B737-300 ACR TAXIES ACROSS ACTIVE RWY WITHOUT CLRNC.

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LNDG AT ZZZ; US; ILS RWY 4R WITH WX RPTED 2 1/2 MI AND 800 FT BROKEN. APCHING OM AT 180 KTS; ADVISED TO SLOW TO LNDG SPD BECAUSE OF 60 KT OVERTAKE ON ACFT Y; B757; ACFT IN FRONT OF US. AT THE SAME TIME ACFT ABOUT 3-4 MI BEHIND US WAS BEING GIVEN VECTORS THROUGH FINAL FOR SPACING. RWY WAS WET WITH LIGHT RAIN. STOPPED STRAIGHT AHEAD ON RWY THEN EXPEDITED OFF ON TXWY K. BECAUSE OF POS OF B757 ACFT HOLDING SHORT OF RWY 4L ON TXWY K WAS WORRIED WE WOULD NOT TOTALLY CLR RWY 4R. AS WE APCHED CLOSE IN BEHIND THE B757; TWR CLRED RJ INTO POS TO HOLD ON RWY 4L AND CLRED ACR TO CROSS RWY 4L. THEN I HEARD TWR GIVE US TURN L ON TXWY P HOLD SHORT OF SOMETHING -- 'ARE YOU GOING TO INTL TERMINAL OR RB?' BECAUSE WE WERE APCHING THE HOLD LINE FOR RWY 4L AND BECAUSE OF NOISE IN THE COCKPIT; I THOUGHT I HAD JUST NOT CLRLY HEARD THE XING CLRNC AT THE BEGINNING OF OUR TAXI INSTRUCTIONS. I ASKED THE FO IF WE WERE CLRED TO CROSS AND I THOUGHT HE GAVE ME AN AFFIRMATIVE RESPONSE. I HAD MY ARPT DIAGRAM OUT BUT DID NOT HAVE THE TIME TO LOOK AT IT BECAUSE OF THE CLOSENESS OF THE B757 AND EVERYTHING ELSE THAT WAS HAPPENING SO QUICKLY. I WAS NOT FAMILIAR WITH WHERE TXWY P WAS; BUT NORMAL TAXI INSTRUCTIONS CLRING RWY 4L WAS A L TURN. CHKED THE FAR END OF RWY 4L AS WE CROSSED THE HOLD BAR FOR RWY 4L. RJ WAS HOLDING IN POS AT FAR END OF RWY 4L. ABOUT 5-10 FT OVER HOLD BAR FOR RWY 4L; FO ASKED TWR IF WE WERE CLRED TO CROSS RWY 4L. TWR RESPONDED 'I NEVER ISSUED SUCH A CLRNC; TAXI TXWY S; TXWY A TO THE RAMP.' WE WERE GOING TOO FAST TO STOP WITHOUT GOING 1/2 WAY ONTO THE RWY; SO BEST COURSE OF ACTION WAS TO KEEP GOING. AFTER WE CLRED RWY 4L; RJ WAS CLRED FOR TKOF. PROCEEDED TO THE RAMP WITHOUT FURTHER INCIDENT. TALKED TO FO AND ASKED HIM ABOUT HIS RESPONSE TO MY ASKING HIM IF WE WERE CLRED TO CROSS RWY 4L. THE FO SAID HE NEVER HEARD ME ASK HIM NOR DID HE RESPOND TO ANY QUESTIONS THAT I MIGHT HAVE ASKED HIM AT THE TIME. I RUSHED MYSELF IN AN EFFORT TO MAKE SURE I WOULD NOT BE IN THE WAY OF ACFT THAT I THOUGHT WERE BEHIND ME. BECAUSE OF THE OUTSTANDING WORK OF THE TWR OPERATOR; THERE WAS NEVER A CONFLICT FROM THE MISUNDERSTANDING I HAD OF THE TWR'S INSTRUCTIONS. SINCE WE WERE PAST TXWY P WHEN WE GOT OUR INSTRUCTIONS; I NEEDED TO STOP AT THE HOLD BAR FOR RWY 4L AND RECONFIRM WITH TWR WHAT THEY WANTED US TO DO. IN THE FUTURE; I WILL NOT PROCEED BEYOND A HOLD BAR UNTIL I AM SURE THAT I HAVE PERSONALLY HEARD THE CLR TO CROSS INSTRUCTIONS FROM TWR. SUPPLEMENTAL INFO FROM ACN 545203: WE TUCKED IN CLOSE BEHIND THE B757 ACFT WITH THE NOSE OF OUR ACFT WHAT I'D CONSIDER WELL PAST TXWY P AND WERE ISSUED THE FOLLOWING TAXI CLRNC: 'ACFT X; L ON TXWY P; HOLD SHORT OF TXWY L; ARE YOU INTL OR GOING TO RB TODAY?' (I WROTE THIS CLRNC ON MY CLIP BOARD.) WHEN I LOOKED UP; WE WERE MOVING FORWARD BEHIND THE B757. AT THAT TIME; I QUERIED THE TWR AND ASKED IF WE WERE CLRED TO CROSS RWY 4L AND THE RESPONSE WAS 'I NEVER ISSUED SUCH A CLRNC; TAXI TXWY S TO TXWY A TO THE RAMP.' THE CAPT AND I WERE CONFUSED BY THE TAXI INSTRUCTIONS. THE NOSE OF THE ACFT WAS PAST TXWY P AND THERE IS NO WAY TO GO L ON TXWY P FROM OUR LOCATION AND HOLD SHORT OF TXWY L. AT NO TIME WERE WE EVER TOLD TO HOLD SHORT OF RWY 4L WHICH WOULD HAVE BEEN VERY CLR TO THE COCKPIT.