Logical conjunction
In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

(ANDs) of predicates from a given theory. This approach is referred to as the lazy approach.

Dubbed DPLL(T) , this architecture gives the responsibility of Boolean reasoning to the DPLL-based SAT solver which, in turn, interacts with a solver for theory T through a well-defined interface. The theory solver need only worry about checking the feasibility of conjunctions of theory predicates passed on to it from the SAT solver as it explores the Boolean search space of the formula. For this integration to work well, however, the theory solver must be able to participate in propagation and conflict analysis, i.e., it must be able to infer new facts from already established facts, as well as to supply succinct explanations of infeasibility when theory conflicts arise. In other words, the theory solver must be incremental and backtrackable
Backtracking
Backtracking is a general algorithm for finding all solutions to some computational problem, that incrementally builds candidates to the solutions, and abandons each partial candidate c as soon as it determines that c cannot possibly be completed to a valid solution.The classic textbook example...

.

## SMT for undecidable theories

Most of the common SMT approaches support decidable
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...

theories. However, many real-world systems can only be modelled by means of non-linear arithmetic over the real numbers involving transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...

s, e.g. an aircraft and its behavior. This fact motivates an extension of the SMT problem to non-linear theories, e.g. determine whether

where

is satisfiable. Then, such problems become undecidable
Undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is impossible to construct a single algorithm that always leads to a correct yes-or-no answer....

in general. (It is important to note, however, that the theory of real closed field
Real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...

s, and thus the full first order theory of the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...

s, are decidable
Decidability (logic)
In logic, the term decidable refers to the decision problem, the question of the existence of an effective method for determining membership in a set of formulas. Logical systems such as propositional logic are decidable if membership in their set of logically valid formulas can be effectively...

using quantifier elimination
Quantifier elimination
Quantifier elimination is a concept of simplification used in mathematical logic, model theory, and theoretical computer science. One way of classifying formulas is by the amount of quantification...

. This is due to Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...

.) The first order theory of the natural numbers with addition (but not multiplication), called Presburger arithmetic
Presburger arithmetic
Presburger arithmetic is the first-order theory of the natural numbers with addition, named in honor of Mojżesz Presburger, who introduced it in 1929. The signature of Presburger arithmetic contains only the addition operation and equality, omitting the multiplication operation entirely...

, is also decidable. Since multiplication by constants can be implemented as nested additions, the arithmetic in many computer programs can be expressed using Presburger arithmetic, resulting in decidable formulas.

Examples of SMT solvers addressing Boolean combinations of theory atoms from undecidable arithmetic theories over the reals are ABsolver, which employs a classical DPLL(T) architecture with a non-linear optimization packet as (necessarily incomplete) subordinate theory solver, and HySAT-2, building on a unification of DPLL SAT-solving and interval constraint propagation called the iSAT algorithm .

## SMT solvers

The table below summarizes some of the features of the many available SMT solvers. The column "SMT-LIB" indicates compatibility with the SMT-LIB language; many systems marked 'yes' may support only older versions of SMT-LIB, or offer only partial support for the language. The column "CVC" indicates support for the CVC language. The column "DIMACS" indicates support for the DIMACS format.

Projects differ not only in features and performance, but also in the viability of the surrounding community, its ongoing interest in a project, and its ability to contribute documentation, fixes, tests and enhancements. Based on these measures, it appears that the most vibrant, well-organized projects are OpenSMT, STP and CVC4.
Platform Features Notes
Name OS License SMT-LIB CVC DIMACS Built-in theories API SMT-COMP http://www.smtcomp.org/
ABsolver Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

CPL
In computing, the CPL is a free software / open-source software license published by IBM. The Free Software Foundation and Open Source Initiative have approved the license terms of the CPL....

linear arithmetic, non-linear arithmetic C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

no DPLL-based
Alt-Ergo Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Mac OS
Mac OS
Mac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

Free software
Free software
Free software, software libre or libre software is software that can be used, studied, and modified without restriction, and which can be copied and redistributed in modified or unmodified form either without restriction, or with restrictions that only ensure that further recipients can also do...

empty theory, linear arithmetic OCaml 2008 based on congruence closure
Barcelogic Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

Proprietary empty theory, difference logic C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

2009 DPLL-based, congruence closure
Beaver Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

BSD
BSD licenses are a family of permissive free software licenses. The original license was used for the Berkeley Software Distribution , a Unix-like operating system after which it is named....

bitvectors OCaml 2009 SAT-solver based
Boolector Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

GPLv3 bitvectors, arrays C 2009 SAT-solver based
CVC3 Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

BSD
BSD licenses are a family of permissive free software licenses. The original license was used for the Berkeley Software Distribution , a Unix-like operating system after which it is named....

empty theory, linear arithmetic, arrays, tuples, types, records, bitvectors, quantifiers C/C++ 2010 proof output to HOL
Higher-order logic
In mathematics and logic, a higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and a stronger semantics...

CVC4 Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

BSD
BSD licenses are a family of permissive free software licenses. The original license was used for the Berkeley Software Distribution , a Unix-like operating system after which it is named....

pre-alpha 2010 active development; pre-alpha
Decision Procedure Toolkit (DPT) Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

Apache
The Apache License is a copyfree free software license authored by the Apache Software Foundation . The Apache License requires preservation of the copyright notice and disclaimer....

OCaml no DPLL-based
iSAT Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

Proprietary non-linear arithmetic no DPLL-based
MathSAT Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

Proprietary empty theory, difference logics, linear arithmetic 2010 DPLL-based
MiniSmt Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

LGPL non-linear arithmetic 2010 Yices-based
OpenSMT Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

GPLv3 empty theory, differences, linear arithmetic, bitvectors C++ 2010 lazy SMT Solver
SatEEn ? Proprietary linear arithmetic, difference logic none 2009
SONOLAR Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

Proprietary bitvectors C 2010 SAT-solver based
Spear Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Mac OS
Mac OS
Mac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

Proprietary bitvectors 2008
STP Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, OpenBSD
OpenBSD
OpenBSD is a Unix-like computer operating system descended from Berkeley Software Distribution , a Unix derivative developed at the University of California, Berkeley. It was forked from NetBSD by project leader Theo de Raadt in late 1995...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

, Mac OS
Mac OS
Mac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

MIT
The MIT License is a free software license originating at the Massachusetts Institute of Technology . It is a permissive license, meaning that it permits reuse within proprietary software provided all copies of the licensed software include a copy of the MIT License terms...

bitvectors, arrays C, C++
C++
C++ is a statically typed, free-form, multi-paradigm, compiled, general-purpose programming language. It is regarded as an intermediate-level language, as it comprises a combination of both high-level and low-level language features. It was developed by Bjarne Stroustrup starting in 1979 at Bell...

, Python
Python (programming language)
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...

, OCaml, Java
Java (programming language)
Java is a programming language originally developed by James Gosling at Sun Microsystems and released in 1995 as a core component of Sun Microsystems' Java platform. The language derives much of its syntax from C and C++ but has a simpler object model and fewer low-level facilities...

2010 SAT-solver based
SWORD Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

Proprietary bitvectors 2009
UCLID Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

BSD
BSD licenses are a family of permissive free software licenses. The original license was used for the Berkeley Software Distribution , a Unix-like operating system after which it is named....

empty theory, linear arithmetic, bitvectors, and constrained lambda (arrays, memories, cache, etc.) no SAT-solver based, written in Moscow ML
Moscow ML
Moscow ML is an implementation of Standard ML.The codebase is derived from Caml Light.The latest release is 2.01. Supported platforms include Unix, Windows, Mac OS and .NET.- External links :*...

. Input language is SMV model checker. Well-documented!
veriT Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

BSD
BSD licenses are a family of permissive free software licenses. The original license was used for the Berkeley Software Distribution , a Unix-like operating system after which it is named....

empty theory, difference logic C/C++ 2010 SAT-solver based
Yices Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Mac OS
Mac OS
Mac OS is a series of graphical user interface-based operating systems developed by Apple Inc. for their Macintosh line of computer systems. The Macintosh user experience is credited with popularizing the graphical user interface...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

Proprietary 2009
Z3 Linux
Linux
Linux is a Unix-like computer operating system assembled under the model of free and open source software development and distribution. The defining component of any Linux system is the Linux kernel, an operating system kernel first released October 5, 1991 by Linus Torvalds...

, Windows
Microsoft Windows
Microsoft Windows is a series of operating systems produced by Microsoft.Microsoft introduced an operating environment named Windows on November 20, 1985 as an add-on to MS-DOS in response to the growing interest in graphical user interfaces . Microsoft Windows came to dominate the world's personal...

Proprietary empty theory, linear arithmetic, bitvectors, arrays, quantifiers C, .Net, OCaml 2008