Introduction

Tau Language is about enabling you to create software that elegantly adapts to meet your requirements in a fully formal, correct-by-construction manner.

Tau Language is uniquely vastly expressive while retaining decidability and allows you to refer to sentences in Tau Language. And, as you'll see, offers many advantages when compared to other common formal languages and programming languages.

Tau Language offers a future where whether you're a developer, end-user, or stakeholder, Tau-built software will be able to accurately adapt to be what you, or a group of users, want it to be.

More precisely, the Tau Language is a logical software specification language. It allows you to write constraints about software, check for satisfiability, and run a candidate program that meets those specifications. All related logical tasks are decidable. It is based on the first-order theory of atomless Boolean algebras.

For a more detailed explanation of the theory behind the Tau Language, please refer to:

TABA book Theories and Applications of Boolean Algebras by Ohad Asor (In works).
Youtube lecture series on Atomless Boolean Algebra by Ohad Asor.

This README.md is structured in the following way: first, we provide a detailed explanation of the Tau Language, including the syntax and semantics of the language. Then, we provide a quick start guide to start using the Tau Language. Finally, we provide a guide to install the Tau Framework in your system, how to use the command line interface and the Tau REPL (Read-Eval-Print-Loop) that allows you to interact with the Tau Language.

We also provide a list of known issues, future work and how to submit issues.

To skip straight to our quick start section click Quick start.

Installing the Tau Framework

Linux

Currently we automatically build the following binaries packages (AMD64 architecture):

deb (Debian/Ubuntu): tau-0.7-Linux.deb
rpm (Fedora): tau-0.7-Linux.rpm

The executable is installed in /usr/bin/tau.

Windows

For windows we provide a convenient installer that includes the tau executable and also a zip file:

Installer: tau-0.7-win64.exe
Zip file: tau-0.7-win64.zip

MacOS (not available yet)

A macOS installer will be available in the future.

Compiling the source code

To compile the source code you need a recent C++ compiler supporting C++23, e.g. GCC 13.1.0. You also need at least a cmake version 3.22.1 installed in your system. The only code dependency is libboost.

After cloning:

git clone https://github.com/IDNI/tau-lang.git

you can run either the release.sh or debug.sh or relwithdebinfo.sh scripts to build the binaries.

To build with doxygen documentation:

# Compiles the source code in release mode and also the documentation
./release.sh -DBUILD_DOC=ON
# Compiles the source code in debug mode and also the documentation
./debug.sh -DBUILD_DOC=ON
# Compiles the source code in release mode with debug information and also the documentation
./relwithdebinfo.sh -DBUILD_DOC=ON

Once you have compiled the source code you can run the tau executable to execute Tau programs. The tau executable is located in either build-Release or build-Debug or build-RelWithDebInfo.

Quick start

To start using the Tau Language, download the latest release from the GitHub page. Once you have downloaded and installed the executable (see the Section Installing the Tau Framework), you can run it from the command line just by typing tau.

The programming model underlying the Tau Language is fully declarative. You specify, possibly only very implicitly, how the current and previous inputs and outputs are related, at each point of time. So what you write in the Tau Language is not a program, but a specification, or spec, which represents all programs that meet the specification. Once you run a specification, you actually run one automatically-chosen representative program from that set.

In the scope of the Tau Language, a program means that for all inputs, at each point in time, exist outputs, that clearly do not depend on future inputs ("time-compatible"). Implied from this definition is that all programs run indefinitely no matter what the inputs are.

For example, the following program:

o1[t] = 0

states that the output o1 at all time points (t) has to be 0. Similarly the following program:

o1[t] = i1[t]

states that the output o1 at time t has to be the same as the input i1 at time t. In the above examples, o1 and i1 are IO variables. They are used to define

the inputs and outputs of the specified programs and also declare their type.

An example of how to define IO variables is the following:

tau i1 = console
tau o1 = console

In the above case we specify that i1 and o1 are of type tau and they take values from the console (lets say stdin for the input and stdout for the output). You can define as IO streams also files:

tau i1 = ifile("input.in")
tau o1 = ofile("output.out")

Note that those Tau specs define only one program, each (there's a caveat in this statement but we shall treat it later on). An example of a Tau spec that specifies infinitely many programs would be:

o1[t] & i1[t] = 0

Here & is conjunction in the Boolean algebra from which the inputs and outputs are taken from. This spec says that the conjunction has to be empty. You can clearly consider more complicated specifications, e.g.:

o1[t] & o1[t-1] & i1[t] = 0 || o1[t] = i1[t]

which states that the current output, and the previous output, and the current input, has to be 0, or, the output has to equal the input. Note the difference between Boolean and Logical operators. The former are &|', and the latter are &&,||,!.

In order to simplify the process of writing and running Tau programs, we allow to define functions and predicates, possibly by means of recurrence relations. The following is a simple predicate defined by a recurrence relation, which takes as argument a Tau formula:

f[0](y) := T
f[n](y) := f[n - 1](y)

which you can use in your program as follows:

o1[t] = 0 && f(i1[t])

Or also, you can use the following recurrence relation definition

g[0](y) := 0
g[n](y) := g[n](y)'

which takes as argument a Boolean function and alternates between 0 and 1 depending on the parity of n.

To get all the details about the Tau Language, please refer to the Section The Tau Language. You can find there all the details about the syntax and semantics of the language.

In the demos folder you can find lots of examples regarding how to use the Tau Language, its semantics and workings.

The Tau Language

In the Tau Language you define how the current and previous inputs and outputs are related over time, using the first-order theory of atomless Boolean algebras extended with a time dimension. For example you can write o1[t] & o1[t-1] & i1[t] = 0 which would mean that the current output, and the previous output, and the current input, have to have an empty intersection. The set-theoretic perspective of Boolean algebra is justfied by Stone's representation theorem for Boolean algebras, but more concretely, when a Tau spec is treated as a BA element, it can be seen as a set of all programs that admit that spec, and the Boolean operations are simply the set-theoretic union/intersection/complementation.

Tau specifications

At the top level, a Tau specification (we also say spec) is a collection of "always" and "sometimes" statements applied to local specifications (expressed by tau, see section Tau formulas), combined by the logical connectives and, or and not, denoted by &&, || and ! respectively. For example a well-formed Tau specification is

(always local_spec1) && (sometimes local_spec2)

where local_spec1 and local_spec2 are formulas as in section Tau formulas. We say local specification because a formula tau can only talk about a fixed (though arbitrary) point in time.

Recall from section Variables that there are input and output stream variables. For example the output stream variable o1[t-2] means "the value in output stream number 1 two time-steps ago". So o1[t] would mean "the value in output stream number 1 at the current time-step". Likewise, there are input stream variables like i1[t]. It means "the input in the input stream 1 at the current time-step". Input streams can also have an offset in order to speak about past inputs. For example i2[t-3] means "the input in the input stream 2 three time-steps ago". As explained in section Variables, input and output streams currently need to be defined before running a Tau specification.

In all above cases, t is a free variable and refers to the current time at each point in time. The key point now is that an always statement will quantify all scoped t universally, while a sometimes statement will quantify them existentially. For example the specification always o1[t] = 0 says that at all time-steps the output stream number 1 will write 0. Similarly, the specification sometimes o1[t] = 0 says that there exists a time-step at which the output stream 1 will write 0.

Formally, the grammar for Tau specifications is

spec         => tau | always tau | sometimes tau | spec && spec | spec || spec | !spec
rr           => (rec_relation)* spec.
rec_relation => tau_rec_relation | term_rec_relation

A Tau specification without a mentioning of "always" or "sometimes" is implicitly assumed to be an "always" statement. The rr in the above grammar describes how to add function and predicate definitions directly to the formula. In REPL they can also be provided separately as explained in the Tau REPL subsection Functions, predicates and input/output stream variables. See the subsection Functions and predicates for the definitions of tau_rec_relation and term_rec_relation.

Note that instead of writing always and sometimes you can also use box [] and diamond <>, respectively.

Using this notation, a slightly bigger example of a Tau spec would be

    ([] o1[t] i1[t] = 0 && (i1[t] != 1 -> o1[t] != 0)) && (<> o1[t] = i1[t]')

which reads: at each point of time, the output should be disjoint from the input. If the input is not 1, then the output is not zero. And, at least once during execution, the output equals the complement of the input.

Tau formulas

In traditional programming languages, we have decisions,... In the case of Tau Language, well formed formulas deal with that. They provide us an extra logical layer on basic computations (given by Boolean formulas) allowing us to use conditional and similar constructions.

Well formed formulas are given in Tau Language by the following grammar:

tau => "(" tau "&&" tau ")" | "!" tau | "(" tau "^" tau ")" | "(" tau "||" tau ")"
	| "(" tau "->" tau ")" | "(" tau "<->" tau ")" | "(" tau "?" tau ":" tau ")"
	| "(" term "=" term ")" | "(" term "!=" term ")" | "("term "<" term")"
	| "("term "!<" term")"	| "(" term "<=" term ")" | "(" term "!<=" term ")"
	| "(" term ">" term ")"	| "(" term "!>" term ")" | "all" var tau
	| "ex" var tau | tau_ref | T | F.

where

tau stands for a well formed sub-formula and the operators &, !, ^, |, ->, <-> and ? stand for conjunction, negation, exclusive-or, disjunction, implication, equivalence and conditional, in the usual sense, (respectively).
the operators =, <, <= and > stand for equality, less than, less or equal than and greater than; the operators !=, !<, !<= and !> denote their negations,
all stands for the universal quantifier and ex for the existential one,
tau_ref is a reference to a predicate (see the Subsection Functions and predicates), and finally,
T and F represent the true and false values.

For example, the following is a valid well formed formula:

(x && y || z) = 0 -> (x = 0 ? y = 0 : z = 0)

where x, y and z are variables.

Boolean functions

One of the key ingredients of the Tau Language are the Boolean functions (Boolean combinations of variables, and constants over some chosen atomless (or finite -to be develop-) Boolean algebra and variables).They are given by the following grammar:

term => "("term "&" term")" | term "'" | "("term "+" term")" | "("term "|" term")"
	 | term_ref | constant | uninterpreted_constant | var | "0" | "1".

where

term stands for a well formed sub-formula and the operators &, ', ^ and | stand for conjunction, negation, exclusive-or and disjunction (respectively).
term_ref is a call to the given recurrence relation (see the Subsection Functions and Predicates),
constant stands for an element of the Boolean algebras (see Subsection Constants for details),
uninterpreted_constant stands for an uninterpreted constant of the Boolean algebra, they are assumed to be existentially quantified in the context of the formula. The syntax is a follows:

uninterpreted_constant => "<:" name ">".

var is a variable of type a Boolean algebra element (see Subsection Variables for details), and
finally, 0 and 1 stands for the given elements in the corresponding Boolean algebra.

For example, the following is a valid expression in terms of a Boolean function:

(x & y | (z ^ 0))

where x, y and z are variables.

Functions and predicates

Another key concept in the Tau Language are functions and predicates. They are given by the following grammar where term_rec_relation defines the syntax for a function and tau_rec_relation defines the syntax for a predicate:

term_rec_relation => term_ref ":=" term.
term_ref          => sym "[" (offset)+  "]" "(" variable+ ")".
tau_rec_relation  => tau_ref ":=" tau.
tau_ref           => sym "[" (offset)+  "]" "(" variable+ ")".

where sym is the name of the function or predicate (it has to be a sequence of letters and numbers starting by a letter) and offset is a positive integer or a variable.

Examples of functions and predicates are:

g[0](Y) := 1.
g[n](Y) := g[n - 1](Y).

or also

g[0](Y) := T.
g[n](Y) := h[n - 1](Y).
h[0](Y) := F.
h[n](Y) := g[n - 1](Y).

Constants

Constants in the Tau Language are elements of the underlying Boolean algebras, usually other than 0 and 1 that have a dedicated syntax.

In the REPL, we support two Boolean algebras: the Tau Boolean algebra and the simple Boolean function algebra. The Tau Boolean algebra is an extensional Boolean algebra that encodes Tau specifications over base algebras (in the REPL case we only support the simple Boolean functions as base one).

The syntax for the first case, the Tau Boolean algebra, is the following:

constant => "{" tau "}" [":" "tau"].

i.e. we may have a Tau formula seen as a Boolean algebra element (you can omit the type, as tau is the default type). For example, the following is a valid constant in the Tau Boolean algebra:

{ ex x ex y ex z (x & y | z) = 0 }:tau

or even

{ { ex x ex y ex z (x & y | z) = 0 }:tau = 0 }:tau

where x, y and z are variables.

Regarding the simple Boolean function algebra, the syntax is the following:

constant => "{" sbf "}" ":" "sbf".

where the grammar for simple Boolean functions is the following:

sbf => "("sbf "&" sbf")" | sbf "'" | "("sbf "^" sbf")" | "("sbf "+" sbf")"
	| "("sbf "|" sbf")"	| var | "0" | "1".

where sbf stands for a simple Boolean function, and the operators &, ', (^|+) and | stand for conjunction, negation, exclusive-or and disjunction; var stands for a variable of type Boolean algebra element, and finally, 0 and 1 stand for the given elements in the simple Boolean algebra.

A constant in the simple Boolean function algebra is for example:

{ (x & y | z) }:sbf

where x, y and z are variables.

Variables

Variables range over Boolean algebra elements. In the REPL you can work with open formulas (i.e. when variables are not quantified), but a specification makes sense only for closed formulas. Their syntax depends on whether the charvar option is enabled or not. If it is enabled, the syntax is a single character followed by digits. Otherwise, the syntax is an arbitrary string of chars.

We also have IO variables, which are actually infinite sequence of Boolean algebra elements, each, indexed by positions in the sequence. They are used to define the inputs and outputs of the program. The name for an input variable is i{num} (e.g. i1) whereas ouput variables are of the form o{num} (in the near future we will allow arbitrary names for IO variables). They should always be referred to with reference to time (i.e. position in the sequence), and the syntax is i2[t] or o1[t] where t always denotes time, but also can be i1[t-1] or o2[t-3] (always a constant lookback).

As commented later on, IO variables need to be defined before the spec is run. For example, the following is a valid definition of IO variables:

tau i1 = console.
tau o1 = console.

where tau points to the type of the variables (in this case, tau formulas) and console stands for the input/output stream of the variable (in this case, the console).

In the near future we will allow arbitrary names for the IO variables.

Finally, we have the uninterpreted constant context, which are implicitly existentially quantified variables. The syntax is <:name>.

Reserved symbols

Tau Language has a set of reserved symbols that cannot be used as identifiers. In particular, we insist that T and F are reserved for true and false values respectively in tau formulas and 0 and 1 stand for the corresponding Boolean algebra elements.

Command line interface

The general form of tau executable command line is:

tau [ options ] [ <program> ]

where [ options ] are the command line options and [ <program> ] is the Tau program you want to run. If you omit the tau program, the Tau REPL will be started.

The general options are the following:

Option	Description
-h, --help	detailed information about options
-l, --license	show the license
-v, --version	show the version of the executable
--------------------	-------------------------------------------------------
-V, --charvar	charvar (enabled by default)
-S, --severity	severity level (trace/debug/info/error)
-I, --indenting	indenting of the formulas
-H, --highlighting	syntax highlighting

whereas the REPL specific options are:

Options	Description
-e, --evaluate	REPL command to be evaluated
-s, --status	display status
-c, --color	use colors
-d, --debug	debug mode

The Tau REPL

The Tau REPL is a command line interface that allows you to interact with the Tau Language. It is a simple and easy to use tool that allows you to write and execute Tau programs on the go.

Basic REPL commands

The Tau REPL provides a set of basic commands that allow you to obtain help, version information, exit the REPL, clear the screen and so on. The syntax of the commands is the following:

help|h [<command>]: shows a general help message or the help message of a specific command.
version|v: shows the version of the Tau REPL. The version of the Tau REPL corresponds to the repo commit.
quit|q or exit: exits the Tau REPL.
clear|c: clears the screen.

REPL options

You have several options at your disposal to configure the Tau REPL. In order to set or get the value of an option you can use the following commands:

get [<option>]: shows all configurable settings and their values or a single one if its name is provided.
set <option> [=] <value>: sets a configurable option to a desired value.
toggle <option>: toggle an option between on/off.

The options you have at your disposal are the following:

c|color|colors: Can be on/off. Controls usage of terminal colors in its output. It's on by default.
s|status: Can be on/off. Controls status visibility in the prompt. It's on by default.
sev|severity: Possible values are trace/debug/info/error. The value determines how much information the REPL will provide. This is set to error by default.
h|hilight|highlight: Can be on/off. Controls usage of highlighting in the output of commands. It's off by default.
i|indent|indentation: Can be on/off. Controls usage of indentation in the output of commands. It's on by default.
charvar|v: Can be on/off. Controls usage of character variables in the REPL. It's on by default.
d|dbg|debug: Can be on/off. Controls debug mode. It's off by default.

Functions, predicates and input/output stream variables

As in other programming languages, you can define functions, predicates (both possibly using recurrence relations) but also input and output stream variables. The syntax of the commands is the following:

definitions|defs: shows all the definitions of the current program. That includes the definitions of functions, predicates and the input/output stream variables.
definitions|defs <number>: shows the definition of the given function or predicate.
rec_relation: defines a function or predicate supporting the usage of recurrence relations. See the Tau Language Section for more information.
<type> i<number> = console | ifile(<filename>): defines an input stream variable. The input variable can read values from the console or from a provided file. <type> can be either tau or sbf (simple Boolean function) at the moment.
<type> o<number> = console | ofile(<filename>): defines an output stream variable. The output variable can write values to the console or into a file. can be eithertauorsbf` (simple Boolean function) at the moment.

Memory related commands

All the results are stored in the REPL memory. You can also store well-formed Tau formulas or Boolean functions for later reference. To do so, you can use the following syntax:

tau|term: store a tau formula or a Boolean function in the REPL memory.

If you want to consult the REPL memory contents, you can use the following commands:

history|hist: show all the previously stored Tau expressions.
history|hist <repl_memory>: show the Tau expression at the specified REPL memory position.

In general, to retrieve a Tau expression from the REPL memory, you can use the following syntax:

%: to retrieve the Tau expression stored at the latest position
%<number>: to retrieve the Tau expression stored at position <number>
%-<number>: to retrieve the Tau expression stored at the latest position minus <number>

Expression manipulation

You can substitute expressions into other expressions or instantiate variables in expressions. The syntax of the commands is the following:

subst|s <repl_memory|tau|term> [<repl_memory|tau|term>/<repl_memory|tau|term>]: substitutes a memory, well-formed formula or Boolean function by another one in the given expression (beeing this one a memory position, well-formed formula or Boolean function).
instantiate|inst|i <repl_memory|tau> [<var>/<repl_memory|term>]: instantiates a variable by a memory position, well-formed formula or Boolean function in the given well-formed or Boolean function expression.
instantiate|inst|i <repl_memory|term> [<var>/<repl_memory|term>]: instantiates a variable by a memory position or Boolean function in the given expression.

Logical procedures

The Tau REPL also provides a set of logical procedures that allow you to check several aspects of the given program/well-formed formulas/Boolean functions. The syntax of the commands is the following:

valid <repl_memory|tau>: checks if the given program is valid.
sat <repl_memory|tau>: checks if the given program is satisfiable.
unsat <repl_memory|tau>: checks if the given program is unsatisfiable.
solve <repl_memory|tau>: solves the given system of equations given by the well-formed formula. It only computes one solution.
normalize|n <repl_memory|rr|ref|tau|term>: normalizes the given expression. See the TABA book for details.
qelim <repl_memory|tau>: performs quantifier elimination on the given expression.

Normal forms

Also, the Tau REPL includes several transformation procedures to standard forms. The syntax of the commands is as follows:

dnf <repl_memory|tau|term>: computes the disjunctive normal form of the given expression.
cnf <repl_memory|tau|term>: computes the conjunctive normal form of the given expression.
nnf <repl_memory|tau|term>: computes the negation normal form of the given expression.
mnf <repl_memory|tau|term>: computes the minterm normal form of the given expression.
snf <repl_memory|tau|term>: computes the strong normal form of the given expression.
onf <var> <repl_memory|tau>: computes the order normal form of the given expression with respect to the given variable.

Program execution

Finally, you can run the given program once you have defined the IO variables as you need. The syntax of the commands is:

run|r <repl_memory|tau>: runs the given program.

Known issues

This is a short list of known issues that will be fixed in a subsequent release:

Issue in Fixed Point Calculations
Incorrect type inference of IO variables in certain cases.
Normalization:
- Error in DNF/CNF conversions in the normalizer
“Sometimes” keyword issues:
- The correctness of the satisfiability algorithm is still not fully verified when input variables appear under "sometimes".
- Allow constants positions under "sometimes".
- Redundant “Sometimes” statements are not detected.
Simplification:
- Simplification of Boolean equations may take longer time in a few cases.
- Path simplification algorithm does not take equalities between variables into account leading to later blow ups.
Minor errors in windows REPL

Future work

Add support for redefinition of recurrence relations.
Add support for arbitrary names for IO variables.
Improve the performance of normalization of Boolean functions.

Submitting issues

Like any other open-source project on GitHub, you can submit issues using the following link: Tau Language issues.

License

Tau Language is licensed under the following terms: Tau Language License

Authors

The Tau Language has been developed by the following authors:

Ohad Asor
David Castro Esteban
Tomáš Klapka
Lucca Tiemens

Name		Name	Last commit message	Last commit date
Latest commit History 1,647 Commits
.github/workflows		.github/workflows
cmake		cmake
demos		demos
docs		docs
examples		examples
external		external
parser		parser
src		src
tests		tests
.gitignore		.gitignore
.gitmodules		.gitmodules
CMakeLists.txt		CMakeLists.txt
DEVELOPMENT.md		DEVELOPMENT.md
LICENSE.txt		LICENSE.txt
README.md		README.md
REWRITING_DAG.md		REWRITING_DAG.md
benchmark.sh		benchmark.sh
build.sh		build.sh
clean.sh		clean.sh
debug.sh		debug.sh
echo		echo
packages.sh		packages.sh
release.sh		release.sh
relwithdebinfo.sh		relwithdebinfo.sh
save_benchmarks.sh		save_benchmarks.sh
tau_grammar.md		tau_grammar.md
test-debug.sh		test-debug.sh
test-release.sh		test-release.sh
test-relwithdebinfo.sh		test-relwithdebinfo.sh
w64-debug.sh		w64-debug.sh
w64-packages.sh		w64-packages.sh
w64-release.sh		w64-release.sh
wine-test.sh		wine-test.sh

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Table of contents

Introduction

Installing the Tau Framework

Linux

Windows

MacOS (not available yet)

Compiling the source code

Quick start

The Tau Language

Tau specifications

Tau formulas

Boolean functions

Functions and predicates

Constants

Variables

Reserved symbols

Command line interface

The Tau REPL

Basic REPL commands

REPL options

Functions, predicates and input/output stream variables

Memory related commands

Expression manipulation

Logical procedures

Normal forms

Program execution

Known issues

Future work

Submitting issues

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Table of contents

Introduction

Installing the Tau Framework

Linux

Windows

MacOS (not available yet)

Compiling the source code

Quick start

The Tau Language

Tau specifications

Tau formulas

Boolean functions

Functions and predicates

Constants

Variables

Reserved symbols

Command line interface

The Tau REPL

Basic REPL commands

REPL options

Functions, predicates and input/output stream variables

Memory related commands

Expression manipulation

Logical procedures

Normal forms

Program execution

Known issues

Future work

Submitting issues

License

Authors

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