### Related work

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## 5.1. Related work

The unfolding of a graph transformation system (GTS) was first proposed in Baldan, 1999Paolo Baldan, Andrea Corradini and Ugo Montanari. 1999. Unfolding and Event Structure Semantics for Graph Grammars. In Foundations of Software Science and Computation Structures, Berlin, Heidelberg. Springer Berlin Heidelberg, 73--89. doi: 10.1007/3-540-49019-1_6 as a generalisation of a well-known construction on Petri nets Winskel, 1987Glynn Winskel. 1987. Event structures. Originally defined for DPO rewriting, the unfolding was later generalised to SPO Baldan, 2007Paolo Baldan, Andrea Corradini, Ugo Montanari and Leila Ribeiro. 2007. Unfolding semantics of graph transformation. Information and Computation 205, 5 (May 2007, 733--782). doi: 10.1016/j.ic.2006.11.004 Baldan, 2014Paolo Baldan, Andrea Corradini, Tobias Heindel, Barbara König and Paweł Sobociński. 2014. Processes and unfoldings: concurrent computations in adhesive categories. Mathematical Structures in Computer Science 24, 4 (June 2014). doi: 10.1017/s096012951200031x and SqPO Behr, 2019Nicolas Behr. 2019. Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra Framework. Electronic Proceedings in Theoretical Computer Science 309 (December 2019, 23--52). doi: 10.4204/eptcs.309.2 in arbitrary adhesive categories. The unfolding is a powerful GTS technique that has found applications in model verification Baldan, 2008Paolo Baldan, Andrea Corradini and Barbara König. 2008. Unfolding Graph Transformation Systems: Theory and Applications to Verification Baldan, 2008Paolo Baldan, Andrea Corradini and Barbara König. 2008. A framework for the verification of infinite-state graph transformation systems. Information and Computation 206, 7 (July 2008, 869--907). doi: 10.1016/j.ic.2008.04.002 Costa, 2012Simone André da Costa and Leila Ribeiro. 2012. Verification of graph grammars using a logical approach. Science of Computer Programming 77, 4 (April 2012, 480--504). doi: 10.1016/j.scico.2010.02.006 and other formal analysis tools such as model-based diagnosis Baldan, 2008Paolo Baldan, Thomas Chatain, Stefan Haar and Barbara König. 2008. Unfolding-Based Diagnosis of Systems with an Evolving Topology and model transformation analysis Bisztr., 2009Dénes András Bisztray. 2009. Compositional verification of model-level refactorings based on graph transformations. PhD Thesis. University of Leicester.

Unfoldings of finite GSTs are often infinite. A lot of work has therefore concerned itself with finding sufficient conditions for finiteness or the existence of finite complete prefixes of unfoldings Baldan, 2008Paolo Baldan, Andrea Corradini and Barbara König. 2008. Unfolding Graph Transformation Systems: Theory and Applications to Verification Baldan, 2004Paolo Baldan, Andrea Corradini and Barbara König. 2004. Verifying Finite-State Graph Grammars: An Unfolding-Based Approach. In CONCUR 2004 - Concurrency Theory, Berlin, Heidelberg. Springer Berlin Heidelberg, 83--98. doi: 10.1007/978-3-540-28644-8_6 Baldan, 2008Paolo Baldan, Andrea Corradini, Barbara König and Stefan Schwoon. 2008. McMillan's Complete Prefix for Contextual Nets Baldan, 2010Paolo Baldan, Alessandro Bruni, Andrea Corradini, Barbara König and Stefan Schwoon. 2010. On the Computation of McMillan's Prefix for Contextual Nets and Graph Grammars. In Graph Transformations, Berlin, Heidelberg. Springer Berlin Heidelberg, 91--106. doi: 10.1007/978-3-642-15928-2_7 Schwoon, 2013Stefan Schwoon. 2013. Efficient verification of sequential and concurrent systems. PhD Thesis. École normale supérieure de Cachan-ENS Cachan. On the other hand, unfoldings of GTSs of quantum computation are expected to be intractably large Yang, 2021Yichen Yang, Mangpo Phitchaya Phothilimtha, Yisu Remy Wang, Max Willsey, Sudip Roy and Jacques Pienaar. 2021. Equality Saturation for Tensor Graph Superoptimization. CoRR abs/2101.01332. doi: 10.48550/ARXIV.2101.01332, with no complete prefixes in general. Rather, our interests lie in finding heuristics that determine the subspace of the unfolding of interest, combined with fast algorithms to expand finite unfolding prefixes into larger ones. This chapter is to our knowledge the first work in this direction.

Persistent data structures on the other hand have a rich history in computer science Drisco., 1989James R. Driscoll, Neil Sarnak, Daniel D. Sleator and Robert E. Tarjan. 1989. Making data structures persistent. Journal of Computer and System Sciences 38, 1 (February 1989, 86--124). doi: 10.1016/0022-0000(89)90034-2 Lagogi., 2005George Lagogiannis, Yannis Panagis, Spyros Sioutas and Athanasios Tsakalidis. 2005. A survey of persistent data structures. In Proceedings of the 9th WSEAS International Conference on Computers, Stevens Point, Wisconsin, USA. World Scientific and Engineering Academy and Society (WSEAS), and particularly within functional programming Okasaki, 1996Chris Okasaki and Peter Lee. 1996. Purely functional data structures. Carnegie Mellon University, USA Okasaki, 1998Chris Okasaki and Andy Gill. 1998. Fast Mergeable Integer Maps. In Workshop on ML, Septempter 1998, 77--86 Hinze, 2005Ralf Hinze and Ross Paterson. 2005. Finger trees: a simple general-purpose data structure. Journal of Functional Programming 16, 02 (November 2005, 197). doi: 10.1017/s0956796805005769. Confluently persistent data structures were first explored in Drisco., 1994James R. Driscoll, Daniel D. K. Sleator and Robert E. Tarjan. 1994. Fully persistent lists with catenation. Journal of the ACM 41, 5 (Septempter 1994, 943--959). doi: 10.1145/185675.185791. A general treatment of the approach was subsequently presented in Fiat, 2003Amos Fiat and Haim Kaplan. 2003. Making data structures confluently persistent. Journal of Algorithms 48, 1 (August 2003, 16--58). doi: 10.1016/s0196-6774(03)00044-0 and improved in Collet., 2012Sébastien Collette, John Iacono and Stefan Langerman. 2012. Confluent Persistence Revisited. In Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, January 2012. Society for Industrial and Applied Mathematics, 593--601. doi: 10.1137/1.9781611973099.50. Chaler., 2018Parinya Chalermsook, Mayank Goswami, László Kozma, Kurt Mehlhorn and Thatchaphol Saranurak. 2018. Multi-Finger Binary Search Trees. In International Symposium on Algorithms and Computation 2018. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi: 10.4230/LIPICS.ISAAC.2018.55 proposed a data structure for confluently persistent tries.

Within the field of graph transformations, there is a well-developed theory for persistent (and confluently persistent) transformations in the form of the concurrent graph transformation formalism of Corradini et al. Corrad., 1996A. Corradini, U. Montanari and F. Rossi. 1996. Graph Processes. Fundamenta Informaticae 26, 3,4 (241--265). doi: 10.3233/fi-1996-263402 and Baldan et al. Baldan, 1999Paolo Baldan, Andrea Corradini, Ugo Montanari, Francesca Rossi, Hartmut Ehrig and Michael Löwe. 1999. Concurrent Semantic of Algebraic Graph Transformations. In Handbook of Graph Grammars and Computing by Graph Transformation, August 1999. World Scientific, 107--187. doi: 10.1142/9789812814951_0003. This categorical formalism has also been extended to include support for overlapping transformations in Echahed, 2017Rachid Echahed and Aude Maignan. 2017. Parallel Graph Rewriting with Overlapping Rules. In Proceedings of the 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning, LPAR, 300--318. doi: 10.29007/576h.

The first practical application of persistent graph rewriting was developed by the graph rewriting engine GRAPE Weber, 2017Jens H. Weber. 2017. GRAPE – A Graph Rewriting and Persistence Engine. In Graph Transformation. Springer International Publishing, 209--220. doi: 10.1007/978-3-319-61470-0_13 and was originally based on ephemeral data structures with transactional ACID semantics. Later, its successor, GrapeVine Weber, 2022Jens H. Weber. 2022. Tool Support for Functional Graph Rewriting with Persistent Data Structures - GrapeVine. In Graph Transformation. ICGT 2022. Springer International Publishing, 195--206. doi: 10.1007/978-3-031-09843-7_11, enhanced this with the first fully persistent data structure for graph rewriting. In this work, the vertices and edges that result from graph rewrites are stored individually in a specialised database. Depending on the requested data version, the graph can be retrieved from the database’s individual vertex and edge entities using the database’s graph query language. To our knowledge, no confluently persistent data structure has been proposed for graph rewriting.

As we will see in section 5.5, confluent persistence is a particularly valuable property in the absence of a rewriting strategy, i.e. a procedure to select and prioritise among possible graph transformations Echahed, 2008Rachid Echahed. 2008. Inductively Sequential Term-Graph Rewrite Systems. In Graph Transformations, ICGT. Springer Berlin Heidelberg, 84--98. doi: 10.1007/978-3-540-87405-8_7. This distinguishes the approach presented in this thesis from most previous work. Rewriting strategies feature prominently in PORGY, a tool for port graph rewriting Andrei, 2011Oana Andrei, Maribel Fernández, Hélène Kirchner, Guy Melançon, Olivier Namet and Bruno Pinaud. 2011. PORGY: Strategy-Driven Interactive Transformation of Graphs. Electronic Proceedings in Theoretical Computer Science 48 (February 2011, 54--68). doi: 10.4204/eptcs.48.7 Ferná., 2010Maribel Fernández and Olivier Namet. 2010. Strategic programming on graph rewriting systems. Electronic Proceedings in Theoretical Computer Science 44 (December 2010, 1--20). doi: 10.4204/eptcs.44.1; the graph rewriting software GROOVE provides the notion of a control program to govern the transformation order Rensink, 2004Arend Rensink. 2004. The GROOVE Simulator: A Tool for State Space Generation. In Applications of Graph Transformations with Industrial Relevance. Springer Berlin Heidelberg, 479--485. doi: 10.1007/978-3-540-25959-6_40; and finally, tools such as GrGen provide advanced control flow primitives to specify rewrite rule execution Geiß, 2006Rubino Geiß, Gernot Veit Batz, Daniel Grund, Sebastian Hack and Adam Szalkowski. 2006. GrGen: A Fast SPO-Based Graph Rewriting Tool. In Graph Transformations. ICGT 2006.. Springer Berlin Heidelberg, 383--397. doi: 10.1007/11841883_27.

Specifying rewriting strategies yields efficient graph transformation procedures and is particularly effective for systems with provable properties such as confluence and termination Verma, 1995Rakesh M. Verma. 1995. Transformations and confluence for rewrite systems. Theoretical Computer Science 152, 2 (December 1995, 269--283). doi: 10.1016/0304-3975(94)00255-0. As a result, rewriting strategies have also been used successfully within classical compiler optimisations Assmann, 2000Uwe Assmann. 2000. Graph rewrite systems for program optimization. ACM Transactions on Programming Languages and Systems 22, 4 (July 2000, 583--637). doi: 10.1145/363911.363914 and quantum circuit optimisation Fagan, 2018Andrew Fagan and Ross Duncan. 2018. Optimising Clifford Circuits with Quantomatic. In Proceedings 15th International Conference on Quantum Physics and Logic, QPL 2018, Halifax, Canada, 3-7th June 2018, 85--105. doi: 10.4204/EPTCS.287.5 Duncan, 2020Ross Duncan, Aleks Kissinger, Simon Perdrix and John van de Wetering. 2020. Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus. Quantum 4 (June 2020, 279). doi: 10.22331/q-2020-06-04-279.

However, such properties of the transition system – or successful heuristic approximations for it – cannot always be derived. In these cases, the space of graphs reachable from an input graph within the transition system must be explored non-deterministically. In the absence of a control program, GROOVE will fall back to an exhaustive exploration of the search space – for an exploration up to depth Δ\DeltaΔ, the result is a search tree of size O(γ Δ)\mathcal O(\gamma^\Delta)O(γ Δ), where γ\gamma γ is the number of possible rewrites at every graph in the search space (assuming γ\gamma γ is constant for every reachable graph).

Exhaustive exploration is used extensively in model checking, typically to verify properties that must hold for all reachable graphs Rensink, 2004Arend Rensink, Ákos Schmidt and Dániel Varró. 2004. Model Checking Graph Transformations: A Comparison of Two Approaches. In Graph Transformations. ICGT 2004. Springer Berlin Heidelberg, 226--241. doi: 10.1007/978-3-540-30203-2_17. It has also proven to be very useful for compiler optimisation, where the constantly evolving rewrite rules, instruction sets and complex, architecture-dependent cost functions render it challenging to fix a deterministic program transformation schedule.

Jia et al. showed in Jia, 2019Zhihao Jia, Oded Padon, James Thomas, Todd Warszawski, Matei Zaharia and Alex Aiken. 2019. TASO: optimizing deep learning computation with automatic generation of graph substitutions. In Proceedings of the 27th ACM Symposium on Operating Systems Principles, October 2019. ACM, 47--62. doi: 10.1145/3341301.3359630 that computation graph optimisation using graph transformations was achievable without predefined rewriting strategies. They discovered new state-of-the-art implementations for computation graphs of interest to the deep learning community using a simple exhaustive search of the space of possible rewrites with backtracking. This approach was then adapted to quantum circuit optimisation in Xu, 2022Mingkuan Xu, Zikun Li, Oded Padon, Sina Lin, Jessica Pointing, Auguste Hirth, Henry Ma, Jens Palsberg, Alex Aiken, Umut A. Acar and Zhihao Jia. 2022. Quartz: Superoptimization of Quantum Circuits. In Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation, June 2022. Association for Computing Machinery, 625--640. doi: 10.1145/3519939.3523433 and Xu, 2023Amanda Xu, Abtin Molavi, Lauren Pick, Swamit Tannu and Aws Albarghouthi. 2023. Synthesizing Quantum-Circuit Optimizers. Proceedings of the ACM on Programming Languages 7, PLDI (June 2023, 835--859). doi: 10.1145/3591254.

These recent results fit within a long line of compiler research called superoptimisation Fraser, 1979Christopher W. Fraser. 1979. A compact, machine-independent peephole optimizer. In Proceedings of the 6th ACM SIGACT-SIGPLAN symposium on Principles of programming languages - POPL ’79. ACM Press, 1--6. doi: 10.1145/567752.567753 Massal., 1987Henry Massalin. 1987. Superoptimizer: a look at the smallest program. In Proceedings of the second international conference on Architectual support for programming languages and operating systems, October 1987. ACM, 122--126. doi: 10.1145/36206.36194 Sands, 2011Duncan Sands. 2011. Super-optimizing LLVM IR. (November 2011). Retrieved on 13/01/2025 (LLVM Developer's meeting) from http://llvm.org/devmtg/2011-11/Sands_Super-optimizingLLVMIR.pdf Bansal, 2006Sorav Bansal and Alex Aiken. 2006. Automatic generation of peephole superoptimizers. ACM SIGARCH Computer Architecture News 34, 5 (October 2006, 394--403). doi: 10.1145/1168919.1168906 Sasnau., 2017Raimondas Sasnauskas, Yang Chen, Peter Collingbourne, Jeroen Ketema, Jubi Taneja and John Regehr. 2017. Souper: A Synthesizing Superoptimizer. CoRR abs/1711.04422. On top of excellent optimisation performance, this approach to compiler optimisation using graph transformation systems (GTS) is exceptionally flexible, as rewrite rules can be generated and tailored on demand to the constraints and instruction set of the target hardware. For any supplied cost function, the compiler can explore all valid program transformations to find the rewrites sequence that minimises cost. This keeps the cost function-specific logic separate from the transformation semantics of the program, making it straightforward to replace or update the optimisation objective.

The adaptation of superoptimisation to quantum optimisation of Xu, 2022Mingkuan Xu, Zikun Li, Oded Padon, Sina Lin, Jessica Pointing, Auguste Hirth, Henry Ma, Jens Palsberg, Alex Aiken, Umut A. Acar and Zhihao Jia. 2022. Quartz: Superoptimization of Quantum Circuits. In Proceedings of the 43rd ACM SIGPLAN International Conference on Programming Language Design and Implementation, June 2022. Association for Computing Machinery, 625--640. doi: 10.1145/3519939.3523433 and Xu, 2023Amanda Xu, Abtin Molavi, Lauren Pick, Swamit Tannu and Aws Albarghouthi. 2023. Synthesizing Quantum-Circuit Optimizers. Proceedings of the ACM on Programming Languages 7, PLDI (June 2023, 835--859). doi: 10.1145/3591254 is, however, showing scaling difficulties: unlike classical superoptimisation, which is usually designed to optimise small subroutines within programs, e.g. focusing on arithmetic instructions, single instruction multiple data (SIMD) etc., the technique should in principle be able to optimise quantum programs in their entirety and requires tens of thousands of rewrite rules. This leads to immense search spaces that superoptimisation does not scale well to.

For the special case of term rewriting, i.e. rewriting of tree expressions, a technique known as equality saturation was introduced in Tate, 2009Ross Tate, Michael Stepp, Zachary Tatlock and Sorin Lerner. 2009. Equality saturation: a new approach to optimization. ACM SIGPLAN Notices 44, 1 (January 2009, 264--276). doi: 10.1145/1594834.1480915 to compress and reduce the size of the search space significantly. Equality saturation can be viewed as a twist on persistent data structures designed to optimise terms through term rewriting. It is persistent insofar as it preserves all data inserted into it, though, unlike persistent data structures, it does not retain the history of transformations. This introduces a new step in the optimisation process known as the extraction phase, where the best term stored within the data structure must be identified and recovered.

An efficient implementation was presented in Willsey, 2021Max Willsey. 2021. Practical and Flexible Equality Saturation. PhD Thesis. University of Washington, and it has recently been adopted in modern compiler optimisation pipelines Fallin, 2022Chris Fallin. 2022. Cranelift: Using E-Graphs for Verified, Cooperating Middle-End Optimizations. (August 2022). Retrieved on 14/01/2025 (RFC) from https://github.com/bytecodealliance/rfcs/blob/main/accepted/cranelift-egraph.md. Though the approach was extended to optimise computation graphs for deep learning in Yang, 2021Yichen Yang, Mangpo Phitchaya Phothilimtha, Yisu Remy Wang, Max Willsey, Sudip Roy and Jacques Pienaar. 2021. Equality Saturation for Tensor Graph Superoptimization. CoRR abs/2101.01332. doi: 10.48550/ARXIV.2101.01332, equality saturation does not generalise to graph rewriting. Equality saturation and the difficulties of adapting it to graph rewriting are interesting (and subtle!) enough to warrant their own section.

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