# Synthesis

See also note on:

- Constrained Horn Clauses
- Invariant Synthesis
- Prolog/Minikanren
- SMT

Sketch

Courses:

People:

- Armando Solar-lezama
- Hila Peleg
- Yotam Feldman
- Nadia Polikarpova
- Isil Dillig
- rahul sharma

https://github.com/logic-and-learning-lab/Popper

synquid - type driven program synthesis

rosetta?

Minikanren - barliman. Inseven examples there is also an interpreter run backwards

https://github.com/TyGuS/suslik syntheiss of heap maniuplation programs from seperation lgoic

https://www.youtube.com/watch?v=mFjSbxV_1vw&ab_channel=Fastly Synthesis in reverse engineering Synesthesia

# Alternating Quantifiers

Exists forall problems

There exists instanitations of holes in your program such that for all possible inputs, the program is correct. Vaguely speaking: \(\exists H \forall x \phi(x)\)

However, what is phi? One approach would be to define it using weakest precondition semantics. Or strongest postcondition

\[\exists H \psi(H) \land \forall x pre(x) \rightarrow WP(Prog,post(x))\]\(\forall\) is kind of like an infinite conjunction. For finite types you’re quantifying it over, you can expand it into conjunction. \(\forall c \phi(c) = \phi(red) \land \phi(blue) \land \phi(green)\)

You can also attempt to partially expand these quantifiers as a method to solve the alternating quantifier problem. This is CEGIS

\[\exists H. finite expansion\]In some sense, constraint solvers/smt/mathematical programming solvers are solving problems with implicit existential quantification over all the free variables.

Fixing the synthesis parameters, we

# inductive logic programming

# Syntax Guided Synthesis (Sygus)

Sygus - syntax guided synthesis demo of sygus sygus 2 compatbile with smtlib2? search based program synthesis

```
(set-option :lang sygus2)
()
;(check-synth)
;(define-fun spec ((x (BitVec 4)) (y BitVec 4)) (BitVec 4)
; (ite (bvslt x y) y x)
;)
```

invariant synthesis: `Spec = Inv /\ body /\ Test' => Inv'`

Proof synthesis?

## Sketch

PDR

Question about survey on synethesizing inductive ivanraints

Yotam Feldman answers:

- Houdini: invariant inference for conjunctive propositional invariants. I like the exposition in https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.153.3511&rep=rep1&type=pdf
- Interpolation: McMillan’s seminal paper on unbounded SAT-based model checking, which is hugely influential.

https://people.eecs.berkeley.edu/~alanmi/courses/2008_290A/papers/mcmillan_cav03.pdf

- IC3/PDR: Another hugely influential technique, originally also for propositional programs. I like the exposition in the extension to universally quantified invariants over uninterpreted domains: https://tau.ac.il/~sharonshoham/papers/jacm17.pdf
- Spacer: PDR for software, with arithmetic, also hugely influential in this domain. https://t.co/gVlNrL6xts
- Syntax-guided methods: less familiar with these ones, but I’d hit the survey https://sygus.org/assets/pdf/Journal_SyGuS.pdf and work by Grigory Fedyukovich (e.g. https://ieeexplore.ieee.org/document/8603011)

https://www.youtube.com/watch?v=-eH2t8G1ZkI&t=3413s syntax guided synthesis sygus-if https://sygus.org/ CVC4 supports it. LoopInvGen, OASES, DryadSynth, CVC4

polikarpova, peleg, isil dillig

https://www.youtube.com/watch?v=h2ZsstWit9E&ab_channel=SimonsInstitute - automated formal program reapir “fault localization” https://github.com/eionblanc/mini-sygus

https://arxiv.org/pdf/2010.07763.pdf refinement types constrained horn lcauses. Describes using simple houdini algorithm,.