Final Report Summary - CORTST-PROGRAMS (Testing and Correcting Programs with Applications to Codes)
General:
--------
To goal of this project is to get new insights into super fast testing and correcting of functions and programs.
Testing means that checking whether a given function satisfies a given global property is done by performing *few* number of of calls to the function. Correcting means that a function that is close to satisfy a global required property, could be automatically corrected to satisfy the property.
We have found new relations between testing and correcting of functions/programs to seemingly unrelated areas:
-------------------------------------------------------------------------------------------------------------------
- Symmetric groups and symmetric codes
- Expander graphs
- High dimensional expanders
- Topology and geometry
A description of the main results
---------------------------------
Edge Transitive Ramanujan Graphs and Highly Symmetric LDPC Good Codes
----------------------------------------------------------------------
Symmetric codes are those in which all coordinates look the same. Namely, there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all low-degree codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a conflict between 1/rate and density (constraint length). In a work with Alexander Lubotzky we present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves an important open problem about the possible existence of such codes.
Locally Testable Codes and Expanders
------------------------------------
Expanders and Error correcting codes are two celebrated notions, coupled together, e.g. in the seminal work on Expander Codes by Sipser and Spielman that yields some of the best codes. In recent years there is a growing interest in locally testable codes (LTCs). It is known that Expander Codes based on *best* expanders can NOT yield LTCs, so it seems that expansion is somewhat contradicting to the notion of LTCs. In a work with Irit Dinur we show that, nevertheless, the graph associated with an LTC *must* be essentially an expander. Namely, every LTC can be decomposed into a constant number of "basic" codes whose associated graph is an expander.
Dense locally testable codes cannot have constant rate and distance
-------------------------------------------------------------------
A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important open question is whether there exist LTCs that have constant rate, constant relative distance, and that can be tested with a constant number of queries. Such LTCs are called "good".
In a work with Irit Dinur we show that dense LTCs cannot be good. The density of a tester is roughly the average number of distinct local views in which a coordinate participates. An LTC is dense if it has a tester with density \omega(1).
Cohomological codes
---------------------
Higher dimensional expanders are natural candidates for obtaining locally testable and correctable codes, because they offer a strong form of redundancy that is essential for testability/correctability. This form of redundancy is lacking by their one-dimensional analogue (namely, expander graphs). Hence, the known expander codes,constructed from one-dimensional expanders are not locally testable. In a work with David Kazhadan and Alex Lubotzky we manage to construct for the first time non linear codes with good distance from two dimensional expanders. We use some topological properties of these complexes to derive local testability of these codes. These new understandings should lead to a new era in testing and correcting of functions and programs. In a work with Shai Evra we generalize this paradigm and obtain LTCs from high dimensional expanders of EVERY degree. The ability to obtain locally testable codes from expanders of higher degrees enable us to get codes of better rate which are locally testable.
High Dimensional expanders and property testing
---------------------------------------------
In this work we show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable.
Using this connection, we derive several testability results.
This draw new connections between testing/correcting to seemingly unrelated fields such as topology and probability.
Bounded degree high dimensional expanders
-----------------------------------------
Expander graphs are bounded degree graphs that are highly connected. These objects had tremendous impact on computer science and mathematics since there invention some 40 years ago. In the last 10 years combinatorists and topologists have searched for bounded degree higher dimensional expanders.
Using the connection we have discovered between high dimensional expanders and property testing, we were able to come up with the FIRST bounded degree high dimensional expanders. Our findings solved an important open question that was studied by different communities of researchers.
This direction was pursued in works with David Kazhdan, Alex Lubotzky, and Shai Evra
Local testing and topological overlapping
----------------------------------------
Our investigation of local testability of functions and its connection to topology was useful is tackling the following open question.
A configuration of n points in the plain (R^2), where each point is connected to all the rest by a strait line, form a O(n^3) triangles.
It is known that for such configurations there always exists a point in R^2 (in the plain) that is covered by constant fraction of the formed triangles. Topologist have asked if this still holds if we consider bounded degree configurations (namely each point participate ONLY in a constant number of triangles so overall there are only O(n) triangles in the configuration). Moreover they were interested in the case that each point is connected to the other points by SOME continuous functions (that is NOT necessarily a strait line). A configuration of points that meet these these requirements is called a "bounded degree configuration with the topological overlapping property". For many years the existences of bounded degree configuration with the topological overlapping property exist was open. It was not clear that such a configuration could at all exist.
In a recent work with Uli Wagner and Dominic Dotterrer we have shown that a configuration of points that form high dimensional expander has the topological overlapping property. Using the bounded degree expanders that we have found, we came up with a complete solution to the question of bounded degree configuration with the topological overlapping property.
Summary of results:
-----------------------
We have found many new relations between testability/correctability of functions to symmetric codes, expansion, high dimensional expansion, topological overlapping and more. All these new relations enabled us to construct new locally testable codes and functions based on topology of some highly symmetric simplicial complexes. These codes/functions that we have obtained form a completely new type of locally testable functions that are very different than the state of the art. It is suggestive that functions that encode in them some topological structure are of good source for local testability/correctability. Moreover, our understating of testability and its properties let us derive new important results in these seemingly unrelated fields as topology and high dimensional expanders.
Following is a concrete list of our achievements:
-------------------------------------------------
We have managed to get new relations between testing/correcting of functions and programs to symmetry, symmetric codes, expander graphs, high dimensional expanders and topology.
We have found better explicit constructions of symmetric codes than were previously known. In particular, we construct for the first time, "good" LDPC symmetric codes. These codes could be potentially testable.
We have managed to get new relations between testing/correcting of functions and programs to high dimensional expanders and topology.
We have managed to find a bounded degree high dimensional expanders of every dimension. This solves a major open problem raised by Gromov and other already more than 10 years ago.
We have managed to show that high dimensional expanders imply locally testable codes. We use the Ramanujan complexes to derive new type of locally testable codes
We have managed to show that high dimensional expanders imply non linear codes with good distance. We have used the Ramanujan complexes to derive a concrete set of codes with such properties
We have managed to show that high dimensional expansion imply a property in topology known as the topological overlapping property.
We have shown that there exist bounded degree high dimensional expanders with the topological overlapping property, this solves a major open problem that was open over 10 years.
We have basically drawn new and important relations between testing and topology that are already influencing new large group of researchers in both communities and beyond.
--------
To goal of this project is to get new insights into super fast testing and correcting of functions and programs.
Testing means that checking whether a given function satisfies a given global property is done by performing *few* number of of calls to the function. Correcting means that a function that is close to satisfy a global required property, could be automatically corrected to satisfy the property.
We have found new relations between testing and correcting of functions/programs to seemingly unrelated areas:
-------------------------------------------------------------------------------------------------------------------
- Symmetric groups and symmetric codes
- Expander graphs
- High dimensional expanders
- Topology and geometry
A description of the main results
---------------------------------
Edge Transitive Ramanujan Graphs and Highly Symmetric LDPC Good Codes
----------------------------------------------------------------------
Symmetric codes are those in which all coordinates look the same. Namely, there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all low-degree codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a conflict between 1/rate and density (constraint length). In a work with Alexander Lubotzky we present a symmetric LDPC code with constant rate and constant distance (i.e. good LDPC code) that its constraint space is generated by the orbit of one constant weight constraint under a group action. Our construction provides the first symmetric LDPC good codes. This solves an important open problem about the possible existence of such codes.
Locally Testable Codes and Expanders
------------------------------------
Expanders and Error correcting codes are two celebrated notions, coupled together, e.g. in the seminal work on Expander Codes by Sipser and Spielman that yields some of the best codes. In recent years there is a growing interest in locally testable codes (LTCs). It is known that Expander Codes based on *best* expanders can NOT yield LTCs, so it seems that expansion is somewhat contradicting to the notion of LTCs. In a work with Irit Dinur we show that, nevertheless, the graph associated with an LTC *must* be essentially an expander. Namely, every LTC can be decomposed into a constant number of "basic" codes whose associated graph is an expander.
Dense locally testable codes cannot have constant rate and distance
-------------------------------------------------------------------
A q-query locally testable code (LTC) is an error correcting code that can be tested by a randomized algorithm that reads at most q symbols from the given word. An important open question is whether there exist LTCs that have constant rate, constant relative distance, and that can be tested with a constant number of queries. Such LTCs are called "good".
In a work with Irit Dinur we show that dense LTCs cannot be good. The density of a tester is roughly the average number of distinct local views in which a coordinate participates. An LTC is dense if it has a tester with density \omega(1).
Cohomological codes
---------------------
Higher dimensional expanders are natural candidates for obtaining locally testable and correctable codes, because they offer a strong form of redundancy that is essential for testability/correctability. This form of redundancy is lacking by their one-dimensional analogue (namely, expander graphs). Hence, the known expander codes,constructed from one-dimensional expanders are not locally testable. In a work with David Kazhadan and Alex Lubotzky we manage to construct for the first time non linear codes with good distance from two dimensional expanders. We use some topological properties of these complexes to derive local testability of these codes. These new understandings should lead to a new era in testing and correcting of functions and programs. In a work with Shai Evra we generalize this paradigm and obtain LTCs from high dimensional expanders of EVERY degree. The ability to obtain locally testable codes from expanders of higher degrees enable us to get codes of better rate which are locally testable.
High Dimensional expanders and property testing
---------------------------------------------
In this work we show that the high dimensional expansion property as defined by Gromov, Linial and Meshulam, for simplicial complexes is a form of testability. Namely, a simplicial complex is a high dimensional expander iff a suitable property is testable.
Using this connection, we derive several testability results.
This draw new connections between testing/correcting to seemingly unrelated fields such as topology and probability.
Bounded degree high dimensional expanders
-----------------------------------------
Expander graphs are bounded degree graphs that are highly connected. These objects had tremendous impact on computer science and mathematics since there invention some 40 years ago. In the last 10 years combinatorists and topologists have searched for bounded degree higher dimensional expanders.
Using the connection we have discovered between high dimensional expanders and property testing, we were able to come up with the FIRST bounded degree high dimensional expanders. Our findings solved an important open question that was studied by different communities of researchers.
This direction was pursued in works with David Kazhdan, Alex Lubotzky, and Shai Evra
Local testing and topological overlapping
----------------------------------------
Our investigation of local testability of functions and its connection to topology was useful is tackling the following open question.
A configuration of n points in the plain (R^2), where each point is connected to all the rest by a strait line, form a O(n^3) triangles.
It is known that for such configurations there always exists a point in R^2 (in the plain) that is covered by constant fraction of the formed triangles. Topologist have asked if this still holds if we consider bounded degree configurations (namely each point participate ONLY in a constant number of triangles so overall there are only O(n) triangles in the configuration). Moreover they were interested in the case that each point is connected to the other points by SOME continuous functions (that is NOT necessarily a strait line). A configuration of points that meet these these requirements is called a "bounded degree configuration with the topological overlapping property". For many years the existences of bounded degree configuration with the topological overlapping property exist was open. It was not clear that such a configuration could at all exist.
In a recent work with Uli Wagner and Dominic Dotterrer we have shown that a configuration of points that form high dimensional expander has the topological overlapping property. Using the bounded degree expanders that we have found, we came up with a complete solution to the question of bounded degree configuration with the topological overlapping property.
Summary of results:
-----------------------
We have found many new relations between testability/correctability of functions to symmetric codes, expansion, high dimensional expansion, topological overlapping and more. All these new relations enabled us to construct new locally testable codes and functions based on topology of some highly symmetric simplicial complexes. These codes/functions that we have obtained form a completely new type of locally testable functions that are very different than the state of the art. It is suggestive that functions that encode in them some topological structure are of good source for local testability/correctability. Moreover, our understating of testability and its properties let us derive new important results in these seemingly unrelated fields as topology and high dimensional expanders.
Following is a concrete list of our achievements:
-------------------------------------------------
We have managed to get new relations between testing/correcting of functions and programs to symmetry, symmetric codes, expander graphs, high dimensional expanders and topology.
We have found better explicit constructions of symmetric codes than were previously known. In particular, we construct for the first time, "good" LDPC symmetric codes. These codes could be potentially testable.
We have managed to get new relations between testing/correcting of functions and programs to high dimensional expanders and topology.
We have managed to find a bounded degree high dimensional expanders of every dimension. This solves a major open problem raised by Gromov and other already more than 10 years ago.
We have managed to show that high dimensional expanders imply locally testable codes. We use the Ramanujan complexes to derive new type of locally testable codes
We have managed to show that high dimensional expanders imply non linear codes with good distance. We have used the Ramanujan complexes to derive a concrete set of codes with such properties
We have managed to show that high dimensional expansion imply a property in topology known as the topological overlapping property.
We have shown that there exist bounded degree high dimensional expanders with the topological overlapping property, this solves a major open problem that was open over 10 years.
We have basically drawn new and important relations between testing and topology that are already influencing new large group of researchers in both communities and beyond.