""" Tools to emit SMTLIB bitvector queries encoding concrete RTLs containing only primitive instructions. """ from .primitives import GROUP as PRIMITIVES, prim_from_bv, prim_to_bv, bvadd,\ bvult, bvzeroext, bvsplit, bvconcat from cdsl.ast import Var from cdsl.types import BVType from .elaborate import elaborate try: from typing import TYPE_CHECKING, Tuple # noqa from cdsl.xform import Rtl, XForm # noqa from cdsl.ast import VarMap # noqa from cdsl.ti import VarTyping # noqa except ImportError: TYPE_CHECKING = False def bvtype_to_sort(typ): # type: (BVType) -> str """Return the BitVec sort corresponding to a BVType""" return "(_ BitVec {})".format(typ.bits) def to_smt(r): # type: (Rtl) -> Tuple[str, VarMap] """ Encode a concrete primitive Rtl r sa SMTLIB 2.0 query. Returns a tuple (query, var_m) where: - query is the resulting query. - var_m is a map from Vars v with non-BVType to their Vars v' with BVType s.t. v' holds the flattend bitvector value of v. """ assert r.is_concrete() # Should contain only primitives primitives = set(PRIMITIVES.instructions) assert set(d.expr.inst for d in r.rtl).issubset(primitives) q = "" m = {} # type: VarMap # Build declarations for any bitvector Vars for v in r.vars(): typ = v.get_typevar().singleton_type() if not isinstance(typ, BVType): continue q += "(declare-fun {} () {})\n".format(v.name, bvtype_to_sort(typ)) # Encode each instruction as a equality assertion for d in r.rtl: inst = d.expr.inst # For prim_to_bv/prim_from_bv just update var_m. No assertion needed if inst == prim_to_bv: assert isinstance(d.expr.args[0], Var) m[d.expr.args[0]] = d.defs[0] continue if inst == prim_from_bv: assert isinstance(d.expr.args[0], Var) m[d.defs[0]] = d.expr.args[0] continue if inst in [bvadd, bvult]: # Binary instructions assert len(d.expr.args) == 2 and len(d.defs) == 1 lhs = d.expr.args[0] rhs = d.expr.args[1] df = d.defs[0] assert isinstance(lhs, Var) and isinstance(rhs, Var) if inst in [bvadd]: # Normal binary - output type same as args exp = "(= {} ({} {} {}))".format(df, inst.name, lhs, rhs) else: # Comparison binary - need to convert bool to BitVec 1 exp = "(= {} (ite ({} {} {}) #b1 #b0))"\ .format(df, inst.name, lhs, rhs) elif inst == bvzeroext: arg = d.expr.args[0] df = d.defs[0] assert isinstance(arg, Var) fromW = arg.get_typevar().singleton_type().width() toW = df.get_typevar().singleton_type().width() exp = "(= {} ((_ zero_extend {}) {}))"\ .format(df, toW-fromW, arg) elif inst == bvsplit: arg = d.expr.args[0] assert isinstance(arg, Var) arg_typ = arg.get_typevar().singleton_type() width = arg_typ.width() assert (width % 2 == 0) lo = d.defs[0] hi = d.defs[1] exp = "(and " exp += "(= {} ((_ extract {} {}) {})) "\ .format(lo, width//2-1, 0, arg) exp += "(= {} ((_ extract {} {}) {}))"\ .format(hi, width-1, width//2, arg) exp += ")" elif inst == bvconcat: assert isinstance(d.expr.args[0], Var) and \ isinstance(d.expr.args[1], Var) lo = d.expr.args[0] hi = d.expr.args[1] df = d.defs[0] # Z3 Concat expects hi bits first, then lo bits exp = "(= {} (concat {} {}))"\ .format(df, hi, lo) else: assert False, "Unknown primitive instruction {}".format(inst) q += "(assert {})\n".format(exp) return (q, m) def equivalent(r1, r2, inp_m, out_m): # type: (Rtl, Rtl, VarMap, VarMap) -> str """ Given: - concrete source Rtl r1 - concrete dest Rtl r2 - VarMap inp_m mapping r1's non-bitvector inputs to r2 - VarMap out_m mapping r1's non-bitvector outputs to r2 Build a query checking whether r1 and r2 are semantically equivalent. If the returned query is unsatisfiable, then r1 and r2 are equivalent. Otherwise, the satisfying example for the query gives us values for which the two Rtls disagree. """ # Sanity - inp_m is a bijection from the set of inputs of r1 to the set of # inputs of r2 assert set(r1.free_vars()) == set(inp_m.keys()) assert set(r2.free_vars()) == set(inp_m.values()) # Note that the same rule is not expected to hold for out_m due to # temporaries/intermediates. # Rename the vars in r1 and r2 with unique suffixes to avoid conflicts src_m = {v: Var(v.name + ".a", v.get_typevar()) for v in r1.vars()} dst_m = {v: Var(v.name + ".b", v.get_typevar()) for v in r2.vars()} r1 = r1.copy(src_m) r2 = r2.copy(dst_m) # Convert inp_m, out_m in terms of variables with the .a/.b suffixes inp_m = {src_m[k]: dst_m[v] for (k, v) in inp_m.items()} out_m = {src_m[k]: dst_m[v] for (k, v) in out_m.items()} # Encode r1 and r2 as SMT queries (q1, m1) = to_smt(r1) (q2, m2) = to_smt(r2) # Build an expression for the equality of real Cretone inputs of r1 and r2 args_eq_exp = "(and \n" for v in r1.free_vars(): args_eq_exp += "(= {} {})\n".format(m1[v], m2[inp_m[v]]) args_eq_exp += ")" # Build an expression for the equality of real Cretone outputs of r1 and r2 results_eq_exp = "(and \n" for (v1, v2) in out_m.items(): results_eq_exp += "(= {} {})\n".format(m1[v1], m2[v2]) results_eq_exp += ")" # Put the whole query toghether q = '; Rtl 1 declarations and assertions\n' + q1 q += '; Rtl 2 declarations and assertions\n' + q2 q += '; Assert that the inputs of Rtl1 and Rtl2 are equal\n' + \ '(assert {})\n'.format(args_eq_exp) q += '; Assert that the outputs of Rtl1 and Rtl2 are not equal\n' + \ '(assert (not {}))\n'.format(results_eq_exp) return q def xform_correct(x, typing): # type: (XForm, VarTyping) -> str """ Given an XForm x and a concrete variable typing for x build the smtlib query asserting that x is correct for the given typing. """ assert x.ti.permits(typing) # Create copies of the x.src and x.dst with their concrete types src_m = {v: Var(v.name, typing[v]) for v in x.src.vars()} src = x.src.copy(src_m) dst = x.apply(src) dst_m = x.dst.substitution(dst, {}) # Build maps for the inputs/outputs for src->dst inp_m = {} out_m = {} for v in x.src.vars(): if v.is_input(): inp_m[src_m[v]] = dst_m[v] elif v.is_output(): out_m[src_m[v]] = dst_m[v] # Get the primitive semantic Rtls for src and dst prim_src = elaborate(src) prim_dst = elaborate(dst) return equivalent(prim_src, prim_dst, inp_m, out_m)