# Program synthesis

In computer science, program synthesis is the task to construct a program that provably satisfies a given high-level formal specification. In contrast to program verification, the program is to be constructed rather than given; however, both fields make use of formal proof techniques, and both comprise approaches of different degrees of automatization. In contrast to automatic programming techniques, specifications in program synthesis are usually non-algorithmic statements in an appropriate logical calculus.[1]

## Origin

During the Summer Institute of Symbolic Logic at Cornell University in 1957, Alonzo Church defined the problem to synthesize a circuit from mathematical requirements.[2] Even though the work only refers to circuits and not programs, the work is considered to be one of the earliest descriptions of program synthesis and some researchers refer to program synthesis as "Church's Problem". In the 1960s, a similar idea for an "automatic programmer" was explored by researchers in artificial intelligence.[citation needed]

Since then, various research communities considered the problem of program synthesis. Notable works include the 1969 automata-theoretic approach by Büchi and Landweber,[3] and the works by Manna and Waldinger (c. 1980). The development of modern high-level programming languages can also be understood as a form of program synthesis.

## 21st century developments

The early 21st century has seen a surge of practical interest in the idea of program synthesis in the formal verification community and related fields. Armando Solar-Lezama showed that it is possible to encode program synthesis problems in Boolean logic and use algorithms for the Boolean satisfiability problem to automatically find programs.[4] In 2013, a unified framework for program synthesis problems was proposed by researchers at UPenn, UC Berkeley, and MIT.[5] Since 2014 there has been a yearly program synthesis competition comparing the different algorithms for program synthesis in a competitive event, the Syntax-Guided Synthesis Competition or SyGuS-Comp.[6] Still, the available algorithms are only able to synthesize small programs.

A 2015 paper demonstrated synthesis of PHP programs axiomatically proven to meet a given specification, for purposes such as checking a number for being prime or listing the factors of a number.[7]

## The framework of Manna and Waldinger

Non-clausal resolution rules (unifying substitutions not shown)
Nr Assertions Goals Program Origin
51 ${\displaystyle E[p]}$
52 ${\displaystyle F[p]}$
53 ${\displaystyle G[p]}$ s
54 ${\displaystyle H[p]}$ t
55 ${\displaystyle E[{\text{true}}]\lor F[{\text{false}}]}$ Resolve(51,52)
56 ${\displaystyle \lnot F[{\text{true}}]\land G[{\text{false}}]}$ s Resolve(52,53)
57 ${\displaystyle \lnot F[{\text{false}}]\land G[{\text{true}}]}$ s Resolve(53,52)
58 ${\displaystyle G[{\text{true}}]\land H[{\text{false}}]}$ p ? s : t Resolve(53,54)

The framework of Manna and Waldinger, published in 1980,[8] starts from a user-given first-order specification formula. For that formula, a proof is constructed, thereby also synthesizing a functional program from unifying substitutions.

The framework is presented in a table layout, the columns containing:

• A line number ("Nr") for reference purposes
• Formulas that already have been established, including axioms and preconditions, ("Assertions")
• Formulas still to be proven, including postconditions, ("Goals"),[9]
• Terms denoting a valid output value ("Program")[10]
• A justification for the current line ("Origin")

Initially, background knowledge, pre-conditions, and post-conditions are entered into the table. After that, appropriate proof rules are applied manually. The framework has been designed to enhance human readability of intermediate formulas: contrary to classical resolution, it does not require clausal normal form, but allows one to reason with formulas of arbitrary structure and containing any junctors ("non-clausal resolution"). The proof is complete when ${\displaystyle {\it {true}}}$ has been derived in the Goals column, or, equivalently, ${\displaystyle {\it {false}}}$ in the Assertions column. Programs obtained by this approach are guaranteed to satisfy the specification formula started from; in this sense they are correct by construction.[11] Only a minimalist, yet Turing-complete, functional programming language, consisting of conditional, recursion, and arithmetic and other operators[12] is supported.

### Proof rules

Proof rules include:

For example, line 55 is obtained by resolving Assertion formulas ${\displaystyle E}$ from 51 and ${\displaystyle F}$ from 52 which both share some common subformula ${\displaystyle p}$. The resolvent is formed as the disjunction of ${\displaystyle E}$, with ${\displaystyle p}$ replaced by ${\displaystyle {\it {true}}}$, and ${\displaystyle F}$, with ${\displaystyle p}$ replaced by ${\displaystyle {\it {false}}}$. This resolvent logically follows from the conjunction of ${\displaystyle E}$ and ${\displaystyle F}$. More generally, ${\displaystyle E}$ and ${\displaystyle F}$ need to have only two unifiable subformulas ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$, respectively; their resolvent is then formed from ${\displaystyle E\theta }$ and ${\displaystyle F\theta }$ as before, where ${\displaystyle \theta }$ is the most general unifier of ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$. This rule generalizes resolution of clauses.[13]
Program terms of parent formulas are combined as shown in line 58 to form the output of the resolvent. In the general case, ${\displaystyle \theta }$ is applied to the latter also. Since the subformula ${\displaystyle p}$ appears in the output, care must be taken to resolve only on subformulas corresponding to computable properties.
• Logical transformations.
For example, ${\displaystyle E\land (F\lor G)}$ can be transformed to ${\displaystyle (E\land F)\lor (E\land G)}$) in Assertions as well as in Goals, since both are equivalent.
• Splitting of conjunctive assertions and of disjunctive goals.
An example is shown in lines 11 to 13 of the toy example below.
This rule allows for synthesis of recursive functions. For a given pre- and postcondition "Given ${\displaystyle x}$ such that ${\displaystyle {\textit {pre}}(x)}$, find ${\displaystyle f(x)=y}$ such that ${\displaystyle {\textit {post}}(x,y)}$", and an appropriate user-given well-ordering ${\displaystyle \prec }$ of the domain of ${\displaystyle x}$, it is always sound to add an Assertion "${\displaystyle x'\prec x\land {\textit {pre}}(x')\implies {\textit {post}}(x',f(x'))}$".[14] Resolving with this assertion can introduce a recursive call to ${\displaystyle f}$ in the Program term.
An example is given in Manna, Waldinger (1980), p.108-111, where an algorithm to compute quotient and remainder of two given integers is synthesized, using the well-order ${\displaystyle (n',d')\prec (n,d)}$ defined by ${\displaystyle 0\leq n' (p.110).

Murray has shown these rules to be complete for first-order logic.[15] In 1986, Manna and Waldinger added generalized E-resolution and paramodulation rules to handle also equality;[16] later, these rules turned out to be incomplete (but nevertheless sound).[17]

### Example

Example synthesis of maximum function
Nr Assertions Goals Program Origin
1 ${\displaystyle A=A}$ Axiom
2 ${\displaystyle A\leq A}$ Axiom
3 ${\displaystyle A\leq B\lor B\leq A}$ Axiom
10 ${\displaystyle x\leq M\land y\leq M\land (x=M\lor y=M)}$ M Specification
11 ${\displaystyle (x\leq M\land y\leq M\land x=M)\lor (x\leq M\land y\leq M\land y=M)}$ M Distr(10)
12 ${\displaystyle x\leq M\land y\leq M\land x=M}$ M Split(11)
13 ${\displaystyle x\leq M\land y\leq M\land y=M}$ M Split(11)
14 ${\displaystyle x\leq x\land y\leq x}$ x Resolve(12,1)
15 ${\displaystyle y\leq x}$ x Resolve(14,2)
16 ${\displaystyle \lnot (x\leq y)}$ x Resolve(15,3)
17 ${\displaystyle x\leq y\land y\leq y}$ y Resolve(13,1)
18 ${\displaystyle x\leq y}$ y Resolve(17,2)
19 ${\displaystyle {\textit {true}}}$ x<y ? y : x Resolve(18,16)

As a toy example, a functional program to compute the maximum ${\displaystyle M}$ of two numbers ${\displaystyle x}$ and ${\displaystyle y}$ can be derived as follows.[citation needed]

Starting from the requirement description "The maximum is larger than or equal to any given number, and is one of the given numbers", the first-order formula ${\displaystyle \forall X\forall Y\exists M:X\leq M\land Y\leq M\land (X=M\lor Y=M)}$ is obtained as its formal translation. This formula is to be proved. By reverse Skolemization,[18] the specification in line 10 is obtained, an upper- and lower-case letter denoting a variable and a Skolem constant, respectively.

After applying a transformation rule for the distributive law in line 11, the proof goal is a disjunction, and hence can be split into two cases, viz. lines 12 and 13.

Turning to the first case, resolving line 12 with the axiom in line 1 leads to instantiation of the program variable ${\displaystyle M}$ in line 14. Intuitively, the last conjunct of line 12 prescribes the value that ${\displaystyle M}$ must take in this case. Formally, the non-clausal resolution rule shown in line 57 above is applied to lines 12 and 1, with

• p being the common instance x=x of A=A and x=M, obtained by syntactically unifying the latter formulas,
• F[p] being truex=x, obtained from instantiated line 1 (appropriately padded to make the context F[⋅] around p visible), and
• G[p] being x ≤ x ∧ y ≤ x ∧ x = x, obtained from instantiated line 12,

yielding ${\displaystyle \lnot (}$truefalse) ∧ (x ≤ x ∧ y ≤ x ∧ true${\displaystyle )}$, which simplifies to ${\displaystyle x\leq x\land y\leq x}$.

In a similar way, line 14 yields line 15 and then line 16 by resolution. Also, the second case, ${\displaystyle x\leq M\land y\leq M\land y=M}$ in line 13, is handled similarly, yielding eventually line 18.

In a last step, both cases (i.e. lines 16 and 18) are joined, using the resolution rule from line 58; to make that rule applicable, the preparatory step 15→16 was needed. Intuitively, line 18 could be read as "in case ${\displaystyle x\leq y}$, the output ${\displaystyle y}$ is valid (with respect to the original specification), while line 15 says "in case ${\displaystyle y\leq x}$, the output ${\displaystyle x}$ is valid; the step 15→16 established that both cases 16 and 18 are complementary.[19] Since both line 16 and 18 comes with a program term, a conditional expression results in the program column. Since the goal formula ${\displaystyle {\textit {true}}}$ has been derived, the proof is done, and the program column of the "${\displaystyle {\textit {true}}}$" line contains the program.

## References

1. ^ Basin, D.; Deville, Y.; Flener, P.; Hamfelt, A.; Fischer Nilsson, J. (2004). "Synthesis of programs in computational logic". In M. Bruynooghe and K.-K. Lau (ed.). Program Development in Computational Logic. LNCS. 3049. Springer. pp. 30–65. CiteSeerX 10.1.1.62.4976.
2. ^ Alonzo Church (1957). "Applications of recursive arithmetic to the problem of circuit synthesis". Summaries of the Summer Institute of Symbolic Logic. 1: 3–50.
3. ^ Richard Büchi, Lawrence Landweber (Apr 1969). "Solving Sequential Conditions by Finite-State Strategies". Transactions of the American Mathematical Society. 138: 295–311. doi:10.2307/1994916. JSTOR 1994916.
4. ^ Solar-Lezama, Armando (2008). Program synthesis by sketching (Ph.D.). University of California, Berkeley.
5. ^ Alur, Rajeev; al., et (2013). "Syntax-guided Synthesis". Proceedings of Formal Methods in Computer-Aided Design. IEEE. p. 8.
6. ^ SyGuS-Comp (Syntax-Guided Synthesis Competition)
7. ^ Charles Volkstorf (7 Jan 2015). "Program Synthesis from Axiomatic Proof of Correctness". arXiv:1501.01363 [cs.LO].
8. ^ Zohar Manna, Richard Waldinger (Jan 1980). "A Deductive Approach to Program Synthesis". ACM Transactions on Programming Languages and Systems. 2: 90–121. doi:10.1145/357084.357090. A preprint appeared in Dec 1978 as SRI Technical Note 177
9. ^ The distinction "Assertions" / "Goals" is for convenience only; following the paradigm of proof by contradiction, a Goal ${\displaystyle F}$ is equivalent to an assertion ${\displaystyle \lnot F}$.
10. ^ When ${\displaystyle F}$ and ${\displaystyle s}$ is the Goal formula and the Program term in a line, respectively, then in all cases where ${\displaystyle F}$ holds, ${\displaystyle s}$ is a valid output of the program to be synthesized. This invariant is maintained by all proof rules. An Assertion formula usually is not associated with a Program term.
11. ^ See Manna, Waldinger (1980), p.100 for correctness of the resolution rules.
12. ^ Only the conditional operator (?:) is supported from the beginning. However, arbitrary new operators and relations can be added by providing their properties as axioms. In the toy example below, only the properties of ${\displaystyle =}$ and ${\displaystyle \leq }$ that are actually needed in the proof have been axiomatized, in line 1 to 3.
13. ^ Manna, Waldinger (1980), p.99
14. ^ Manna, Waldinger (1980), p.104
15. ^ Manna, Waldinger (1980), p.103, referring to: Neil V. Murray (Feb 1979). A Proof Procedure for Quantifier-Free Non-Clausal First Order Logic (Technical report). Syracuse Univ. 2-79.
16. ^ Zohar Manna, Richard Waldinger (Jan 1986). "Special Relations in Automated Deduction". Journal of the ACM. 33: 1–59. doi:10.1145/4904.4905.
17. ^ Zohar Manna, Richard Waldinger (1992). "The Special-Relations Rules are Incomplete". Proc. CADE 11. LNCS. 607. Springer. pp. 492–506.
18. ^ While ordinary Skolemization preserves satisfiability, reverse Skolemization, i.e. replacing universally quantified variables by functions, preserves validity.
19. ^ Axiom 3 was needed for that; in fact, if ${\displaystyle \leq }$ wasn't a total order, no maximum could be computed for uncomparable inputs ${\displaystyle x,y}$.
• Zohar Manna, Richard Waldinger (1975). "Knowledge and Reasoning in Program Synthesis". Artificial Intelligence. 6 (2): 175–208. doi:10.1016/0004-3702(75)90008-9.
• Jonathan Traugott (1986). "Deductive Synthesis of Sorting Programs". Proceedings of the International Conference on Automated Deduction. LNCS. 230. Springer. pp. 641–660.