Parametric Presburger Arithmetic Tristram Bogart Universidad de los Andes 13 March 2018
Quasi-polynomials A function g : N ! Z is: I quasi-polynomial (QP) if there exists a period m and polynomials f 0 , . . . , f m � 1 2 Q [ t ] such that g ( t ) = f i ( t ) , for t ⌘ i mod m . I eventually quasi-polynomial (EQP) if it agrees with a quasi-polynomial for all su ffi ciently large t .
Quasi-polynomials A function g : N ! Z is: I quasi-polynomial (QP) if there exists a period m and polynomials f 0 , . . . , f m � 1 2 Q [ t ] such that g ( t ) = f i ( t ) , for t ⌘ i mod m . I eventually quasi-polynomial (EQP) if it agrees with a quasi-polynomial for all su ffi ciently large t . Example 3 t 2 � 2 8 1 for t ⌘ 0 (mod 3) 3 t > < j k t 2 � 2 t +1 3 t 2 � 2 1 3 t + 1 = for t ⌘ 1 (mod 3) 3 3 3 t 2 � 2 > 1 3 t for t ⌘ 2 (mod 3) :
Ehrhart’s Theorem Theorem (Ehrhart, 1962) Let A 2 Z m ⇥ d , b 2 Z m , and suppose the rational polyhedron P = { x 2 R d : A x b } is a polytope (i.e., that P is bounded.) For each t 2 N , let S t = tP \ Z d = { x 2 Z d : A x b t } . Then the function L P ( t ) = | S t | is quasi-polynomial.
Ehrhart’s Theorem Theorem (Ehrhart, 1962) Let A 2 Z m ⇥ d , b 2 Z m , and suppose the rational polyhedron P = { x 2 R d : A x b } is a polytope (i.e., that P is bounded.) For each t 2 N , let S t = tP \ Z d = { x 2 Z d : A x b t } . Then the function L P ( t ) = | S t | is quasi-polynomial. 8 9 2 � 2 0 3 2 1 3 > > > > 0 � 2 x � 1 < = ( x , y ) 2 R 2 : 6 7 6 7 Example P = 6 7 6 7 2 0 y 1 4 5 4 5 > > > > 0 2 1 : ; ( ( t + 1) 2 if t is even; L P ( t ) = t 2 if t is odd.
Parametric Polytopes Theorem (Chen-Li-Sam, 2012) Let A ( t ) 2 Z [ t ] m ⇥ d , b ( t ) 2 Z [ t ] m . For each t 2 N , let S t = { x 2 Z d : A ( t ) x b ( t ) } . Then the function g ( t ) = | S t | (if finite) is eventually quasi-polynomial. Ehrhart’s Theorem is the case where A is constant and b is linear of the form b ( t ) = b t .
An Example of the Chen-Li-Sam Theorem Example (Kevin Woods): ( ( t 2 � 2 t + 2 ) | 2 x + (2 t � 2) y | ( x , y ) 2 Z 2 : S t = t 2 � 2 t + 2 | (2 � 2 t ) x 1 + 2 x 2 | ( t 2 � 2 t + 2 for t odd | S t | = t 2 � 2 t + 5 for t even
The Frobenius problem Suppose a 1 , . . . , a s 2 N and gcd( a 1 , . . . , a s ) = 1. Find the maximum element of S = { x 2 N : ¬ 9 y 1 , . . . , y s 2 N [ x = y 1 a 1 + · · · + y s a s ] } , Example: a 1 = 3 , a 2 = 8. S C = { 0 , 3 , 6 , 8 , 9 , 11 , 12 , 14 , 15 , 16 , . . . } . g (3 , 8) = 13.
The Frobenius problem Suppose a 1 , . . . , a s 2 N and gcd( a 1 , . . . , a s ) = 1. Find the maximum element of S = { x 2 N : ¬ 9 y 1 , . . . , y s 2 N [ x = y 1 a 1 + · · · + y s a s ] } , Example: a 1 = 3 , a 2 = 8. S C = { 0 , 3 , 6 , 8 , 9 , 11 , 12 , 14 , 15 , 16 , . . . } . g (3 , 8) = 13. Parametric version: for each t 2 N , find the maximum of S t = { x 2 N : ¬ 9 y 1 , . . . , y s 2 N [ x = y 1 a 1 ( t ) + · · · + y s a s ( t )] } , the complement of the projection of the integer points in a parametric polyhedron.
The Frobenius problem Suppose a 1 , . . . , a s 2 N and gcd( a 1 , . . . , a s ) = 1. Find the maximum element of S = { x 2 N : ¬ 9 y 1 , . . . , y s 2 N [ x = y 1 a 1 + · · · + y s a s ] } , Example: a 1 = 3 , a 2 = 8. S C = { 0 , 3 , 6 , 8 , 9 , 11 , 12 , 14 , 15 , 16 , . . . } . g (3 , 8) = 13. Parametric version: for each t 2 N , find the maximum of S t = { x 2 N : ¬ 9 y 1 , . . . , y s 2 N [ x = y 1 a 1 ( t ) + · · · + y s a s ( t )] } , the complement of the projection of the integer points in a parametric polyhedron. Theorem (Bobby Shen, 2015) Let a 1 ( t ) , . . . , a s ( t ) 2 Z [ t ] be such that for t � 0 , a i ( t ) > 0 and gcd( a 1 ( t ) , . . . , a s ( t )) = 1 . Then g ( a 1 ( t ) , . . . , a s ( t )) is eventually quasi-polynomial.
A Common Framework A parametric Presburger set (as defined by Woods) is a family of sets S t ✓ Z d , one for each natural number t , defined using a Boolean combination of linear inequalities of the form a ( t ) · x b ( t ) where a ( t ) 2 Z [ t ] d , b ( t ) 2 Z [ t ], plus quantifiers 8 x i , 9 x j over variables other than t . All sets S t covered by the Chen-Li-Theorem as well as parametric Frobenius sets (i.e. subsemigroups of N , or even of N k ) are parametric Presburger sets.
Properties of integer point set families Let S t , for t 2 N , be a family of subsets of Z d . Consider the following properties that S t might or might not have.
Properties of integer point set families Let S t , for t 2 N , be a family of subsets of Z d . Consider the following properties that S t might or might not have. (1) The set of t such that S t is nonempty is eventually periodic. (2) There exists an EQP g : N ! N such that, if S t has finite cardinality, then g ( t ) = | S t | . (3) There exists a function x : N ! Z d , whose coordinate functions are EQPs, such that, if S t is nonempty, then x ( t ) 2 S t .
Properties of integer point set families Let S t , for t 2 N , be a family of subsets of Z d . Consider the following properties that S t might or might not have. (1) The set of t such that S t is nonempty is eventually periodic. (2) There exists an EQP g : N ! N such that, if S t has finite cardinality, then g ( t ) = | S t | . (3) There exists a function x : N ! Z d , whose coordinate functions are EQPs, such that, if S t is nonempty, then x ( t ) 2 S t . (4) (Assuming S t ✓ N d ) There exists a period m such that, for su ffi ciently large t ⌘ i mod m , P n i j =1 α ij z q ij ( t ) z x = X , (1 � z b i1 ( t ) ) · · · (1 � z b iki ( t ) ) x 2 S t where α ij 2 Q , and the coordinate functions of q ij , b ij : N ! Z d are polynomials with the b ij ( t ) eventually lexicographically positive.
Main Theorems Theorem (Woods, 2014) 1. Let S t be any family of subsets of N d . If S t satisfies (4), then it also satisfies (1), (2), and (3). 2. If S t ✓ N d is defined by a quantifier-free parametric Presburger formula, then S t satisfies all four of the properties.
Main Theorems Theorem (Woods, 2014) 1. Let S t be any family of subsets of N d . If S t satisfies (4), then it also satisfies (1), (2), and (3). 2. If S t ✓ N d is defined by a quantifier-free parametric Presburger formula, then S t satisfies all four of the properties. Theorem (B-Goodrick-Woods, 2017) Let S t ✓ Z d be any parametric Presburger family. Then Properties (1), (2), and (3) all hold. Furthermore, if S t ✓ N d , then (4) holds.
Quantifier elimination? Theorem (Presburger, 1929) The language ( Z , + , 0 , ) of ordinary Presburger arithmetic, extended by divisibility predicates D c for each positive integer c , admits quantifier elimination. That is, every Presburger set S can be defined by a quantifier-free formula, possibly involving divisibility predicates.
Quantifier elimination? Theorem (Presburger, 1929) The language ( Z , + , 0 , ) of ordinary Presburger arithmetic, extended by divisibility predicates D c for each positive integer c , admits quantifier elimination. That is, every Presburger set S can be defined by a quantifier-free formula, possibly involving divisibility predicates. If the same were to hold for parametric Presburger arithmetic, then our theorem would immediately follow from Woods’ result.
Quantifier elimination? Theorem (Presburger, 1929) The language ( Z , + , 0 , ) of ordinary Presburger arithmetic, extended by divisibility predicates D c for each positive integer c , admits quantifier elimination. That is, every Presburger set S can be defined by a quantifier-free formula, possibly involving divisibility predicates. If the same were to hold for parametric Presburger arithmetic, then our theorem would immediately follow from Woods’ result. However, we do not know of any reasonable language for PPA that admits quantifier elimination.
A ffi ne reduction t ✓ Z d 0 be parametric Presburger families. An Let S t ✓ Z d and S 0 a ffi ne reduction from S 0 t to S t is an EQP-a ffi ne-linear function F : Z d 0 ⇥ N ! Z d such that for every t 2 Z , F restricts to a bijection from S 0 t to S t .
A ffi ne reduction t ✓ Z d 0 be parametric Presburger families. An Let S t ✓ Z d and S 0 a ffi ne reduction from S 0 t to S t is an EQP-a ffi ne-linear function F : Z d 0 ⇥ N ! Z d such that for every t 2 Z , F restricts to a bijection from S 0 t to S t . Proposition A ffi ne reductions preserve Properties (1), (2), (3), and (4).
Proof of the Main Theorem: Step 1 Using logical equivalence, S t can be defined by a parametric Presburger formula with only polynomially-bounded quantifiers and possibly predicates for divisibility by EQP functions.
Proof of the Main Theorem: Step 1 Using logical equivalence, S t can be defined by a parametric Presburger formula with only polynomially-bounded quantifiers and possibly predicates for divisibility by EQP functions. Example S t = { ( x , z ) : 9 y [ x + 1 ty z ^ ty 3 z � x ] }
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