q129/random_k2/random_1.md

|     |     | PAL: | Program-aided | Language |     | Models |     |     |     |
| --- | --- | ---- | ------------- | -------- | --- | ------ | --- | --- | --- |
LuyuGao*1 AmanMadaan*1 ShuyanZhou*1 UriAlon1 PengfeiLiu12 YimingYang1 JamieCallan1
GrahamNeubig12
{luyug,amadaan,shuyanzh,ualon,pliu3,yiming,callan,gneubig}@cs.cmu.edu
|     | Abstract |     |     | 1.Introduction |     |     |     |     |     |
| --- | -------- | --- | --- | -------------- | --- | --- | --- | --- | --- |
3202 naJ 72  ]LC.sc[  2v53401.1122:viXra
Untilasrecentlyastwoyearsago,reasoningwasconsidered
| Large language | models | (LLMs) | have recently |     |     |     |     |     |     |
| -------------- | ------ | ------ | ------------- | --- | --- | --- | --- | --- | --- |
tobeoneofthemostsignificantchallengesthatlargelan-
demonstrated an impressive ability to perform guagemodels(LLMs)hadnotyetovercome(Marcus,2018;
| arithmetic | and symbolic | reasoning | tasks, when |                         |     |     |                        |     |     |
| ---------- | ------------ | --------- | ----------- | ----------------------- | --- | --- | ---------------------- | --- | --- |
|            |              |           |             | 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 |     |     |     |     |
| --- | --- | --- | --- | ------------ | ----------------------------------- | --- | --- | --- | --- |
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 |
| --- | --- | --- | --- | -------- | ------- | --------- | -------------- | ----- | -------- |
solving
it intosteps, as well as each stepof the prompting(Brownetal.,2020).
| problem. | WhileLLMsseemtobeadeptatthis |     |     |     |     |     |     |     |     |
| -------- | ---------------------------- | --- | --- | --- | --- | --- | --- | --- | --- |
sortofstep-by-stepdecomposition,LLMsoften Thisprocesshasbeenacceleratedbymethodsthatrequire
|     |     |     |     | LLMs to | generate | their | explicit reasoning | steps, | such as |
| --- | --- | --- | --- | ------- | -------- | ----- | ------------------ | ------ | ------- |
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 |
| --- | --- | --- | --- | ------------------------------------------ | --- | --- | --- | --- | -------- |
lemsandgenerateprogramsastheintermediate
modelisexpectedtoapplyasimilardecompositiontotheac-
reasoningsteps,butoffloadsthesolutionsteptoa
runtimesuchasaPythoninterpreter. WithPAL, tualtestexample,andconsecutivelyreachanaccuratefinal
|     |     |     |     | answer(Lingetal.,2017;Aminietal.,2019). |     |     |     | Nevertheless, |     |
| --- | --- | --- | --- | --------------------------------------- | --- | --- | --- | ------------- | --- |
decomposingthenaturallanguageprobleminto
whileLLMscandecomposenaturallanguageproblemsinto
runnablestepsremainstheonlylearningtaskfor
stepsandperformsimplearithmeticoperations,theirperfor-
theLLM,whilesolvingisdelegatedtotheinter-
mancefallsdramaticallywhendealingwithcomplexarith-
| preter. We | demonstrate | this | synergy between a |     |     |     |     |     |     |
| ---------- | ----------- | ---- | ----------------- | --- | --- | --- | --- | --- | --- |
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 |
| --------- | --------- | ---------------- | --------- | ------- | ------ | ----------- | ------------ | -------- | ------- |
| 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 |
| --- | --- | --- | --- | ------------ | --- | -------------- | --------- | ------ | ------- |
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 |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | ------ | ----- | --- |
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   |     |     |     |     |     |     |     |     |     |     |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
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 |     |     |     |     |
| --- | --- | --- | --- | --- | --- | ---- | --- | --- | ---- | --- | --- | --- | --- |
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 |     |     |     |     |     |     |     |
| ------------- | --- | ---------------- | --- | ----- | --- | ------------ | --- | --- | --- | --- | --- | --- | --- |
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  |
| --- | --- | --- | --- | --- | --- | --- | -------------------------------- | --- | --- | --- | --- | -------------- | --- |
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 |              |       |
| --- | --- | --- | --- | --- | --- | --- | ------------------------- | --- | --------- | ------ | --------------------- | ------------ | ----- |
|     |     |     |     |     |     |     | 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 11
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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 totalweightis4*50=200pounds. totalweightis4*50=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. Soyouget3*2=6 youwillhavetopay9412763*3+
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. |     |
| --- | --- | ---------------------------------------------------- | --- |
Wefoundthisespeciallycommonwhenthetimedeltasareacrossthemonthboundary. WeshowanexampleinFigure25.
HerewithCOTprompting,theLLMexpressestheknowledgeofthe28-day-longFebruary,yetitstilloutputs02/29/2017as
thefinalanswer. WithPAL,theactualcalendarisaccurateasaprogramhandlestheoperation.