bap-core-theory 2.0.0 · OCaml Package

The Basic theory of bitvector machines.

The Basic theory could be derived from the Minimal theory using the Basic functor.

include Minimal

include Init

val var : 'a var -> 'a pure

var v is the value of the variable v.

val unk : 'a Value.sort -> 'a pure

unk s an unknown value of sort s.

This term explicitly denotes a term with undefined or unknown value.

val let_ : 'a var -> 'a pure -> 'b pure -> 'b pure

let_ v exp body bind the value of exp to v body.

val ite : bool -> 'a pure -> 'a pure -> 'a pure

ite c x y is x if c evaluates to b1 else y.

include Bool

val b0 : bool

b0 is false aka 0 bit

val b1 : bool

b1 is true aka 1 bit

val inv : bool -> bool

inv x inverts x.

val and_ : bool -> bool -> bool

and_ x y is a conjunction of x and y.

val or_ : bool -> bool -> bool

or_ x y is a disjunction of x and y.

include Bitv

val int : 's Bitv.t Value.sort -> word -> 's bitv

int s x is a bitvector constant x of sort s.

val msb : 's bitv -> bool

msb x is the most significant bit of x.

val lsb : 's bitv -> bool

lsb x is the least significant bit of x.

val neg : 's bitv -> 's bitv

neg x is two-complement unary minus

val not : 's bitv -> 's bitv

not x is one-complement negation.

val add : 's bitv -> 's bitv -> 's bitv

add x y addition modulo 2^'s

val sub : 's bitv -> 's bitv -> 's bitv

sub x y subtraction modulo 2^'s

val mul : 's bitv -> 's bitv -> 's bitv

mul x y multiplication modulo 2^'s

val div : 's bitv -> 's bitv -> 's bitv

div x y unsigned division modulo 2^'s truncating towards 0.

The division by zero is defined to be a vector of all ones of size 's.

val sdiv : 's bitv -> 's bitv -> 's bitv

sdiv x y is signed division of x by y modulo 2^'s,

The signed division operator is defined in terms of the div operator as follows:

                             /
                             | div x y : if not mx /\ not my
                             | neg (div (neg x) y) if mx /\ not my
                 x sdiv y = <
                             | neg (div x (neg y)) if not mx /\ my
                             | div (neg x) (neg y) if mx /\ my
                             \

              where mx = msb x, and my = msb y.

val modulo : 's bitv -> 's bitv -> 's bitv

modulo x y is the remainder of div x y modulo 2^'s.

val smodulo : 's bitv -> 's bitv -> 's bitv

smodulo x y is the signed remainder of div x y modulo 2^'s.

This version of the signed remainder where the sign follows the dividend, and is defined via the % = modulo operation as follows

                           /
                           | x % y : if not mx /\ not my
                           | neg (neg x % y) if mx /\ not my
            x smodulo y = <
                           | neg (x % (neg y)) if not mx /\ my
                           | neg (neg x % neg y) mod m if mx /\ my
                           \

            where mx = msb x  and my = msb y.

val logand : 's bitv -> 's bitv -> 's bitv

logand x y is a bitwise logical and of x and y.

val logor : 's bitv -> 's bitv -> 's bitv

logor x y is a bitwise logical or of x and y.

val logxor : 's bitv -> 's bitv -> 's bitv

logxor x y is a bitwise logical xor of x and y.

val shiftr : bool -> 's bitv -> 'b bitv -> 's bitv

shiftr s x m shifts x right by m bits filling with s.

val shiftl : bool -> 's bitv -> 'b bitv -> 's bitv

shiftl s x m shifts x left by m bits filling with s.

val sle : 'a bitv -> 'a bitv -> bool

sle x y binary predicate for singed less than or equal

val ule : 'a bitv -> 'a bitv -> bool

ule x y binary predicate for unsigned less than or equal

val cast : 'a Bitv.t Value.sort -> bool -> 'b bitv -> 'a bitv

cast s b x casts x to sort s filling with b.

If m = size s - size (sort b) > 0 then m bits b are prepended to the most significant part of the vector.

Otherwise, if m <= 0, i.e., it is a narrowing cast, then the value of b doesn't affect the result of evaluation.

val concat : 'a Bitv.t Value.sort -> 'b bitv list -> 'a bitv

concat s xs concatenates a list of vectors xs.

All vectors are concatenated from left to right (so that the most significant bits of the resulting vector are defined by the first element of the list and so on).

The result of concatenation are then casted to sort s with all extra bits (if any) set to zero.

val append : 'a Bitv.t Value.sort -> 'b bitv -> 'c bitv -> 'a bitv

append s x y appends x and y and casts to s.

Appends x and y so that in the resulting vector the vector x occupies the most significant part and y the least significant part. The resulting vector is then casted to the sort s with extra bits (if any) set to zero.

include Memory

val load : ('a, 'b) mem -> 'a bitv -> 'b bitv

load m k is the value associated with the key k in the memory m.

val store : ('a, 'b) mem -> 'a bitv -> 'b bitv -> ('a, 'b) mem

store m k x a memory m in which the key k is associated with the word x.

include Effect

val perform : 'a Effect.sort -> 'a eff

perform s performs a generic effect of sort s.

val set : 'a var -> 'a pure -> data eff

set v x changes the value stored in v to the value of x.

val jmp : _ bitv -> ctrl eff

jmp dst passes the control to a program located at dst.

val goto : label -> ctrl eff

goto lbl passes the control to a program labeled with lbl.

val seq : 'a eff -> 'a eff -> 'a eff

seq x y performs effect x, after that perform effect y.

val blk : label -> data eff -> ctrl eff -> unit eff

blk lbl data ctrl a labeled sequence of effects.

val repeat : bool -> data eff -> data eff

repeat c data repeats data effects until the condition c holds.

val branch : bool -> 'a eff -> 'a eff -> 'a eff

branch c lhs rhs if c holds then performs lhs else rhs.

val zero : 'a Bitv.t Value.sort -> 'a bitv

zero s creates a bitvector of all zeros of sort s.

val is_zero : 'a bitv -> bool

is_zero x holds if x is a bitvector of all zeros

val non_zero : 'a bitv -> bool

non_zero x holds if x is not a zero bitvector.

val succ : 'a bitv -> 'a bitv

succ x is the successor of x.

val pred : 'a bitv -> 'a bitv

pred x is the predecessor of x.

val nsucc : 'a bitv -> int -> 'a bitv

nsucc x n is the nth successor of x.

val npred : 'a bitv -> int -> 'a bitv

npred x is the nth predecessor of x.

val high : 'a Bitv.t Value.sort -> 'b bitv -> 'a bitv

high s x is the high cast of x to sort s.

if m = size (sort x) - size s > 0 then high s x evaluates to m most significant bits of x; else if m = 0 then high s x evaluates to the value of x; else m zeros are prepended to the most significant part of x.

val low : 'a Bitv.t Value.sort -> 'b bitv -> 'a bitv

low s x = cast s b0 x is the low cast of x to sort s.

if m = size (sort x) - size s < 0 then low s x evaluates to size s least significant bits of x; else if m = 0 then high s x evaluates to the value of x; else m zeros are prepended to the most significant part of x.

val signed : 'a Bitv.t Value.sort -> 'b bitv -> 'a bitv

signed s x = cast s (msb x) x is the signed cast of x to sort s.

if m = size s - (size (sort x)) > 0 then extends x on its most significant side with m bits equal to msb x; else if m = 0 then signed s x evaluates to x else it evaluates to size s least significant bits of x.

val unsigned : 'a Bitv.t Value.sort -> 'b bitv -> 'a bitv

unsigned s x = cast s b0 x is the unsigned cast of x to sort s.

if m = size s - (size (sort x)) > 0 then extends x on its most significant side with m zeros; else if m = 0 then unsigned s x evaluates to x else it evaluates to size s least significant bits of x.

val extract : 'a Bitv.t Value.sort -> 'b bitv -> 'b bitv -> _ bitv -> 'a bitv

extract s hi lo x extracts bits of x between hi and lo.

Extracts hi-lo+1 consecutive bits of x from hi to lo, and casts them to sort s with any excessive bits set to zero.

Note that hi-lo+1 is taken modulo 2^'b, so this operation is defined even if lo is greater or equal to hi.

val loadw : 'c Bitv.t Value.sort -> bool -> ('a, _) mem -> 'a bitv -> 'c bitv

loadw s e m k loads a word from the memory m.

if e evaluates to b1 (big endian case), then the term evaluates to low s (m[k] @ m[k+1] @ ... @ m[k+n] ), else the term evaluates to low s (m[k+n] @ m[k+n-1] @ ... @ m[k] ) where x @ y is append (vals m) x y, and n is the smallest natural number such that n * size (vals (sort m)) >= size s, an m[i] is the i'th value of memory m. .

This is a generic function that loads words using specified endianess (e could be read as is_big_endian) with arbitrary byte size.

val storew : bool -> ('a, 'b) mem -> 'a bitv -> 'c bitv -> ('a, 'b) mem

storew e m k x stores a word in the memory m.

Splits the word x into chunks of size equal to the size of memory elements and lays them out in the big endian order, if e evaluates to b1, or little endian order otherwise.

val arshift : 'a bitv -> 'b bitv -> 'a bitv

arshift x m = shiftr (msb x) m arithmetically shifts x right by m bits.

val rshift : 'a bitv -> 'b bitv -> 'a bitv

rshift x m = shiftr b0 x m shifts x right by m bits

val lshift : 'a bitv -> 'b bitv -> 'a bitv

lshift x y = shiftl b0 x m shifts x left by m bits.

val eq : 'a bitv -> 'a bitv -> bool

eq x y holds if x and y are bitwise equal.

val neq : 'a bitv -> 'a bitv -> bool

eq x y holds if x and y are not bitwise equal.

val slt : 'a bitv -> 'a bitv -> bool

slt x y signed strict less than.

val ult : 'a bitv -> 'a bitv -> bool

ult x y unsigned strict less than.

val sgt : 'a bitv -> 'a bitv -> bool

sgt x y signed strict greater than.

val ugt : 'a bitv -> 'a bitv -> bool

ugt x y unsigned strict greater than.

val sge : 'a bitv -> 'a bitv -> bool

sge x y signed greater or equal than.

val uge : 'a bitv -> 'a bitv -> bool

sge x y signed greater or equal than.

package bap-core-theory