q129/random_k2/random_1.md

		PAL:	Program-aided	Language		Models
LuyuGao1 AmanMadaan1 ShuyanZhou*1 UriAlon1 PengfeiLiu12 YimingYang1 JamieCallan1
GrahamNeubig12
{luyug,amadaan,shuyanzh,ualon,pliu3,yiming,callan,gneubig}@cs.cmu.edu
	Abstract			1.Introduction
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3202 naJ 72 ]LC.sc[ 2v53401.1122:viXra
Untilasrecentlyastwoyearsago,reasoningwasconsidered
Large language	models	(LLMs)	have recently
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tobeoneofthemostsignificantchallengesthatlargelan-
demonstrated an impressive ability to perform guagemodels(LLMs)hadnotyetovercome(Marcus,2018;
arithmetic	and symbolic	reasoning	tasks, when
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				2020;Garcez&Lamb,2020).			Recently,LLMshaveshown
providedwithafewexamplesattesttime(“few- impressivesuccessonawiderangeoftasks,includingcom-
shot prompting”). Much of this success can be monsense (Wei et al., 2021; Sanh et al., 2021; Madaan
attributedtopromptingmethodssuchas“chain-
				etal.,2022),	mathematical(Lewkowyczetal.,2022;Wu
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of-thought”,whichemployLLMsforbothunder- et al., 2022; Mishra et al., 2022), and symbolic reason-
standingtheproblemdescriptionbydecomposing
				ing (Yao	et al.,	2022; Ahn	et al., 2022),	using	few-shot
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solving
it intosteps, as well as each stepof the prompting(Brownetal.,2020).
problem.	WhileLLMsseemtobeadeptatthis
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sortofstep-by-stepdecomposition,LLMsoften Thisprocesshasbeenacceleratedbymethodsthatrequire
				LLMs to	generate	their	explicit reasoning	steps,	such as
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makelogicalandarithmeticmistakesinthesolu-
“chain-of-thought”(Weietal.,2022),“scratchpads”(Nye
tionpart,evenwhentheproblemisdecomposed
correctly. In this paper, we present Program- etal.,2021),and“least-to-most”(Zhouetal.,2022)prompt-
ing. Inparticular,thewidelyusedchain-of-thought(COT)
AidedLanguagemodels(PAL):anovelapproach
methodpresentsthemodelwiththeexplicitintermediate
thatusestheLLMtoreadnaturallanguageprob-
				stepsthatarerequiredtoreachthefinalanswer.					Then,the
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lemsandgenerateprogramsastheintermediate
modelisexpectedtoapplyasimilardecompositiontotheac-
reasoningsteps,butoffloadsthesolutionsteptoa
runtimesuchasaPythoninterpreter. WithPAL, tualtestexample,andconsecutivelyreachanaccuratefinal
				answer(Lingetal.,2017;Aminietal.,2019).				Nevertheless,
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decomposingthenaturallanguageprobleminto
whileLLMscandecomposenaturallanguageproblemsinto
runnablestepsremainstheonlylearningtaskfor
stepsandperformsimplearithmeticoperations,theirperfor-
theLLM,whilesolvingisdelegatedtotheinter-
mancefallsdramaticallywhendealingwithcomplexarith-
preter. We	demonstrate	this	synergy between a
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neuralLLMandasymbolicinterpreteracross13 metic(Hendrycksetal.,2021;Madaan&Yazdanbakhsh,
2022)orlargenumbers(Nogueiraetal.,2021;Qianetal.,
mathematical,symbolic,andalgorithmicreason-
2022). Infact,evenwhenfine-tuningaPaLM-basedmodel
ingtasksfromBIG-BenchHardandotherbench-
				on 164B	tokens	of explicit	mathematical	content,	its two
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marks. In	all these	natural language	reasoning
mostcommonfailuresarereportedly“incorrectreasoning”
tasks, generating	code	using	an LLM and rea-
-----------------	----	-----	---------------	---	---	---	---	---	---
soningusingaPythoninterpreterleadstomore and“incorrectcalculation”(Lewkowyczetal.,2022).
accurateresultsthanmuchlargermodels. Forex- In this paper, we propose Program-Aided Language
ample,PALusingCODEXachievesstate-of-the-
				model (PAL):		a novel method	that uses	an LLM	to read
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artfew-shotaccuracyontheGSM8Kbenchmark naturallanguageproblemsandgenerateprogramsasrea-
ofmathwordproblems,surpassingPaLM-540B soningsteps,butoffloadsthesolutionsteptoaPythoninter-
whichuseschain-of-thoughtbyabsolute15%top-
preter,asillustratedinFigure1.Thisoffloadingleveragesan

1. Our code and data are publicly available at LLMthatcandecomposeanaturallanguageprobleminto http://reasonwithpal.com.

   				programmatic		steps, which	is fortunately	available	using
   contemporarystate-of-the-artLLMsthatarepre-trainedon
   				bothnaturallanguageandprogramminglanguages					(Brown

| ---------------------------------------- | --- | --- | -------------- | ------------------------------------------ | ---- | ------- | ---------------- | --- | -------------- | | *Thefirstthreeauthorscontributedequally. | | | 1LanguageTech- | | | | | | | | | | | | et al., 2020; | Chen | et al., | 2021a; Chowdhery | | et al., 2022). | 2Inspired nologies Institute, Carnegie Mellon University, USA Whilenaturallanguageunderstandinganddecomposition Cognition,USA.

PAL:Program-aidedLanguageModels 2 Chain-of-Thought (Wei et al., 2022) Program-aided Language models (this work) Input Input Q: Roger has 5 tennis balls. He buys 2 more cans of tennis balls. Each can has 3 tennis balls. How many Q: Roger has 5 tennis balls. He buys 2 more cans of tennis balls does he have now? tennis balls. Each can has 3 tennis balls. How many tennis balls does he have now? A: Roger started with 5 tennis balls. tennis_balls = 5 A: Roger started with 5 tennis balls. 2 cans of 3 tennis 2 cans of 3 tennis balls each is balls each is 6 tennis balls. 5 + 6 = 11. The answer is 11. bought_balls = 2 * 3 tennis balls. The answer is Q: The bakers at the Beverly Hills Bakery baked 200 answer = tennis_balls + bought_balls loaves of bread on Monday morning. They sold 93 loaves in the morning and 39 loaves in the afternoon. A grocery Q: The bakers at the Beverly Hills Bakery baked 200 store returned 6 unsold loaves. How many loaves of loaves of bread on Monday morning. They sold 93 loaves bread did they have left? in the morning and 39 loaves in the afternoon. A grocery store returned 6 unsold loaves. How many loaves of bread did they have left? Model Output Model Output A: The bakers started with 200 loaves. They sold 93 in A: The bakers started with 200 loaves the morning and 39 in the afternoon. So they sold 93 + loaves_baked = 200 39 = 132 loaves. The grocery store returned 6 loaves. So They sold 93 in the morning and 39 in the afternoon they had 200 - 132 - 6 = 62 loaves left. loaves_sold_morning = 93 The answer is 62. ❌ loaves_sold_afternoon = 39 The grocery store returned 6 loaves. loaves_returned = 6 The answer is answer = loaves_baked - loaves_sold_morning

• loaves_sold_afternoon + loaves_returned

  print(answer) 74 Figure1: AdiagramillustratingPAL:Givenamathematicalreasoningquestion,Chain-of-thought(left)generatesinterme- diatereasoningstepsoffree-formtext. Incontrast,Program-aidedLanguagemodels(PAL,right)generateintermediate stepsandPythoncode. ThisshiftstheroleofrunningthereasoningstepsfromthelanguagemodeltothePythoninterpreter. Thefinalanswerisobtainedbyrunningthegeneratedreasoningchain. Chain-of-thoughtreasoningis highlightedinblue; PALstepsare highlighted in gray and pink;thePythoninterpreterrunis highlightedinblackandgreen. requireLLMs,solvingandreasoningcanbedonewiththe 2.Background: Few-shotPrompting externalsolver. Thisbridgesanimportantgapinchain-of- Few-shotpromptingleveragesthestrengthoflarge-language thought-likemethods,wherereasoningchainscanbecorrect modelstosolveataskwithasetofkexamplesthatarepro- butproduceanincorrectanswer. vided as part of the test-time input (Brown et al., 2020; We demonstrate the effectiveness of PAL across 13 arith- Liuetal.,2021;Chowdheryetal.,2022),wherek isusu- metic and symbolic reasoning tasks. In all these tasks, allyanumberinthelowsingledigits. Theseinput-output PAL usingCodex(Chenetal.,2021a)outperformsmuch examples {(x ,y )}k are concatenated in a prompt p i i i=1 largermodelssuchasPaLM-540Busingchain-of-thought ≡(cid:104)x ·y (cid:105)(cid:107)(cid:104)x ·y (cid:105)(cid:107)...(cid:107)(cid:104)x ·y (cid:105). where“·”denotes 1 1 2 2 k k prompting. For example, on the popular GSM8K bench- theconcatenationofaninputandoutput,and“(cid:107)”indicate mark, PAL achieves state-of-the-art accuracy, surpassing theconcatenationofdifferentexamples. Duringinference, PaLM-540B withchain-of-thoughtbyabsolute15%top- atestinstancex isappendedtotheprompt,andp(cid:107)x test test 1accuracy. Whenthequestionscontainlargenumbers,a ispassedtothemodelwhichattemptstocompletep(cid:107)x , test datasetwecallGSM-HARD,PALoutperformsCOTbyanab- andtherebygenerateananswery . Notethatsuchfew- test solute40%. Webelievethatthisseamlesssynergybetween shotpromptingdoesnotmodifytheunderlyingLLM. aneuralLLMandasymbolicinterpreterisanessentialstep towardsgeneralandrobustAIreasoners.

					PAL:Program-aidedLanguageModels								3
Weietal.(2022)additionallyaugmenteachin-contextexam- 2cansof3tennisballseachis6, in PAL we also aug-
plewithchainofthought(COT)intermediatesteps. Specifi- ment each such NL step with its corresponding pro-
											tennis	balls	= 5
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cally,eachin-contextexampleintheCOTsetupisatriplet grammaticstatementsuchas and
(cid:104)x ,t ,y (cid:105),wherex andy areinput-outputpairasbefore, bought balls = 2 * 3. Thisway,themodellearns
i i	i	i	i
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andt isanaturallanguagedescriptionofthestepsthatare togenerateaprogramthatwillprovidetheanswerforthe
i
neededtoarriveattheoutputy fromtheinputx . SeeFig- test question, instead of relying on LLM to perform the
				i		i
-----------------	---	----------------------------	---	---	---	----	---------------------	---	---	---	---	---	---
ure1foranexample.		Withtheadditional“thoughts”t				,the	calculationcorrectly.
i
promptissettop≡(cid:104)x		1	·t 1 ·y 1 (cid:105)(cid:107)(cid:104)x	2 ·t	2 ·y 2 (cid:105)(cid:107)...(cid:107)(cid:104)x	k ·t k ·y k (cid:105).
-------------------------	---	---	--------------------------------------	----	-----------------------------------------------	----------------------	---	---	---	---	---	---	---
WepromptthelanguagemodeltogenerateNLintermediate
During inference, the new question x is appended to steps using comment syntax (e.g. “# ...” in Python)
test
thepromptasbeforeandsuppliedtotheLLM.Crucially, suchtheywillbeignoredbytheinterpreter. Wepassthe
themodelistaskedwithgeneratingboththethoughtt generatedprogramt toitscorrespondingsolver,werun
						test			test
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andthefinalanswery . Thisapproachofpromptingthe it,andobtainthefinalrunresulty . Inthisworkweuse
		test									test
---	---	----	---	---	---	---	---	---	---	---	----	---	---
modeltofirstgenerateareasoningprocesst test improves a standard Python interpreter, but this can be any solver,
the accuracy of the answer y across a wide range of interpreteroracompiler.
test
tasks (Wang	et	al., 2022a;	Wei	et al.,	2022;	Zhou et al.,
-----------	---	-----------	---	-------	-----	------------	---	---	---	---	---	---	---
2022;Wangetal.,2022b).
							CraftingpromptsforPAL				Inourexperiments,welever-
---	---	---	---	---	---	---	---------------------	---	---	---	-------------------------	---	---
agedthepromptsofexistingworkwheneveravailable,and
3.Program-aidedLanguageModels
otherwiserandomlyselectedthesamenumber(3-6)ofex-
In a Program-aided Language model, we propose to gen- amples as previous work for creating a fixed prompt for
erate the thoughts t for a given natural language prob- everybenchmark. Inallcases,weaugmentedthefree-form
lemxasinterleavednaturallanguage(NL)andprogram- textpromptsintoPAL-styledprompts,leveragingprogram-
mingconstructssuchasforloopsanddictionarieswhen
ming language		(PL) statements.		Since	we	delegate the
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solution step to an interpreter, we do not provide the fi- needed. Generally,writingPALpromptsiseasyandquick.
nal answers	to	the examples		in our	prompt.	That is, ev-
--------------	---------	------------	------	----------------	---------------	--------------------	------------------------------------------------	-------	-------------	---	--------	----------	-----
							We also	ensure	that variable		names in	the prompt	mean-
ery in-context		example	in PAL	is a	pair (cid:104)x	, t (cid:105), where
						i i	ingfully	reflect	their roles.	For	example,	a variable	that
t =[s	,s ,...,s	]witheachs		∈NL∪PL,asequence
j 1	2	N		i			describesthenumberofapplesinthebasketshouldhavea
oftokensineitherNLorPL.Thecompletepromptisthusp namesuchasnum apples in basket. Thiskeepsthe
≡(cid:104)x ·t	(cid:105)(cid:107)(cid:104)x	·t (cid:105)(cid:107)...(cid:107)(cid:104)x		·t	(cid:105).
--------------	----------------------------	-------------------------------------------	---	---	----------	---	---	---	---	---	---	---	---
1 1 2 2 k k generated code linked to the entities in the question. In
Section6weshowthatsuchmeaningfulvariablenamesare
Given a	test instance		x test , we	append	it to	the prompt,
-------	-------------	---	-----------	------	-----	-----------	---	---	---	---	---	---	---
and p(cid:107)x is fed to the LM. We let the LM generate a critical. Notably, it is also possible to incrementally run
test
predictiont ,whichcontainsboththeintermediatesteps thePLsegmentsandfeedtheexecutionresultsbacktothe
test
							LLMtogeneratethefollowingblocks.					Forsimplicity,	in
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andtheircorrespondingprogrammaticstatements.
ourexperiments,weusedasingle,post-hoc,execution.
							This work	focuses	on	COT-style	reasoning	chain,	but in
---	---	---	---	---	---	---	---------	-------	---	---------	---------	------	------
A: Roger started with 5 tennis balls.
							Appendix	I we	show	that PAL	also	improves	Least-to-
---	---	---	---	---	---	---	--------	----	----	--------	----	--------	---------
tennis_balls = 5
							Most (Zhou	et	al., 2022)	prompts,	which	introduce	rea-
---	---	---	---	---	---	---	----------	---	----------	--------	-----	---------	----
2 cans of 3 tennis balls each is
soningchainsthatdecomposeaquestionintosub-questions.
bought_balls = 2 * 3
tennis balls. The answer is
4.ExperimentalSetup
answer = tennis_balls + bought_balls
							Dataandin-contextexamples				Weexperimentwiththree
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							broad classes	of	reasoning	tasks:	(1)	mathematical	prob-
Figure 2: A close-up of a single example from lems (§4.1) from a wide range of datasets including
							GSM8K	(Cobbe	et al.,	2021),		(Patel et	al., 2021),
-----	-------	---	----------------	---	---------	---	-----	------	-------	------	-----	---------	-----------
a PAL	prompt.		Chain-of-thought		reasoning	is					SVAMP
highlightedinblue, and PAL programmatic steps ASDIV(Miaoetal.,2020),andMAWPS(Koncel-Kedziorski
are highlighted in gray and pink. etal.,2016);(2)symbolicreasoning(§4.2)fromBIG-Bench
Hard(Suzgunetal.,2022);(3)andalgorithmicproblems
							(§4.3)fromBIG-BenchHardaswell.				Detailsofalldatasets
---	---	---	---	---	---	---	------------------------------	---	---	---	--------------------	---	---
Example A close-up of the example from Figure 1 is areshowninAppendixH. Foralloftheexperimentsfor
shown in Figure 2. While chain-of-thought only de- which COT prompts were available, we use the same in-
composes the solution in the prompt into natural lan- contextexamplesasusedbypreviouswork. Otherwise,we
guage steps such as Rogerstartedwith5tennisballs and randomlysampledafixedsetofin-contextexamples,and

			PAL:Program-aidedLanguageModels								4
Q:Oliviahas$23. Sheboughtfivebagelsfor$3each. Howmuchmoneydoesshehaveleft?
		money_initial		= 23
---	---	-------------	---------------	------	------------	-------------	---	---	---	---	---
		bagels	= 5
		bagel_cost	= 3
		money_spent	=	bagels	* bagel_cost
		money_left	= money_initial			- money_spent
		answer	= money_left
Figure3: Examplepromptforthemathematicalreasoningtasks,fromtheGSM8Kbenchmark.
Q:Onthetable,youseeabunchofobjectsarrangedinarow: apurplepaperclip,apinkstressball,
abrownkeychain,agreenscrunchiephonecharger,amauvefidgetspinner,andaburgundypen.
Whatisthecoloroftheobjectdirectlytotherightofthestressball?
...
		stress_ball_idx		= None
---	---	---------------	---------------	----------------------	-------	------	---	---	---	---	---
		for i,	object	in enumerate(objects):
		if	object[0]	==	'stress	ball':
			stress_ball_idx		=	i
break
		# Find	the directly		right	object
---	---	------------	-----------------	--------------------------	-----	--------	-----	---	---	---	---
		direct_right	=	objects[stress_ball_idx+1]
		# Check	the directly		right	object's	color
		answer	= direct_right[1]
Figure4: AnexampleforaPALpromptintheCOLOREDOBJECTStask. Forspaceconsiderations,weomitthecodethat
createsthelistobjects.
and8.
usedthesamesetforPALandCOT. ThisraisesthequestionofwhetherLLMscangener-
					alizetolargerandnon-integernumbers?					Weconstructeda
---	---	---	---	---	-----------------------------------	---	---	---	---	--------------	---
Baselines We consider three prompting strategies: DI- harderversionofGSM8K,whichwecallGSM-HARD,byre-
RECT prompting – the standard prompting approach us- placingthenumbersinthequestionsofGSM8Kwithlarger
ingpairsofquestionsandimmediateanswers(e.g., 11)as numbers. Specifically, one of the numbers in a question
					wasreplacedwitharandomintegerofupto7digits.						More
--------	-------------------------------	---	-----	-------	-------------------------------------------	---	---	---	---	---	----
in Brown	et al. (2020); chain-of-thought		(COT)	prompt-
ing (Wei et al., 2022); and our PAL prompting. We per- detailsregardingthethisnewdatasetareprovidedinH.1.
formed greedy	decoding	from the language	model	using
-------------	--------	-----------------	-----	-----	---	---	---	---	---	---	---
a temperature of 0. Unless stated otherwise, we used 4.2.SymbolicReasoning
CODEX(code-davinci-002)asourbackendLLMfor
					We	applied	PAL to three	symbolic	reasoning		tasks from
---	---	---	---	---	---	-------	------------	--------	---------	---	----------
bothPAL,DIRECT,andCOT.Indatasetswhereresultsfor
BIG-BenchHard(Suzgunetal.,2022),whichinvolverea-
additionalbaseLMs,suchasPaLM-540B,wereavailable
					s	o n in g ab	o u to b j e cts a	n d c o n c e p ts :	( 1 ) C		O
------------------------------------	---	---	---	---	---	-----------	------------------	--------------------	-------	--------	----------
frompreviouswork,weincludedthemasCOT					.				O	LO R E D	B JE C T S
PaLM-540B
					re	q u ir es	a n sw e ri n g qu	e s ti o n s a b o u t	c o lo r ed	ob j e cts	o n a s ur -
---	---	---	---	---	-----	------------------------------------------------	------------------	----------------------	-----------	----------	------------
					face.	Thistaskrequireskeepingtrackofrelativepositions,
4.1.MathematicalReasoning
					absolutepositions,andthecolorofeachobject.						Figure4
---	---	---	---	---	------------------------------------------	---	---	---	---	---	-------
We evaluate PAL on eight mathematical word problem showsanexampleforaquestionandexamplePALprompt.
datasets. Eachquestioninthesetasksisanalgebraword (2)PENGUINSdescribesatableofpenguinsandsomead-
problematgrade-schoollevel. Anexampleforaquestion ditionalinformationinnaturallanguage,andthetaskisto
andPALexamplepromptisshowninFigure3. Wefound answeraquestionabouttheattributesofthepenguins,for
example,“howmanypenguinsarelessthan8yearsold?”.
thatusingexplicitNLintermediatestepsdoesnotfurther
benefitthesemathreasoningtasks,hencewekeptonlythe While both PENGUINS and COLORED OBJECT tasks re-
meaningfulvariablenamesintheprompt. quire tracking objects, PENGUINS describes dynamics as
					well,	since	the penguins	in the problem		can be	added or
---	---	---	---	---	-----	-----	------------	--------------	---	------	--------
GSM-HARD LLMscanperformsimplecalculationswith removed. Figure17inAppendixJ.2showsanexamplefor
					a	question,	a chain-of-thought	prompt,		and PAL	prompt.
-------------	---------------------------------	---	---	---	---	---------	------------------	-------	---	-------	-------
smallnumbers.	However,Madaan&Yazdanbakhsh(2022)
found that 50% of the numbers in the popular GSM8K (3)DATEisadateunderstandingtaskthatinvolvesinferring
datasetofmathreasoningproblemsareintegersbetween0 datesfromnaturallanguagedescriptions,performingaddi-

				PAL:Program-aidedLanguageModels								5
Q:Ihaveachair,twopotatoes,acauliflower,alettucehead,twotables,acabbage,twoonions,and
threefridges. HowmanyvegetablesdoIhave?
			# note:	I'm	not counting		the	chair,	tables,
---	---	---	-------	---	------------	---	---	------	-------	---	---	---
or fridges
			vegetables_to_count			= {
---	---	---	-------------------	--------------	------	---	---	---	---	---	---	---
				'potato':	2,
				'cauliflower':		1,
				'lettuce	head':	1,
				'cabbage':	1,
				'onion':	2
}
			answer	=	sum(vegetables_to_count.values())
---	---	---	------	---	---------------------------------	---	---	---	---	---	---	---
Figure5: AnexampleforaPALpromptintheOBJECTCOUNTINGtask. ThebaseLMisexpectedtoconverttheinputinto
adictionarywherekeysareentitiesandvaluesaretheirquantities,whilefilteringoutnon-vegetableentities. Finally,the
answeristhesumofthedictionaryvalues.
		GSM8K	GSM-HARD
-----------	---	-----	--------	---	------	-----	--------	----	--------	----	-----------------	----
					SVAMP	ASDIV	SINGLEEQ		SINGLEOP		ADDSUB MULTIARITH
DIRECTCodex		19.7		5.0	69.9	74.0		86.8		93.1	90.9	44.0
COT		4.1			- 12.6	16.9			-		- 18.2	10.7
UL2-20B
COT		17.1			- 39.9	49.0			-		- 52.9	51.8
---	---	----	---	---	------	----	---	---	---	---	------	----
LaMDA-137B
COT		65.6		23.1	74.8	76.9		89.1		91.9	86.0	95.9
---	---	----	---	----	----	----	---	----	---	----	----	----
Codex
COT		56.9			- 79.0	73.9		92.3		94.1	91.9	94.7
---	---	----	---	---	------	----	---	----	---	----	----	----
PaLM-540B
COT Minerva540B		58.8			- -		-		-		- -	-
---------------	---	----	---	----	----	----	---	----	---	----	----	----
PAL		72.0		61.2	79.4	79.6		96.1		94.6	92.5	99.2
Table1: Problemsolverate(%)onmathematicalreasoningdatasets. Thehighestnumberoneachtaskisinbold. The
resultsfor DIRECT and PaLM-540B arefromWeietal.(2022), theresultsforLaMDAandUL2arefromWangetal.
(2022b),andtheresultsforMinervaarefromLewkowyczetal.(2022). WeranPALoneachbenchmark3timesandreport
theaverage;thestandarddeviationisprovidedinTable7.
5.Results
tionandsubtractionofrelativeperiodsoftime,andhaving
someglobalknowledgesuchas“howmanydaysarethere
5.1.MathResults
inFebruary”,andperformingthecomputationaccordingly.
AppendixJ.3showsexampleprompts. Table 1 shows the following results: across all tasks,
PALusingCodexsetsanewfew-shotstate-of-the-arttop-
4.3.AlgorithmicTasks 1 decoding across all datasets, outperforming COT ,
Codex
							COT	PaLM-540B	, and	COT Minerva540B	which was	fine-tuned
---	---	---	---	---	---	---	---	---------	-----	---------------	---------	----------
Finally,wecomparePALandCOTonalgorithmicreason-
onexplicitmathematicalcontent.
ing. Thesearetaskswhereahumanprogrammercanwrite
adeterministicprogramwithpriorknowledgeoftheques-
							Interestingly,		COT	alsobenefitsfromCodexover		PaLM-
---	---	---	---	---	---	---	--------------	---	---	-------------------------	---	-----
tion. Weexperimentwithtwoalgorithmictasks: OBJECT 540BinsomeofthedatasetssuchasASDIV,butperforms
COUNTING and REPEAT COPY. OBJECT COUNTING in- worse than PaLM-540B in others such as SVAMP. Yet,
volvesansweringquestionsaboutthenumberofobjectsbe- usingPALfurtherimprovesthesolverateacrossalldatasets.
longingtoacertaintype.Forexample,asshowninFigure5:
“Ihaveachair,twopotatoes,acauliflower,alettucehead,
twotables,... HowmanyvegetablesdoIhave?”). REPEAT GSM-HARD On GSM-HARD (Table1), theaccuracyof
COPYrequiresgeneratingasequenceofwordsaccording DIRECTdropsdramaticallyfrom19.7%to5.0%(arelative
							dropof74%),theaccuracyof				COT dropsfrom65.6%to
----------------	---	--------	--------	---	--------	----	------------------------	---	---	---	--------------------	---
to instructions.	For	example,	as shown	in	Appendix	J.6:
“Repeatthewordduckfourtimes,buthalfwaythroughalso 20.1%(arelativedropofalmost70%),whilePALremains
sayquack”). stable at 61.5%, dropping by only 14.3%. The results of
COTonGSM-HARDdidnotimproveevenwhenwereplaced
itspromptswithpromptsthatincludelargenumbers(Ap-
							pendixB).Thisshowshow				PAL providesnotonlybetter
---	---	---	---	---	---	---	---------------------	---	---	---	-------------------------	---

						PAL:Program-aidedLanguageModels									6
				COLOREDOBJECT			PENGUINS	DATE		REPEATCOPY		OBJECTCOUNTING
	DIRECTCodex				75.7		71.1	49.9		81.3			37.6
	COT				-		-	26.8			-		-
LaMDA-137B
	COT				-		65.1	65.3			-		-
---	---	---	---	---	---	---	----	----	---	---	---	---	---	---	---
PaLM-540B
	COT				86.3		79.2	64.8		68.8			73.0
---	---	---	---	---	----	---	----	----	---	----	---	---	----	---	---
Codex
	PAL				95.1		93.3	76.2		90.6			96.7
---	---	---	---	---	----	---	----	----	---	----	---	---	----	---	---
Codex
Table2: Solverateonthreesymbolicreasoningdatasetsandtwoalgorithmicdatasets,Inalldatasets,PALachievesamuch
higheraccuracythanchain-of-thought. ResultswithclosedmodelsLaMDA-137BandPaLM-540Bareincludedifavailable
topublic(Weietal.,2022;Suzgunetal.,2022).
resultsonthestandardbenchmarks,butisalsomuchmore 5.2.SymbolicReasoning&AlgorithmicTasksResults
robust. Infact,sincePALoffloadsthecomputationtothe
								Results	for	symbolic	reasoning		and algorithmic	tasks	are
------	------------	---	-----------	---	-----------	---	-------	--------------	---	--------	----------------------------	---	---------------	-----	---
Python	interpreter,		any complex		computation	can	be per-
								showninTable2.			InCOLOREDOBJECTS,PALimproves
formedaccuratelygiventhecorrectlygeneratedprogram.
overthestrongCOTby8.8%,andby19.4%overthestan-
								darddirectprompting.			InPENGUINS,PALprovidesagain
---	---	---	---	---	---	---	---	--------------------	---	---	---------------------------	---	---	---	---
Large Numbers or Incorrect Reasoning? Are the fail- ofabsolute14.1%overCOT.InDATE,PALfurtherprovides
								11.4%gainoverbothCOT					, ,and		.
---	---	---	---	---	---	---	---	--------------------	---	---	---	---	------	---	---
uresonGSM-HARDprimarilyduetotheinabilityofLLMs Codex PaLM-540B LaMDA-137B
to do	arithmetic,		or do the	large	numbers	in the	question
-----	-----------	---	---------	-----	-------	------	--------	---	-------------	---	-------	---	------------	--------	---
								The	two rightmost		columns	of	Table 2 show	that PAL	is
“confuse”theLMwhichgeneratesirrationalintermediate
closetosolvingOBJECTCOUNTING,reaching96.7%and
steps?	To	investigate	this,	we evaluated		the outputs	gen-
--------	---	----------------------------------	-----	------------	---	-----------	----	---------	-----------	--------	---	-----------	-----------------	---------	---
								improving		over COT	by	absolute	23.7%. Similarly,		PAL
eratedby	COT	forthetwoversionsofthesamequestion
								vastly	outperforms		COT	by absolute	21.8%	on REPEAT
(withandwithoutlargenumbers). Wefindthatin16out COPY. Surprisingly, DIRECT prompting performs better
of 25	cases	we analyzed,		COT generates		nearly	identical
-----	-----	------------	---	-------------	---	------	---------	----	---	---------	---	-----	-----------------	---	----
								than	COT	on REPEAT		COPY.	Yet, PAL improves		over
naturallanguage“thoughts”,indicatingthattheprimaryfail-
DIRECTby9.3%inREPEATCOPY.
uremodeistheinabilitytoperformarithmeticaccurately.
SampleoutputsareprovidedintheAppendix,Table11.
1
ycaruccA
					GSM8K				0.8
---	---	---	---	---	-----	---	---	---	---	---	---	---	---	---	---
		COT			7.3
UL2-20B
		COT	LaMDA-137B		27.7						PaL
---	---	---	----------	---	----	---	---	---	---	---	---	---	---	---	---
0.6
		COT			78.0
---	---	---	-----	---	----	---	---	---	---	---	---	---	---	---	---
			Codex								CoT
		COT			74.4
PaLM-540B
									[0,2]	[3,5]	[6,8]	[9,11] [12,14]	[15,17] [18,20]	[21,23]	[24,26]
---	---	---	---	---	----	---	---	---	-----	-----	-----	--------------	---------------	-------	-------
		COT			78.5
Minerva540B
		PAL			80.4							NumberofObjects
---	---	---	---	---	----	---	---	---	---	---	---	---------------	---	---	---
Codex
Table 3: Problem solve rate (%) on GSM8K using Figure6: Thesolverateon COLORED OBJECTS withre-
majority@40(Wangetal.,2022b) specttothenumberofobjectsincludedinthetestquestion.
Multi-sample		Generation		As	found	by Wang	et al.
------------	---	----------	---	---	-----	-------	------	---	---	---	---	---	---	---	---
(2022b),chain-of-thought-stylemethodscanbefurtherim- IsPALsensitivetothecomplexityofthequestion? We
provedbysamplingk >1outputs,andselectingthefinal examinedhowtheperformanceofPALandCOTchangeas
answerusingmajorityvoting. Wethusrepeatedthegreedy- thecomplexityoftheinputquestiongrows,measuredasthe
decodingexperimentsusingnucleussampling(Holtzman numberofobjectsinthequestionofCOLOREDOBJECTS.
etal.,2019)withp = 0.95andk = 40asinLewkowycz AsshowninFigure6,PALissuperiorCOTacrossallinput
etal.(2022)andtemperatureof0.7. AsshowninTable3, lengths. Asthenumberofobjectsinthequestionincreases,
thisfurtherincreasestheaccuracyof PAL from72.0%to COT’saccuracyisunstableanddrops,whilePALremains
80.4% on GSM8K, obtaining 1.9% higher accuracy than consistentlycloseto100%.Moreanalysisonthetoken-level
Minerva-540Busingthesamenumberofsamples. predictionscanbefoundinAppendixG.

PAL:Program-aidedLanguageModels 7

80						80
	PAL			72.0			COT	PAL		69.8
									65.8	65.3
COT
60						60
---	---	---	---	---	---	---	---	---	---	---
RelativeImprovement
etarevloS				60.1					46.9
---------	---	---	---	----	---	---	---	---	----	---
40
40
		31.8					26.5
---	----	----	----	---	---	---	----	---	---	---
	21.7		26.0
20
20	19.1
---	----	-----	---	-----	---	---	---	---	---	---
		22.3%		19.8%			8.6
13.6%
0
0
code-cushman-001 code-davinci-001 code-davinci-002 text-davinci-001 text-davinci-002 text-davinci-003
Figure7: PALwithdifferentmodelsonGSM8K:though
						Figure	8: PAL	with	NL LMs on GSM8K:	though
---	---	---	---	---	---	------	------	----	----------------	------
the absolute accuracies with code-cushman-001 COToutperformsPALwithtext-davinci-001,once
and code-davinci-001			are	lower	than
--------------------	---	---	---	-----	----	---	----------	------------	-----------	-------------
						the	base LM is	sufficiently	strong, PAL	is beneficial
code-davinci-002,
the relative improvement of with text-davinci-002 and text-davinci-003
PALoverCOTisconsistentacrossmodels. aswell. Thatis,PALisnotlimitedtocode-LMsonly.
6.Analysis not only from having a better prompt. Additional details
areprovidedinAppendixB.Foradditionaldiscussionon
Does PAL	work with	weaker	LMs?	In all our	experi-
--------	---------	------	----	----------	-------	---	---	---	---	---
theadvantagesofcode-promptsovertextual-prompts,see
mentsinSection5,PALusedthecode-davinci-002
AppendixG.
model;butcanPALworkwithweakermodelsofcode?					We
------------------------------------------	---	---	---	---	---	---	---	---	---	---
comparedPALwithCOTwhenbothpromptingapproaches
						Dovariablenamesmatter?			Inallourexperiments,we
---	---	---	---	---	---	----------------------	---	---	----------------------	---
code-cushman-001
use the same weaker base LMs usedmeaningfulvariablenamesinthePALprompts,toease
andcode-davinci-001.			AsshowninFigure7,		even
--------------------	---	---	-----------------	---	----	---	---	---	---	---
themodel’sgroundingofvariablestotheentitiestheyrep-
thoughtheabsoluteaccuraciesofcode-cushman-001
						resent.	ForthePythoninterpreter,however,variablenames
---	---	---	---	---	---	-------	---------------------------------------------	---	---	---
andcode-davinci-001arelower,therelativeimprove-
						aremeaningless.		Tomeasuretheimportanceofmeaningful
---	---	---	---	---	---	---------------	---	----------------------------------	---	---
mentofPALoverCOTremainsconsistentacrossmodels.
variablenames,weexperimentedwithtwopromptsvariants:
ThisshowsthatPALcanworkwithweakermodels,while
itsbenefitscaleselegantlytostrongermodelsaswell.
						1.	PAL	–thePALpromptwithoutintermediate
---	---	---	---	---	---	---	---	--------------------------------	---	---
−comment
NLcomments.
Does PAL	work with	LMs	of natural	language?	We
--------	---------	---	----------	---------	---	---	---	---	---	---
also experimented with PAL using the text-davinci 2. PAL−var –thePALpromptwithoutintermediate
−comment
and
series. Figure 8 shows the following interesting re- NL comments with variable names substituted
sults: when the base LM’s “code modeling ability” is withrandomcharacters.
weak(usingtext-davinci-001),COTperformsbetter
than PAL. However, once the LM’s code modeling abil- TheresultsareshowninFigure9. InCOLOREDOBJECTED
ityissufficientlyhigh(usingtext-davinci-002and andDATE,removingintermediateNLcommentsbutkeep-
text-davinci-003),PALoutperformsCOT,andPAL
						ingmeaningfulvariablenames(PAL			−comment	)–slightlyre-
---	---	---	---	---	---	------------------------------	---	---	--------	-------------
performsalmostasPAL . ducestheresultscomparedtothefullPALprompt,butitstill
text-davinci-003				code-davinci-002
----------------	---	---	---	----------------	---	---	---	---	---	---
ThisshowsthatPAL isnotlimitedtoLMsofcode,butit achieveshigheraccuracythanthebaselinesCOT.Remov-
ingvariablenamesaswell(PAL−var
can work with LMs that were mainly trained for natural )furtherdecreases
−comment
language,iftheyhaveasufficientlyhighcodingability. accuracy, and performs worse than COT. Since variable
nameshaveanimportantpartincodequality(Gellenbeck
IsPALbetterbecauseofthePythonpromptorbecause & Cook, 1991; Takang et al., 1996), meaningful variable
of the interpreter? We experimented with generating namesareonlyexpectedtoeasereasoningforCodex,which
Pythoncode,whilerequiringtheneuralLMto“execute”it wastrainedonmostlymeaningfulnames,aswasalsofound
aswell, withoutusinganinterpreter, followingNyeetal. byMadaanetal.(2022).
(2021);Madaanetal.(2022).			Wecreatedpromptsthatare
-----------------------------------------------------	---	---	-----------------------	---	---	-------------	---	---	---	---
similartoPAL’s,exceptthattheydoincludethefinalanswer.						7.RelatedWork
Thisresultedina23.2solverateon			GSM8K,muchlower
thanPAL(72.0),andonly4.5pointshigherthanDIRECT. Prompting Few-shotprompting(Brownetal.,2020)has
Theseresultsreinforceourhypothesisthatthemainbenefit been shown to be an effective approach for a variety of
of PAL comes from the synergy with the interpreter, and tasks(Liuetal.,2021)rangingfromtext-(Gehrmannetal.,

						PAL:Program-aidedLanguageModels						8
	100				COT	PAL	PAL		PAL− v ar
			95.2				−comment		− co mment	93.3
				91.1							91.3 91.9
90
84.4
79.9
79.2
	80						76.2
---	---	---	---	---	---	---	----	---	---	---	---	---
69.1
70
64.8
63.4
60
			ColoredObjects				Date			Penguins
---	---	---	--------------	---	---	---	----	---	---	--------	---	---
Figure9: AblationstudyofPALpromptformats. WeconsidertheoriginalPALprompt,itwithnaturallanguagecomments
removed(PAL ),andfurthervariablenamesreplacedwithrandomcharacter(PAL− v ar ). Asareference,wealso
		−comment							−	co mment
---	---	--------	---	---	---	---	---	---	---	--------	---	---
showtheCOTperformance(blue).
2021;Reifetal.,2021;Weietal.,2021;Sanhetal.,2021) model,whilePALachieves79.4%onthesamebenchmark
tocode-generation(Chenetal.,2021b). Methodssuchas withoutanyspecializedpretraining.
chain-of-thoughtprompting(COT)havefurtherunlockeda
ShortlyafterapreprintofourworkwassubmittedtoarXiv,
varietyofreasoningtasks,boostingtheperformanceofmod-
							another related	work	on “program	of	thought	prompting”
--------------------------	---	---	---	------------------------	---	---	---------------	----------	-----------	--------------	---------	----------
elsonavarietyofbenchmarks.				Nevertheless,allprevious
							(Chen et	al., 2022)	was	also submitted	to arXiv.	Their
approachessufferfrominaccuracyinarithmeticcalculation
							method is	conceptually	similar	to ours,	but PoT	(1) only
---	---	---	---	---	---	---	---------	------------	-------	--------	-------	--------
andincorrectreasoning(Lewkowyczetal.,2022;Hendrycks
demonstratesefficacyonmathematicalproblems,whereas
etal.,2021;Madaan&Yazdanbakhsh,2022).						PALavoids
-------------------------------------	---	---	---	---	---	---------	---	---	---	---	---	---
wedemonstrategainsonsymbolicandalgorithmicbench-
theseproblemsbyoffloadingthecalculationandsomeof
							marks as	well, and	(2) chose	benchmark-specific		prompt
---------------------------------	---	---	---	---	----------------	---	--------	---------	---------	------------------	---	------
thereasoningtoaPythoninterpreter,					whichiscorrectby
examples,whileweusedthesamepromptexamplesaspre-
construction,		given	the right	program.	Further,	not only
-------------	---	-----	---------	--------	--------	--------	---	---	---	---	---	---
viouswork,todisentangledthebenefitofourapproachfrom
thatPALcanimprovethestandardchain-of-thought,itcan
thebenefitofthechoiceofexamples.
improveleast-to-mostprompting(Zhouetal.,2022)aswell,
asweshowinAppendixI.
LMs	with	external	tools	Several	prior	works have
-------------------------------------------	----	--------	-----	-------	-----	----------	---	---	---	---	---	---
equippedneuralmodelswithspecializedmodules.						Forex-
ample, Cobbe et al. (2021) employ a calculator for arith- Semantic parsing Our work can also be seen as a very
metic operations as a post hoc processing, and Demeter generalformofsemanticparsing,whereinsteadofparsing
&Downey(2020)addspecializedmodulesforgenerating intostrictdomain-specificlanguages,themodelgenerates
citiesanddates. Unliketheseworks, PALgeneratescode free-formPythoncode. Someworksconstrainthedecoder
foraPythoninterpreter,whichisgeneralenoughtohandle usingaContext-FreeGrammar(CFG)togenerateadomain-
botharithmeticcalculationsanddates,withoutspecialized specificmeaningrepresentation(Shin&VanDurme,2021)
modulesandad-hocfixes. Chowdheryetal.(2022)andWei oracanonicalutterance,whichcanbeconvertedtoaLisp-
etal.(2022)havealsoexperimentedwithexternalcalcula- likemeaningrepresentation(Shinetal.,2021). Incontrast,
tors;however,thecalculatorhadimprovedCodexbyonly PALdoesnotrequireanyconstrainingordomain-specific
2.3%(absolute)onGSM8KandimprovedPaLM-540Bby representationsotherthanPythoncode. Further,LMsthat
1.7%, while PAL improves Codex by 6.4% on the same werepretrainedonPythonareabundantcomparedtoother
benchmark (Section 5.1). Similarly to our work, Chowd- domain-specific languages, making Python code a much
heryetal.(2022)havealsoexperimentedwithgenerating morepreferablerepresentation.Andoretal.(2019)generate
PythoncodeforsolvingtheGSM8Kbenchmark,buttheir task-specificarithmeticoperationsforreadingcomprehen-
experimentsresultedinlower accuracythanthestandard siontasks;Guptaetal.(2019)designneuralmodulessuch
PaLM-540B that uses chain-of-thought. Pi et al. (2022) ascounttodealwitharithmeticoperations. PAL gener-
pretrainthemodelonexecutionresultsofrandomexpres- alizestheseworksbygeneratinggeneralPythonprograms,
sionsonacalculator,insteadofusingthesolverattesttime withouttheneedfordefiningspecializedmodules. Theclos-
aswell. Whiletheirmodelcanhypotheticallyperformarith- estworktoourstechnicallymaybeBinder(Chengetal.,
meticbetterthanotherpretrainedLMs,theirresultsonthe 2022),butitaddressedmostlyansweringquestionsabout
SVAMPbenchmarkaremuchlower:57.4%usingaT5-11B tablesusingSQLandSQL-likePython.

					PAL:Program-aidedLanguageModels								9
8.Conclusion C., Tillet, P., Such, F. P., Cummings, D., Plappert, M.,
Chantzis,F.,Barnes,E.,Herbert-Voss,A.,Guss,W.H.,
WeintroducePAL,anewmethodfornaturallanguagerea-
Nichol,A.,Paino,A.,Tezak,N.,Tang,J.,Babuschkin,
soning, using	programs		as intermediate		reasoning	steps.
-------------	--------	---	---------------	---	---------	------	-----------	---	---------	---------	---	------	---------
							I., Balaji,	S.,	Jain, S.,	Saunders,	W.,	Hesse,	C., Carr,
DifferentlyfromexistingLM-basedreasoningapproaches,
							A. N., Leike,		J., Achiam,	J.,	Misra,	V., Morikawa,	E.,
--------	-------	----------	-------	---	-----------	-----	-------------	-----------	-----------	-------------	------	-------------	---
the main	idea is	to offload	solving	and	calculating	to an
							Radford,	A., Knight,		M., Brundage,		M., Murati,	M.,
externalPythoninterpreter,insteadofusingtheLLMfor
Mayer,K.,Welinder,P.,McGrew,B.,Amodei,D.,Mc-
bothunderstandingtheproblemandsolving.						Thisresults
--------------------------------------	---	---	---	---	---	-----------	--------------------------------------	---	---	---	---	----------	---
							Candlish,S.,Sutskever,I.,andZaremba,W.					Evaluating
inafinalanswerthatisguaranteedtobeaccurate,giventhe
							LargeLanguageModelsTrainedonCode.					arXivpreprint
---------	---------	------------	---	------	--------------	---	---------------------------------	---	---	---	---	-------------	---
correctly	predicted	programmatic		steps.	We demonstrate
arXiv:2107.03374,2021a.
this seamless	synergy	between		an LLM	and a	Python in-
-------------	-------	-------	---	------	-----	----------	---	---	---	---	---	---	---
terpreteracross13tasksfromBIG-BenchHardandother Chen,M.,Tworek,J.,Jun,H.,Yuan,Q.,Pinto,H.P.d.O.,
benchmarks. In all these benchmarks, PAL outperforms Kaplan,J.,Edwards,H.,Burda,Y.,Joseph,N.,Brockman,
largerLLMssuchas PaLM-540B whichusethepopular G., etal. Evaluatinglargelanguagemodelstrainedon
“chain-of-thought”methodandsetsnewstate-of-the-artac- code. arXivpreprintarXiv:2107.03374,2021b.
curacyonallofthem.		Webelievethattheseresultsunlock
------------------	---	-------------------------------	---	---	---	---	-----------------------------------	---	---	---	---	---	-------
							Chen,W.,Ma,X.,Wang,X.,andCohen,W.W.						Program
excitingdirectionsforfutureneuro-symbolicAIreasoners.
							ofthoughtsprompting:			Disentanglingcomputationfrom
----------	---	---	---	---	---	---	------------------------------------	---	---	----------------------------	---	-------------	---
							reasoningfornumericalreasoningtasks.					arXivpreprint
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PAL:Program-aidedLanguageModels 12 Part I Appendix Table of Contents A AlternativePromptswithoutMeaningfulVariableNames 13 B AdditionalanalysisonArithmeticReasoning 13 C EffectofUsingLanguageModelsofCode 14 D AnalyzingtheEffectofIncreasingNumberofSamplesonPAL 14 E StandardDeviationsAcrossMultipleOrderofPrompts 17 F PALBeyondBenchmarks 17 G CloserLookintoToken-levelBehaviorsofDifferentMechanisms 20 H Datasets 20 H.1 CreatingGSM-HARD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 H.2 GSM-HARDAnalysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 I GeneralizationofPALtoLeast-to-MostPrompting 24 J Prompts 26 J.1 ReasoningaboutColoredObjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 J.2 PenguinsinaTable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 J.3 DateUnderstanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 J.4 Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 J.5 ObjectCounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 J.6 RepeatCopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 K SuccessandFailureModesinSymbolicTasks 33 K.1 ColoredObjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 K.2 PenguinsinaTable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 K.3 DateUnderstanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

PAL:Program-aidedLanguageModels 13 A.AlternativePromptswithoutMeaningfulVariableNames a = 23 b = 5 c = 3

d = b *	c
e = a -	d
print(e)
(a)Structuredexplanationwithuninformativevariablenames(PAL-var)
# Olivia	has $23
--------	-------	---	---	---
a = 23
# number	of bagels	bought
--------	---------	------	---	---
b = 5
# price	of each bagel
-------	-------------	---	---	---
c = 3
# total	price of bagels
-------	---------------	---	---	---
d = b	c
*
# money	left
-------	----	---	---	---
e = a -	d
print(e)
(b)Structuredexplanationwithuninformativevariablenames,butusefulcomments(PAL-var+comms)
money initial	= 23
-------------	---------------	-------------	---	---
bagels	= 5
bagel cost	= 3
money spent	= bagels	* bagel cost
money left	= money initial	- money spent
result	= money left
print(result)
(c)PALprompts
RoleoftextinPAL:threedifferentreasoningstepsforthequestionOliviahas$23. Sheboughtfivebagelsfor
Figure10:
$3each. Howmuchmoneydoesshehaveleft? Uninformativevariablenames(left),Uninformativevariablenameswith
usefulcomments(left),andPAL.Includingtextdescription
	Setting	COT PAL-var	PAL-var+comms	PAL
---	---------	-----------	-------------	----
	SolveRate	63.1 59.0	69.0	71.8
Table4: Roleoftext: includingtexteitherasinformativevariablenames(PAL)orcommentsisimportant(PAL-var+
comms). UninformativevariablenamesPAL-varcauseadrasticdropinperformance,indicatingthatjuststructureisnot
sufficient. ThecorrespondingpromptsareshowninFigure10.
Formathematicalproblems,sinceourstandardpromptsdonotusemuchcomment,westartbycreatingalternativeprompts
wheretheinformativevariablenamesarereplacedwithsingle-letters(Figure10).TheresultsinTable4showsaconsiderable
performance drop: from an average of 71.8% to 59%. Note that the ablation where structured outputs are completely
removedinfavorofpurelytextexplanationsispreciselytheCOTsetting,whichachievesasolverateof63%. Theseresults
underscoretheimportanceoftextbutmoreimportantlyshowthatcombiningbothtextandproceduralstatementsleadsto
higherperformancegains—eitherissub-optimal.
B.AdditionalanalysisonArithmeticReasoning
GSM-hardwithhardprompts TheGSM-HARDexperimentsusedpromptsthatweresampledfromtheGSM8Ktraining
set. WillCOTbehelpedbyusinglargernumbersinthepromptsaswell? Toinvestigatethis,wecreatepromptswherethe
numbersarechangedtolargernumbers,matchingthedistributionofnumbersinGSM-HARD. TheresultsinTable5shows

PAL:Program-aidedLanguageModels 14 thatevenwithapromptthatmatchesthenumbers,thereareonlymodestgainsinperformance. Theseresultsshowthatthe gainsachievedbyusingcode-basedreasoningchainsmaynotbeachievedsimplybyusingbetterfew-shotexamplesfor COT. RegularPrompt PromptwithLargerNumbers

COT	23.3	23.8
Table5: GSM-hardresults,whenthepromptsalsohadexamplesoflargernumbers.
SuccinctCode Theprogramsusedinfew-shotexamplesbyPALaremulti-step,andshowastep-by-stepbreakdownof
thereasoningprocess. Isthisbreakdownnecessary? Alternatively,canwereturnasinglelineexpression(seeFigure11b)to
calculatetheresult? ResultsinTable6(4throw)showsthatisnotthecase. Withsingle-lineexpressions,theperformanceof
PALfallstothelevelofdirectprompting.
Generatingtheanswerdirectly PALfirstgeneratesareasoningchainintheformofaPythonprogram,andpassesthe
generatedprogramtoaruntimetoobtainananswer. IsPALbetteronlybecauseoftheprogram-styleintermediatereasoning
chains,oraretheimprovementsderivedfromoffloadingexecutiontothePythonruntime? Toinvestigatethis,weexperiment
withavariantthatforcestheLLMtogeneratetheansweraftergeneratingthereasoningchain(Figure11e). Thissetting
compelstheLLMtoconditiononthegeneratedcode-basedreasoningtogenerateananswer,simulatingtheruntime. The
resultsinTable6(5throw)showthatthesolveratedropstonearDIRECTlevels.
Thisreinforcesourhypothesisthatwhile
currentLLMscanbeexcellentatspecifyingahigh-levelplantosolveatask—theyarestillincapableofexecutingthem.
Ablation		SolveRate
-------------------------------	---	---------
DIRECT(nointermediatereasoning)		19.7
COT		65.6
PAL		72.0
SuccinctCode		47.8
LLMSimulatingRuntime		23.2
Table6: SolveRatesforAblations
C.EffectofUsingLanguageModelsofCode
Inourexperiments, wefocusedonevaluatingtheperformanceofalanguagemodelforcode. Weaimedtoinvestigate
whethertheadditionalperformanceboostobservedinourresultswasduetotheuseofmodelslikeCodex,orwhetherour
formulationwasusefulevenfortext-basedmodels. Tothisend,weconductedadditionalexperimentsusingtext-based
languagemodels. OurfindingsindicatethatthePALapproachisnotrestrictedtoworkingsolelywithCodex,butcanalso
beappliedtonaturallanguage(NL)models,aslongasthemodelissufficientlystrong. Specifically,ourresultsshowedthat
inthetext-davinci-001model,theuseoftheCoTapproachresultedinbetterperformance.
	Model CoT	PaL
---	---------------------	----
	text-davinci-001 26.5	8.6
	text-davinci-002 46.9	65.8
	text-davinci-003 65.3	69.8
D.AnalyzingtheEffectofIncreasingNumberofSampleson PAL
InSection5.1,weshowthatPALoutperformsstrongbaselinesbothforasinglesampleandbydrawing40samplesand
usingmajorityvoting. Figure12illustratesthetrendsforcaseswhenthenumberofsamplesdrawnarebetween1and40,
andtheinterpolationestimatesdemonstratethatPALremainscompetitivethroughoutthenumberofsamples.

PAL:Program-aidedLanguageModels 15 def solution(): """Shawn has five toys. For Christmas, he got two toys each from his

; mom	and dad.	How many	toys does	he have now?"""
toys_initial	=	5
mom_toys	= 2
dad_toys	= 2
total_received		= mom_toys	+ dad_toys
total_toys	= toys_initial		+ total_received
result =	total_toys
return result
(a)OriginalExample
def solution():
return 5	+ 2 +	2
--------	-----	---	---	---	---
(b)SuccinctCode
def solution():
"""Shawn has 10312864 toys. For Christmas, he got 13267894 toys each
from	his mom	and dad.	How many	toys does he	have now?"""
--------------	--------------	----------	----------------	------------	------------
toys_initial	=	10312864
mom_toys	= 13267894
dad_toys	= 13267894
total_received		= mom_toys	+ dad_toys
total_toys	= toys_initial		+ total_received
result =	total_toys
return result
(c)HardExamplesinPrompt(PAL)
Example(
question="Shawn has 10312864 toys. For Christmas, he got 13267894 toys
each	from	his mom and	dad. How	many toys does	he have now?",
----	----	-----------	--------	--------------	--------------
thought="Shawn started with 10312864 toys. If he got 13267894 toys each
from his mom and dad, then that is 26535788 more toys. 10312864 +
26535788	=	36848652.",
--------	---	-----------	---	---	---
answer="36848652",
),
(d)HardExamplesinPrompt(CoT)
def solution():
"""Shawn has five toys. For Christmas, he got two toys each from his
; mom	and dad.	How many	toys does	he have now?"""
--------------	--------------	----------	----------------	---------------	---
toys_initial	=	5
mom_toys	= 2
dad_toys	= 2
total_received		= mom_toys	+ dad_toys
total_toys	= toys_initial		+ total_received
result =	total_toys
return result
ans = 9
(e)GeneratingAnswersDirectly
Figure11: Ablationsoftheoriginalexamplesolutionforthefew-shotpromptingexperiment.

PAL:Program-aidedLanguageModels 16 85 80 70 60 50 1 8 15 40 Numberofsampledgenerationsforeachquestion )%(etaRevloS PAL COT Minerva PaLM Figure12: ComparisonofsolveratesbetweenPALandbaselinesasthenumberofsamplesincreasesfrom1to40. Notethat thesolveratesforthebaselines(PaLM,COT,Minerva)areobtainedthroughlogisticinterpolationofsolveratesat1and40

PAL:Program-aidedLanguageModels 17 E.StandardDeviationsAcrossMultipleOrderofPrompts Foreachmathreasoningtask,weruninferenceusingthreerandomorderingsoftheprompts. AsshowninTable7,the standarddeviationbetweentheresultsobtainedfromthethreedifferentseedsisminimal.

	COT		PAL
	Average StandardDeviation		Average StandardDeviation
GSM8K	65.6	1.10	72.0	0.16
SVAMP	74.8	0.19	79.4	0.20
	76.9	0.65	79.6	0.14
ASDIV
GSM-HARD	23.3	0.49	61.2	0.91
------------------------------------------------------------------	----	----	----	----
MAWPS-SingleEq	89.1	0.54	96.1	0.30
MAWPS-SingleOp	91.9	0.55	94.6	0.36
MAWPS-AddSub	86.0	0.62	92.5	0.34
MAWPS-MultiArith	95.9	0.51	99.2	0.48
Table7: Standarddeviationsforthreerunsforthemathreasoningdatasets.
F. PAL BeyondBenchmarks
Wearguethatsymbolicreasoningisacrucialcomponentinsolvingawiderangeoftasks. Inthissection,wedemonstrate
examples of tasks that may not initially appear to require using programs as intermediate reasoning steps, but can be
WedemonstratetheseexamplesusingtheChatGPTtool.1
improvedthroughtheuseofPAL-stylereasoning. Incontrastto
thein-context-learningmethodsweusedinthemainpaper,hereweinstructChatGPTtoperformprogram-aidedreasoning
throughoneoftheuserutterances.
InFigure13,inCOT-stylereasoning,whilethereasoningchainiscorrect,thefinalansweriswrong. Incontrast,PAL-style
reasoningcouldnotonlyaccuratelyextractthecolorofobjectsfromthequestionbutalsoproducethecorrectlinesofcode
tobranchtodifferentsituationsthatyieldtheircorrespondingcorrectanswers.
AmoreintriguingexampleislettinganLLMcountthenumberoflettersintheword“intriguing”. InFigure14a,whilethe
step-by-stepexplanationappearsreasonablebysplittingthelettersbyspaces,ChatGPTdoesnotchangetheanswerafter
thisexplicitreasoningandinsistsonthewronganswer. Explicitlyinstructingthemodeltoperformstep-by-stepreasoning
beforeansweringthequestionstillyieldsthewronganswer. Incontrast,PAL-stylereasoningonlytakesafewlinesofcode,
andtheexecutiondoesproducethecorrectanswer,inthiscase. Theseexamplesindicatethat PAL canbenefitevenan
ostensiblypowerfulmodellikeChatGPT.
1chat.openai.com

PAL:Program-aidedLanguageModels 18 (a)InCOTstylereasoning,thecorrectintermediatereasoningchainleadstowronganswers. (b)InPAL,theexecutionofthecodewillproducethecorrectanswer. Figure13: ChatGPTwithPALandCOTtoanswerauser-postedquestion

PAL:Program-aidedLanguageModels 19 (a)Step-by-stepreasoningstruggleoncountingthenumberoflettersintheword“intrigu- ing”whichhastenletters. (b)ExplicitlyinstructingChatGPTtoreasonstep-by-stepbeforegeneratinganswerstill leadstothewronganswer. (c)PALtakesafewlinesofcodeandtheexecutioncouldresultinthecorrectanswer. Figure14: ChatGPTwithPALandCOTtoanswerauser-postedquestion

PAL:Program-aidedLanguageModels 20 G.CloserLookintoToken-levelBehaviorsofDifferentMechanisms Beyondempiricalresults,wemakeinitialattemptstogainadeeperunderstandingofthebehaviorofLLMswithdifferent reasoning mechanisms by looking into the token-level log-likelihood of reasoning chains produced by COT and PAL. Werandomlyselected20questionsfromtheCOLORED OBJECTSdataset,alongwiththeircorrespondingCOTandPAL solutions. Wethenmanuallycomparedthetwomechanismsbyfocusingontokenswithalowlog-likelihood. OuranalysisrevealsthatCOToftenhaslowerconfidenceintokensrelatedtonumbersandquantitativeinformation,the groundedpositionofspatialadjectives(e.g.,right-most),propertiessuchasthecolorofobjects,andnounsthatrefertothe objects. Specifically,wefoundthatthisoccurredinseven,six,two,andsixexamplesoutofthe20weexamined. Incontrast, PALuseslistmanipulations,suchaslen(objects),andaccessesobjectsandtheirassociatedpropertiesthroughlist indexing(e.g.,object[3][0]). WefoundthattheLLMistypicallyconfidentinproducingtheseprograms. Furthermore, weobservedthatwhileCOTrequiresdifferentexpressionsforthesameconceptindifferentcontexts,PALalmostalways usesthesameexpression,whichispresumablymorerobust. Forexample,whentherearefiveobjects,COTpredicts“the right-mostthingisthefifthitemonthelist”,and“theright-mostthingisthethirditemonthelist”whenthenumberof objectsisthree. Occasionally,COTalsopredicts“theright-mostthingislastitemonthelist”whichdoesnotprovidemore concreteinformation. Onthecontrary,PALconfidentlypredictsobjects[-1]consistently. Themoreconsistentand uniformuseofexpressionsinPALcanbeattributedtotheexplicitanddefinednatureofprogramminglanguages,which allowsforclearandaccurateexpressions. H.Datasets Inthefollowingtables(Table8,Table9,Table10),wepresentsstatisticsandexamplesforthedatasetsweconsidered. Dataset N Example ReasoningaboutColoredObjects 2000 On the table, you see a bunch of objects arranged in a row: a purple paperclip,apinkstressball,abrownkeychain,agreenscrunchiephone charger,amauvefidgetspinner,andaburgundypen. Whatisthecolor oftheobjectdirectlytotherightofthestressball? PenguinsinaTable 149 Hereisatablewherethefirstlineisaheaderandeachsubsequentlineis apenguin: name,age,height(cm),weight(kg)Louis,7,50,11Bernard, 5,80,13Vincent,9,60,11Gwen,8,70,15Forexample: theageof Louis is 7, the weight of Gwen is 15 kg, the height of Bernard is 80 cm. Wenowaddapenguintothetable: James,12,90,12Howmany penguinsarelessthan8yearsold? DateUnderstanding 369 2015iscomingin36hours. Whatisthedateoneweekfromtodayin MM/DD/YYYY? Table8: Reasoningdatasetsabouteverydayobjectsandconcepts. Dataset N Example ObjectCounting 1000 Ihaveachair,twopotatoes,acauliflower,alettucehead,twotables,a cabbage,twoonions,andthreefridges. HowmanyvegetablesdoIhave? RepeatCopy 32 Repeatthewordduckfourtimes,buthalfwaythroughalsosayquack. Table9: Reasoningdatasetsaboutalgorithmicproblems.

PAL:Program-aidedLanguageModels 21 Dataset N Example GSM8K(Cobbeetal.,2021) 1319 Oliviahas$23. Sheboughtfivebagelsfor$3each. How muchmoneydoesshehaveleft? SVAMP(Pateletal.,2021) 1000 Eachpackofdvdscosts76dollars. Ifthereisadiscount of25dollarsoneachpack. Howmuchdoyouhavetopay tobuyeachpack? ASDIV(Miaoetal.,2020) 2096 EllenhassixmoreballsthanMarin. Marinhasnineballs. HowmanyballsdoesEllenhave? SINGLEOP(Koncel-Kedziorskietal.,2016) 562 Ifthereare7bottlecapsinaboxandLindaputs7more bottlecapsinside,howmanybottlecapsareinthebox? SINGLEEQ(Koncel-Kedziorskietal.,2016) 508 Bennyboughtasoftdrinkfor2dollarsand5candybars. Hespentatotalof27dollars. Howmuchdideachcandy barcost? ADDSUB(Koncel-Kedziorskietal.,2016) 395 Therewere6rosesinthevase. Marycutsomerosesfrom her flower garden. There are now 16 roses in the vase. Howmanyrosesdidshecut? MULTIARITH(Koncel-Kedziorskietal.,2016) 600 The school cafeteria ordered 42 red apples and 7 green applesforstudentslunches. But,ifonly9studentswanted fruit,howmanyextradidthecafeteriaendupwith? Table10: Examplesfrommathreasoningdatasetsusedinthiswork. TablefromWeietal.(2022). Datasetsvaryinnumber ofwordsperquestion,with47wordsperquestioninGSM8K,21wordsinSINGLEOP,and∼30wordsintheotherdatasets.

	PAL:Program-aidedLanguageModels				22
Question	COT:GSM		COT:GSMHard		Identical?
90singleusecontactscomein1box 90 contacts come in 1 box. Each 90singleusecontactscomein1box. Yes
andwilllastPete45days.Eachbox boxis$100.00.So90contactscost So1boxwilllastPete45days.Each
is $100.00 and currently 10% off. $100.00. Ifhebuys2boxes, then boxis$5451440andcurrently10%
Ifhebuys2boxesofcontact,how hewillhave2x90=180contacts. off.Soeachboxis$5451440*0.9=
muchwilleachpairofcontactscost? 180 contacts cost 2 x $100.00 = $4906296.Ifhebuys2boxesofcon-
	$200.00.	So each pair of	contacts tact,hewillhave2*90=180con-
---	--------------	---------------	------------------------------------	-----------------	---
	costs $200.00	/ 180 = $1.11.	The tacts. Soeachpairofcontactswill
	answeris$1.11.		cost $4906296	/ 180 = $2725849.
Theansweris$2725849
A chef bought 4 bags of onions. The chef bought 4 bags of onions. The chef bought 4 bags of onions. Yes
Each bag weighs 50 pounds. A Eachbagweighs50pounds.Sothe Eachbagweighs50pounds.Sothe
pound of onions cost $1.50. How totalweightis450=200pounds. totalweightis450=200pounds.
muchdidthechefspend? Apoundofonionscost$1.50. So A pound of onions cost $409201.
	200 pounds	of onions cost	200 * So200poundsofonionscost200*
---	----------------	--------------	---------------------------------	---------	---
	1.50=300dollars.	Theansweris	409201=$81840200.	Theanswer
	300.		is$81840200.
A bag has a 5% discount. If it is Thediscountis5%.Soyouwillpay The discount is 5%. So you will Yes
marked $140, how much will you 95%oftheprice.95%of140is133. pay 95% of the price. 95% of
payafterthediscount? Theansweris133. $3921772.0722 is $3730654.0682.
Theansweris$3730654.0682
A basket of green food costs $25 3basketsofgreenfoodwillcost3 A basket of green food costs No
andabasketofredfoodcosts$18. *25=75dollars. 3basketsofred 9412763 dollars. A basket of red
Ifyoubuy3basketsofgreenfood foodwillcost3*18=54dollars. foodcosts18dollars. Ifyoubuy3
and red food, how much will you Butyouget$2offforeachbasket basketsofgreenfoodandredfood,
havetopayintotalifyouget$2off ofredfood. Soyouget32=6 youwillhavetopay94127633+
foreachbasketofredfood? dollarsoff. Soyoupay75+54-6 18*3=28238280dollars.Ifyouget
	=123dollars.Theansweris123.		2dollarsoffforeachbasketofred
---	---------------------------	---	-----------------------------	---	---
food,youwillhavetopay28238280
-18*2=28238244dollars.Thean-
sweris28238244
Abumpercarrinkhas12redcars. Thereare12redcars. Thereare2 Therinkhas9318738redcars.They Yes
Theyhave2fewergreencarsthan fewergreencarsthanredcars. So have2fewergreencarsthanredcars.
they have red cars. They have 3 there are 12 - 2 = 10 green cars. Sotheyhave9318738-2=9318736
timesthenumberofbluecarsasthey Thereare3timesasmanybluecars green cars. They have 3 times
havegreencars. Therinkalsohas asgreencars. Sothereare3*10 the number of blue cars as green
yellowcars.Iftherinkhas75carsin =30bluecars.Thereare12+10+ cars. So they have 3 * 9318736
totalhowmanyyellowcarsdothey 30=52carsthatarenotyellow.So = 27956208 blue cars. The total
have? thereare75-52=23yellowcars. numberofcarsis75. Sothenum-
	Theansweris23.		berofyellowcarsis75-9318738-
---	--------------	---	----------------------------	---	---
9318736-27956208=-55,828,829.
Theansweris-55,828,829
Table11: AnalysisofgenerationsfromCODEX. Manualanalysisof25randomlygeneratedthoughtsrevealsthat16outof
25thoughtswereidentical,whereasotherswerecloseparaphrases.

PAL:Program-aidedLanguageModels 23 H.1.CreatingGSM-HARD Whilereplacingnumbersinthequestioniseasyusingpatternmatching,amorechallengingaspectisrecalculatingthecorrect answer. GSM8Kevaluationsetcontains1319samples,whichisprohibitivelyexpensivetoperformmanualre-calculation. Instead, we leverage PAL to assist obtaining the correct answers. For 71% of the examples where PAL is correct on GSM8K,weutilizethegeneratedprogramandreplacetheinitialvaluewiththelargervalues. Forexample,ifwecreate a harder version of the problem in Figure 3 by replacing $23 dollars with $15687 dollars, we correspondingly replace money initial=23 to money initial=15678. Running the program could automatically produce the correct answeroftheharderquestion. Notably,thisannotationprocessassumesthataprogramthatproducesacorrectanswerto aGSM8Kquestionindicatesthecorrectnessoftheprogramitself. Whilethisisnotguaranteedduetopossiblespurious correlations,wemanuallychecked25programsandfoundallofthemarecorrect. Fortheincorrect29%ofthecases,we runPALagainandperformnucleussampling(Holtzmanetal.,2019)withtemperature0.7,andrepeattheaboveprocessif anycorrectsolutionisfound. Finally,theauthorsmanuallyannotatethecorrectanswerfor50remainingcasesthatPALwas notabletosolveafter100iterations. H.2.GSM-HARDAnalysis Table11showsthoughtsgeneratedwithCOTonGSM8KandGSM-HARD. Amanualanalysisrevealsthatamajorityofthe generatedthoughts(16/25)wereidenticalforGSM8KandGSM-HARD,indicatingthatlargernumbersprimarilydiminish performanceduetofailureofLLMtodoarithmetic..

PAL:Program-aidedLanguageModels 24 I.GeneralizationofPALtoLeast-to-MostPrompting Q: Four years ago, Kody was only half as old as Mohamed. If Mohamed is currently

(cid:44)→ twice	30	years	old,	how old	is Kody?
A: To answer the question "How old is Kody?", we need to know: "How old is Mohamed?",
(cid:44)→ "How old was Mohamed four years ago?", "How old was Kody four years ago?".
(a)Least-to-MostMathReducingPrompt
Four years ago, Kody was only half as old as Mohamed. If Mohamed is currently twice
30	years	old,	how old	is Kody?
---	-----	----	-------	--------	---	---	---	---	---
(cid:44)→
Q: How	old was	Mohamed		four years	ago?
------	-------	-------	---	----------	----	---	---	---	---
A: We were told that Mohamed is currently twice 30 years old, so he is currently 30
*
2 = 60 years old. That means that four years ago he must have been 60 - 4 = 56
(cid:44)→
years	old.	The	answer	is 56.
-----	----	---	------	------	---	---	---	---	---
(cid:44)→
Q: How	old is	Kody?
------	------	-----	---	---	---	---	---	---	---
A: Four years ago, Kody was half as old as Mohamed, so Kody must have been 56 / 2 =
28 years old then. Since Kody was 28 years old four years ago, she must now be 28
(cid:44)→
+ 4	= 32	years	old.	The answer	is	32.
---	----	-----	----	----------	---	---	---	---	---
(cid:44)→
(b)Least-to-MostMathSolvingPrompt

Four years ago, Kody was only half as old as Mohamed. If Mohamed is currently twice

30	years	old,	how old	is Kody?
# How old	was	Mohamed	four	years	ago?
mohamed_age_current			= 30	* 2
mohamed_age_4_years_ago				= mohamed_age_current			-	4
# Final	Question:		How old	is Kody?
kody_age_4_years_ago			=	mohamed_age_4_years_ago			/	2
kody_age_current		=	kody_age_4_years_ago			+ 4
answer	= kody_age_current
(c)PALMathSolvingPrompt
				Figure15:		PromptsforMathdatasets.
---	---	---	---	---------	---	-----------------------	---	---	---
PreviousexperimentsfocusontheCOTtechnique. ThissectionexaminesifPALgeneralizestootherprompttypes. We
considerastrongalternativepromptingstrategyLEAST-TO-MOST(Zhouetal.,2022). LEAST-TO-MOSTsolvesproblems
intwostages,problem-reducingandproblem-solving. Problemreducingstageturnstheproblemintosub-problems,and
thesolvingstagesolvesthemsequentially. Itkeepstwoprompts,eachforanindividualstage. TopatchLEAST-TO-MOST
promptswith PAL,weadoptasimpleandstraightforwardapproach: wenotethatproblemreductionrequireslogically
thinkinginNLwhilesolvingrequirestheprecisionthatPLoffers. Wethereforekeeptheoriginalreducingpromptswhile
onlyturningsolutionsegmentsinthesolvingscriptsinPL.Weshowanexamplereducingprompt,originalsolvingprompt,
andPALsolvingpromptinFigure15. NotethatoneuniquepropertyofPALsolvingcannaturallyusepreviousquestions’
answersasthesymbolvaluesareshared. Incomparison,theoriginalsolvingscriptneedstoexplicitlyre-citeanswersfrom
previousanswers.
		Dataset(500examples)			LEAST-TO-MOST			LEAST-TO-MOST+PAL
---	---	--------------------	---	---	-------------	----	---	-----------------	----
		GSM8K				67.2			72.8
		SVAMP				75.2			78.2
Table12: ResultsonGSM8KandSVAMPwithLEAST-TO-MOSTandLEAST-TO-MOSTwithPALsolvingprompt.
Forouranalysis,weconsidertheMathdatasetsGSM8K,andSVAMPasZhouetal.(2022)foundLeast-to-Mosthelpssolve
complexmathproblems. WepatchtheGSM8KpromptfromtheZhouetal.(2022)intoPAL.Notethattheothertasksin

PAL:Program-aidedLanguageModels 25 Zhouetal.(2022),like“concatenatinglastletters”fromseveralwords,requiresimpleroutinesandaretriviallysolvableby PAL.Weexperimentwithsubsetsof500examplesandrecordresultsinTable12. HereweseePALcantakeadvantageof theproblemdecompositionofferedbytheLEAST-TO-MOSTreducingandfurtherleveragethearithmeticcapabilityinthe Pythonruntimetoachieveadditionalperformancegains.

PAL:Program-aidedLanguageModels 26 J.Prompts WeshowhereexamplePALpromptsweusedforeachdataset. Weshowoneexampleforeachofthefew-shotprompts. Thefullspromptcanbefoundinourreleasedcode. J.1.ReasoningaboutColoredObjects

Q: On the table, you see a bunch of objects arranged in a row: a purple paperclip,

a pink stress ball, a brown keychain, a green scrunchiephone charger, a mauve fidget spinner, and a burgundy pen. What is the color of the object directly to

the	right	of the	stress	ball?
# Put objects		into a	list	to record	ordering
objects	= []
objects	+= [('paperclip',			'purple')]	*	1
objects	+= [('stress		ball',	'pink')]	*	1
objects	+= [('keychain',		'brown')]		* 1
objects	+= [('scrunchiephone			charger',		'green')]	* 1
objects	+= [('fidget		spinner',	'mauve')]		* 1
objects	+= [('pen',	'burgundy')]			* 1
# Find the	index	of the	stress	ball
stress_ball_idx		= None
for i, object		in enumerate(objects):
if object[0]		== 'stress		ball':
	stress_ball_idx		=	i
break
# Find the	directly	right		object
------------------	------------------	--------------------------	---------------	--------	-----	---	---
direct_right	=	objects[stress_ball_idx+1]
# Check	the directly		right	object's	color
direct_right_color		=	direct_right[1]
answer =	direct_right_color

PAL:Program-aidedLanguageModels 27 J.2.PenguinsinaTable """Q: Here is a table where the first line is a header and each subsequent line is a penguin: name, age, height (cm), weight (kg) Louis, 7, 50, 11 Bernard, 5, 80, 13 Vincent, 9, 60, 11 Gwen, 8, 70, 15 For example: the age of Louis is 7, the weight of Gwen is 15 kg, the height of Bernard is 80 cm. We now add a penguin to the

table:	James,	12,	90, 12
How many	penguins	are	less	than 8 years old?
"""
# Put the	penguins	into	a list.
---------------------------	--------	------	-------	-----------
penguins	= []
penguins.append(('Louis',			7,	50, 11))
penguins.append(('Bernard',				5, 80, 13))
penguins.append(('Vincent',				9, 60, 11))
penguins.append(('Gwen',			8,	70, 15))
# Add penguin		James.
penguins.append(('James',			12,	90, 12))
# Find	penguins	under	8 years	old.
penguins_under_8_years_old = [penguin for penguin in penguins if penguin[1] < 8]
# Count	number	of perguins		under 8.
-------------------	---------------------	-----------	-------------------------------	--------
num_penguin_under_8		=	len(penguins_under_8_years_old)
answer	= num_penguin_under_8
Figure17

PAL:Program-aidedLanguageModels 28 J.3.DateUnderstanding

Q: 2015 is coming in 36 hours. What is the date one week from today in MM/DD/YYYY?

If 2015 is coming in 36 hours, then today is 36 hours before.

today = datetime(2015,		1, 1) - relativedelta(hours=36)
# One week	from today,
one_week_from_today		= today + relativedelta(weeks=1)
# The answer	formatted	with %m/%d/%Y	is
one_week_from_today.strftime('%m/%d/%Y')

PAL:Program-aidedLanguageModels 29 J.4.Math #Q: Olivia has $23. She bought five bagels for $3 each. How much money does she have left?

money_initial		= 23
bagels =	5
bagel_cost	= 3
money_spent	=	bagels	bagel_cost
*
money_left	= money_initial		- money_spent
----------	---------------	---	-------------	---
print(money_left)
#Q: Michael had 58 golf balls. On tuesday, he lost 23 golf balls. On wednesday, he lost
2 more. How many golf balls did he have at the end of wednesday?
golf_balls_initial		=	58
-------------------------	---	---	----	---
golf_balls_lost_tuesday			= 23
golf_balls_lost_wednesday			= 2
golf_balls_left = golf_balls_initial - golf_balls_lost_tuesday -
golf_balls_lost_wednesday
print(golf_balls_left)
#Q: There were nine computers in the server room. Five more computers were installed
each day, from monday to thursday. How many computers are now in the server room?
computers_initial		=	9
-----------------	---	-------------------	--------------	-----------------
computers_per_day		=	5
num_days	= 4	# 4 days	between monday	and thursday
computers_added		= computers_per_day		* num_days
computers_total		= computers_initial		+ computers_added
print(computers_total)
#Q: If there are 3 cars in the parking lot and 2 more cars arrive, how many cars are in
the	parking	lot?
------------	--------------	----	--------------	---
cars_initial	=	3
cars_arrived	=	2
total_cars	= cars_initial		+ cars_arrived
print(total_cars)
#Q: Leah had 32 chocolates and her sister had 42. If they ate 35, how many pieces do
they	have	left in	total?
-----------------	----	------------------	------	-------------------
leah_chocolates		= 32
sister_chocolates		=	42
total_chocolates		= leah_chocolates		+ sister_chocolates
chocolates_eaten		= 35
chocolates_left		= total_chocolates		- chocolates_eaten
print(chocolates_left)
			Figure19:	Promptusedformathematicalreasoning(1/2)
---	---	---	---------	---------------------------------------

PAL:Program-aidedLanguageModels 30 #Q: Jason had 20 lollipops. He gave Denny some lollipops. Now Jason has 12 lollipops.

How	many lollipops	did Jason	give to Denny?
jason_lollipops_initial		= 20
jason_lollipops_after		= 12
denny_lollipops = jason_lollipops_initial - jason_lollipops_after
print(denny_lollipops)
#Q: There are 15 trees in the grove. Grove workers will plant trees in the grove today.
After they are done, there will be 21 trees. How many trees did the grove workers
plant	today?
-------------	-------------	---------------	---
trees_initial	= 15
trees_after	= 21
trees_added	= trees_after	- trees_initial
print(trees_added)
#Q: Shawn has five toys. For Christmas, he got two toys each from his mom and dad. How
many	toys does he	have now?
--------------	--------------	----------------	---
toys_initial	= 5
mom_toys	= 2
dad_toys	= 2
total_received	= mom_toys	+ dad_toys
total_toys	= toys_initial	+ total_received
print(total_toys)
		Figure20:	Promptusedformathematicalreasoning(2/2)
---	---	---------	---------------------------------------

PAL:Program-aidedLanguageModels 31 J.5.ObjectCounting

Q: I have a chair, two potatoes, a cauliflower, a lettuce head, two tables, a

cabbage, two onions, and three fridges. How many vegetables do I have?

# note: I'm	not counting	the chair,	tables,	or fridges
vegetables_to_count	=	{
'potato':	2,
'cauliflower':	1,
'lettuce	head': 1,
'cabbage':	1,
'onion':	2
}
print(sum(vegetables_to_count.values()))

Q: I have a drum, a flute, a clarinet, a violin, four accordions, a piano, a

trombone, and a trumpet. How many musical instruments do I have?

musical_instruments_to_count		= {
'drum':	1,
'flute':	1,
'clarinet':	1,
'violin':	1,
'accordion':	4,
'piano':	1,
'trombone':	1,
'trumpet':	1
}
print(sum(musical_instruments_to_count.values()))

Q: I have a chair, two ovens, and three tables. How many objects do I have?

objects_to_count	= {
'chair':	1,
'oven':	2,
'table':	3
}
print(sum(objects_to_count.values()))
		Figure21:	PromptusedforOBJECTCOUNTING.
---	---	---------	----------------------------	---

PAL:Program-aidedLanguageModels 32 J.6.RepeatCopy

Q: Repeat the word duck four times, but halfway through also say quack

result = []

for i in range(1,	5):
result.append("duck")
if i == 2:
----------	---	---
result.append("quack")
print(" ".join(result))

Q: Print boolean eleven times, but after the 3rd and 8th also say correct

result = []

for i in range(1,	12):
result.append("boolean")
if i == 3 or	i == 8:
------------	-------	---
result.append("correct")
print(" ".join(result))

Q: say java twice and data once, and then repeat all of this three times.

result = []

tmp = ["java",	"java", "data"]
for i in range(3):
result.extend(tmp)
print(" ".join(result))

Q: ask a group of insects in what family? four times. after the fourth time say The

happy family result = [] tmp = []

for i in range(1,	5):
tmp.append("a	group of insects	in what family?")
tmp.append("The	happy family")
result.extend(tmp)
print(" ".join(result))
Figure22: PromptusedforREPEATCOPY.

PAL:Program-aidedLanguageModels 33 K.SuccessandFailureModesinSymbolicTasks K.1.ColoredObjects

Find non-gold items to the right of the pencil

non_gold = [object for object in objects[i+1:] if object[1] != 'gold'] (a)SnippetofPALdoingafilteroperation.

Remove all pink objects

non_pink = [object for object in objects if object[1] != 'pink']

Count number of crayons

crayons = [object for object in non_pink if object[0] == 'crayon'] answer = len(crayons) (b)SnippetofPALcomposingmultipleoperations. Figure23: ExamplemodelgenerationsnippetsonReasoningaboutColorObject. Bymanuallyexaminingthethemodeloutputs,weobservethatPALoftenperformsbetteronquestionsthatinvolvecounting objectsthatsatisfyoneorseveralconditions. Forexample,COTfailsinthefollowingexample: “Onthedesk,youseea bunchofitemsarrangedinarow: agoldtextbook,apurplepuzzle,atealnecklace,andasilverpencil. Howmanynon-gold itemsdoyouseetotherightofthepencil?”. WithpureNLreasoning,aLLMcaneasilylosetrackoftheobjectsandoutput thewronganswerof“Thenumberofnon-golditemstotherightofthepencilistwo. Sotheansweristwo.” Incomparison,PALisabletoaccuratelyconstructtheobjectlistswithcorrectorderandattributes. Further,itcanprecisely leverage the simple yet complete PL syntax: it composes routines with functional operators from elementary builtin operation/operators. Figure4liststhelastafewlinesofthesolutiontothequestiondescribedabovegeneratedbyPAL, whichperformfilteringoverobjects. PALcanfurthercomposesuchoperationsacrossmultiplereasoningstepsasshownin Figure23b,whereitfirstfiltersoutallpinkobjectsandthenscanstheremainingtocountcrayons. Otherreasoningsuchas “howmanyobjectsareleftafterfiltering”canbeeasilytackledpreciselywithanruntime. K.2.PenguinsinaTable ...This question focuses on age. We know the following: Louis is 7 years old, Bernard (cid:44)→ is 5 years old, Vincent is 9 years old, and Gwen is 8 years old.Now, we add James (cid:44)→ to this table: James is 12 years old. We then delete the penguin named Bernard (cid:44)→ from the table.The penguins that are less than 8 years old are Louis and Bernard. (cid:44)→ There are 2 penguins less than 8 years old. So the answer is 2. ...

Delete penguin Bernard.

penguins = [penguin for penguin in penguins if penguin[0] != 'Bernard']

Find penguins under 8 years old.

penguins_under_8_years_old = [penguin for penguin in penguins if penguin[1] < 8]

Count number of perguins under 8.

num_penguin_under_8 = len(penguins_under_8_years_old) answer = num_penguin_under_8 Figure24: GeneratedSnippetsforonequestion(...WethendeletethepenguinnamedBernardfromthetable. Howmany penguinsarelessthan8yearsold?) inPenguinsinaTable . Inthistask,wefoundatypicalfailuremodeofthelanguageisitsinabilitytostablytracksstatesinitspastgeneration. Figure24liststhegenerationsofCOTandPALtothequestionthatcontainstheremovalofapenguin(“...Wethendelete thepenguinnamedBernardfromthetable. Howmanypenguinsarelessthan8yearsold?”). AlthoughCOTpicksupthe criticalinformationthat“penguinsthatarelessthen8yearsoldareLouisandBernard”,andthat“Bernardisdeleted”. It stillfailstoaggregatetheinformationproperlyandinferthatthereisonepenguinlessthen8leftintheend. Incomparison, PALexpressesthisdynamicthroughmanipulatingapenguinslistbyfilteringoutthepenguinwhosenameis“Bernard”,

PAL:Program-aidedLanguageModels 34 andmaintainingapenguins under 8 years oldlist. Itoffloadstrackingexactthevalues(inthiscase,thelengthof thelist)tothedeterministicPythonruntime. K.3.DateUnderstanding Q: Jane was born on the last day of Feburary in 2001. Today is her 16-year-old

(cid:44)→ birthday.	What is	the date 24	hours later in MM/DD/YYYY?
A: The last day of February is the 28th, so Jane was born on 02/28/2001. Today is her
(cid:44)→ 16-year old birthday, so today is 02/28/2017. So 24 hours later is 02/29/2017. So
(cid:44)→ the answer	is 02/29/2017.
--------------------	--------------	---	---

Q: Jane was born on the last day of Feburary in 2001. Today is her 16-year-old

birthday.	What is	the date 24	hours later in MM/DD/YYYY?

If Jane was born on the last day of Feburary in 2001 and today is her 16-year-old

birthday,	then today	is 16 years	later.
today = datetime(2001,		2, 28) + relativedelta(years=16)
# 24 hours	later,
later = today	+ relativedelta(hours=24)
# The answer	formatted	with %m/%d/%Y	is
later.strftime('%m/%d/%Y')
		Figure25: ExamplemodelgenerationonDateUnderstanding.
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Wefoundthisespeciallycommonwhenthetimedeltasareacrossthemonthboundary. WeshowanexampleinFigure25.
HerewithCOTprompting,theLLMexpressestheknowledgeofthe28-day-longFebruary,yetitstilloutputs02/29/2017as
thefinalanswer. WithPAL,theactualcalendarisaccurateasaprogramhandlestheoperation.