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### Skolem Functions for Factored Formulas [article]

Ajith K. John, Shetal Shah, Supratik Chakraborty, Ashutosh Trivedi, S. Akshay
2015 arXiv   pre-print
We present a new CEGAR style algorithm for generating Skolem functions from factored propositional formulas.  ...  Given a propositional formula F(x,y), a Skolem function for x is a function \Psi(y), such that substituting \Psi(y) for x in F gives a formula semantically equivalent to \exists F.  ...  CONCLUSION AND FUTURE WORK We presented a CEGAR algorithm for generating Skolem functions from factored propositional formulas.  ...

### Skolem functions for factored formulas

Ajith K. John, Shetal Shah, Supratik Chakraborty, Ashutosh Trivedi, S. Akshay
2015 2015 Formal Methods in Computer-Aided Design (FMCAD)
Solution: Find Skolem function for the input variables X in: 2. Generate Skolem functions matching a given template. ∧ i NotCovered i (X, Y ).  ...  assumption Composition based approaches -Quantifier elimination via functional composition by Jiang [Jia09] -Techniques in Symbolic model checking by Trivedi [Tri03] -Work well for small-sized formulas  ...  . · · · · · · -Skolem functions are in factored form: How to avoid such blowup . . . . . . · · · · · · -Skolem functions are in factored form: ψ i = ∧ψ k i  ...

### What's hard about Boolean Functional Synthesis [article]

S. Akshay, Supratik Chakraborty, Shubham Goel, Sumith Kulal, and Shetal Shah
2018 arXiv   pre-print
In this paper, we first show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must necessarily generate exponential-sized Skolem functions, thereby requiring  ...  We present a two-phase algorithm for Boolean functional synthesis, where the first phase is efficient both in terms of time and sizes of synthesized functions, and solves an overwhelming majority of benchmarks  ...  The function ψ i is called a Skolem function for x i in F , and Ψ = (ψ 1 , . . . ψ n ) is called a Skolem function vector for X in F .  ...

### What's Hard About Boolean Functional Synthesis? [chapter]

S. Akshay, Supratik Chakraborty, Shubham Goel, Sumith Kulal, Shetal Shah
2018 Lecture Notes in Computer Science
In this paper, we first show that unless some hard conjectures in complexity theory are falsified, Boolean functional synthesis must generate large Skolem functions in the worst-case.  ...  Detailed experimental evaluation shows our algorithm to perform better than other techniques for a large number of benchmarks.  ...  We also thank the anonymous reviewers for insightful comments.  ...

### Translation of resolution proofs into short first-order proofs without choice axioms

Hans de Nivelle
2005 Information and Computation
We present a way of transforming a resolution-style proof containing Skolemization into a natural deduction proof without Skolemization.  ...  I thank Marc Bezem for discussions on the subject of this paper and Dimitri Hendriks for the pleasant cooperation in  , which is one of the roots of the current paper.  ...  Acknowledgments It was due to a question asked by Georg Moser during the Deduktionstreffen 2002 in Freiburg that I started thinking about the problem of eliminating Skolemization without eliminating cuts  ...

### Translation of Resolution Proofs into Short First-Order Proofs without Choice Axioms [chapter]

Hans de Nivelle
2003 Lecture Notes in Computer Science
The clausal proof can contain resolution, paramodulation, factoring) into a purely first-order proof of the same formula.  ...  I present a way of transforming a resolution proof on the first-order level (The CNF-transformation can contain Skolemization, subformula replacements.  ...  A Skolem function f originates from Skolemizing some formula ∀x 1 · · · x n ∃y A(x 1 , . . . , x n , y). Simply take A as the serial relation for f : 1.  ...

### Manthan: A Data-Driven Approach for Boolean Function Synthesis [chapter]

Priyanka Golia, Subhajit Roy, Kuldeep S. Meel
2020 Lecture Notes in Computer Science
Manthan views functional synthesis as a classification problem, relying on advances in constrained sampling for data generation, and advances in automated reasoning for a novel proof-guided refinement  ...  Motivated by the progress in machine learning, we propose Manthan, a novel data-driven approach to Boolean functional synthesis.  ...  We are grateful to Mate Soos for tweaking BiasGen to support Manthan. We are indebted to S.  ...

### Manthan: A Data Driven Approach for Boolean Function Synthesis [article]

Priyanka Golia, Subhajit Roy, Kuldeep S. Meel
2020 arXiv   pre-print
Manthan views functional synthesis as a classification problem, relying on advances in constrained sampling for data generation, and advances in automated reasoning for a novel proof-guided refinement  ...  Motivated by the progress in machine learning, we propose Manthan, a novel data-driven approach to Boolean functional synthesis.  ...  We are grateful to Mate Soos for tweaking BiasGen to support Manthan. We are indebted to S.  ...

### Automated reasoning with quantified formulae

David Greve
2009 Proceedings of the Eighth International Workshop on the ACL2 Theorem Prover and its Applications - ACL2 '09
We present an ACL2 framework that employs tables, computed hints and clause processing to identify quantified formulae and to skolemize or instantiate them when possible.  ...  ACL2 allows users to define predicates whose logical behavior mimics that of universally or existentially quantified formulae. Proof support for such quantification, however, is quite limited.  ...  It is possible for skolemization to prepare the way for induction by replacing a function call with a symbol.  ...

### Synthesis from Assume-Guarantee Contracts using Skolemized Proofs of Realizability [article]

Andreas Katis, Grigory Fedyukovich, Andrew Gacek, John Backes, Arie Gurfinkel, Michael W. Whalen
2017 arXiv   pre-print
The proof of re- alizability is performed over a set of forall-exists formulas, and synthesis is per- formed by extracting Skolem functions witnessing the existential quan- tification.  ...  These Skolem functions can then be combined into an imple- mentation.  ...  Given a formula ∃ . (⃗, ), a Skolem function for , sk (⃗) is a function such that ∃ . (⃗, ) ⇐⇒ (⃗, sk (⃗)).  ...

### Skolem functions of arithmetical sentences

Shih Ping Tung
2003 Information and Computation
We define the Skolem function f (x) for ∀x∃y ψ(x, y) as follows. For any a in N (or Z) let f (a) be the least (or least absolute value of) b such that ψ(a, b) is true in N (or Z).  ...  For every arithmetical sentence ∀x∃y∀z ψ(x, y, z) true in N (or Z) there is a polynomial g(x) over Z such that the corresponding Skolem function f (x) < g(|x|) for any x in N (or Z).  ...  Skolem functions Suppose that ∀x∃y φ(x, y) is true in N (or Z). We define the Skolem function f (x) for ∀x∃y φ(x, y) as follows.  ...

### A Normal Form Characterization for Efficient Boolean Skolem Function Synthesis [article]

Preey Shah, Aman Bansal, S. Akshay, Supratik Chakraborty
2021 arXiv   pre-print
Boolean Skolem function synthesis concerns synthesizing outputs as Boolean functions of inputs such that a relational specification between inputs and outputs is satisfied.  ...  Thus, SAUNF provides the right trade-off in knowledge representation for Boolean functional synthesis.  ...  Finding Skolem functions for X 1 and X 2 in terms of I effectively asks us to solve the (n-bit) factorization problem.  ...

### Distributed agent-based automated theorem proving in order-sorted first-order logic [article]

Dohan Kim
2016 arXiv   pre-print
In this paper we provide a refutation-complete report procedure for automated theorem proving in order-sorted first-order logic in a subclass of distributed agent-based networks.  ...  Function symbols h, i, and j in Figure 1 are Skolem symbols employed for the Skolemization procedure.  ...  Since functions h and i are acceptable for our un-Skolemization procedure (see Appendix A), agent y can report the un-Skolemized Σ-formulae ∀c 1 :C∃p 1 :P . E(c 1 :C p 1 :P ) and ∀s 1 :S∃p 2 :P .  ...

### Automated Proof Construction in Type Theory Using Resolution [chapter]

Marc Bezem, Dimitri Hendriks, Hans de Nivelle
2000 Lecture Notes in Computer Science
Abstract We present a way of transforming a resolution-style proof containing Skolemization into a natural deduction proof without Skolemization.  ...  This paper discusses the problem of how to transform a firstorder formula into clausal normal form, and to simultaneously construct a proof that the clausal normal form is correct.  ...  Pliuskevicius for insisting on a complexity analysis.  ...

### Incremental Determinization [chapter]

Markus N. Rabe, Sanjit A. Seshia
2016 Lecture Notes in Computer Science
The algorithm incrementally adds new constraints to the formula until the constraints describe a unique Skolem function -or until the absence of a Skolem function is detected.  ...  We present the algorithm in analogy to search algorithms for SAT and explain how propagation, decisions, and conflicts are lifted from values to Skolem functions.  ...  We thank our anonymous reviewers for their comments and Marcell Vazquez-Chanlatte, Leander Tentrup, Ashutosh Trivedi, and Christoph Wintersteiger for fruitful discussions on this work.  ...
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