Deductive synthesis of recursive plans in linear logic

Centre for Intelligent Systems and their ApplicationsConventionally, the problem of plan formation in Artificial Intelligence deals with the generation of plans in the form of a sequence of actions. This thesis describes an approach to extending the expressiveness of plans to include conditional branches and recursion. This allows problems to be solved at a higher level, such that a single plan in such a language is capable of solving a class of problems rather than a single problem instance. A plan of fixed size may solve arbitrarily large problem instances. To form such plans, we take a deductive planning approach, in which the formation of the plan goes hand-in-hand with the construction of the proof that the plan specification is realisable. The formalism used here for specifying and reasoning with planning problems is Girard's Institutionistic Linear Logic (ILL), which is attractive for planning problems because state change can be expressed directly as linear implication, with no need for frame axioms. We extract plans by means of the relationship between proofs in ILL and programs in the style of Abramsky. We extend the ILL proof rules to account for induction over inductively defined types, thereby allowing recursive plans to be synthesised. We also adapt Abramsky's framework to partially evaluate and execute the plans in the extended language. We give a proof search algorithm tailored towards the fragment of the ILL employed (excluding induction rule selection). A system implementation, Lino, comprises modules for proof checking, automated proof search, plan extraction and partial evaluation of plans. We demonstrate the encodings and solutions in our framework of various planning domains involving recursion. We compare the capabilities of our approach with the previous approaches of Manna and Waldinger, Ghassem-Sani and Steel, and Stephen and Biundo. We claim that our approach gives a good balance between coverage of problems that can be described and the tractability of proof search

Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed.This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin’s cutfree LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1–1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called ‘permutation-free’ calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a ‘permutation-free’ calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to besound and complete with respect to a proof-theoretic semantics and (weak) cutelimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter

Proof Search Issues in Some Non-Classical Logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli (citeandreoli-92) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut-free LJT (citeherb-95, here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1--1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called `permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding emphet al (citeheu-sey-zim-96). It is applied to two calculi for intuitionistic logic and also to two modal logics: Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a `permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut-elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Mechanised Uniform Interpolation for Modal Logics K, GL, and iSL

The uniform interpolation property in a given logic can be understood as the definability of propositional quantifiers. We mechanise the computation of these quantifiers and prove correctness in the Coq proof assistant for three modal logics, namely: (1) the modal logic K, for which a pen-and-paper proof exists; (2) Gödel-Löb logic GL, for which our formalisation clarifies an important point in an existing, but incomplete, sequent-style proof; and (3) intuitionistic strong Löb logic iSL, for which this is the first proof-theoretic construction of uniform interpolants. Our work also yields verified programs that allow one to compute the propositional quantifiers on any formula in this logic

Refinement of Classical Proofs for Program Extraction

The A-Translation enables us to unravel the computational information in classical proofs, by first transforming them into constructive ones, however at the cost of introducing redundancies in the extracted code. This is due to the fact that all negations inserted during translation are replaced by the computationally relevant form of the goal. In this thesis we are concerned with eliminating such redundancies, in order to obtain better extracted programs. For this, we propose two methods: a controlled and minimal insertion of negations, such that a refinement of the A-Translation can be used and an algorithmic decoration of the proofs, in order to mark the computationally irrelevant components. By restricting the logic to be minimal, the Double Negation Translation is no longer necessary. On this fragment of minimal logic we apply the refined A-Translation, as proposed in (Berget et al., 2002). This method identifies further selected classes of formulas for which the negations do not need to be substituted by computationally relevant formulas. However, the refinement imposes restrictions which considerably narrow the applicability domain of the A-Translation. We address this issue by proposing a controlled insertion of double negations, with the benefit that some intuitionistically valid \Pi^0_2-formulas become provable in minimal logic and that certain formulas are transformed to match the requirements of the refined A-Translation. We present the outcome of applying the refined A-translation to a series of examples. Their purpose is two folded. On one hand, they serve as case studies for the role played by negations, by shedding a light on the restrictions imposed by the translation method. On the other hand, the extracted programs are characterized by a specific behaviour: they adhere to the continuation passing style and the recursion is in general in tail form. The second improvement concerns the detection of the computationally irrelevant subformulas, such that no terms are extracted from them. In order to achieve this, we assign decorations to the implication and universal quantifier. The algorithm that we propose is shown to be optimal, correct and terminating and is applied on the examples of factorial and list reversal.Die A-Übersetzung ermöglicht es, die rechnerische Information aus klassischen Beweisen einzuholen. Dennoch hat sie den Nachteil, dass die Programme, die man aus auf diese Weise transformierten Beweisen extrahiert, viele redundante Teile enthalten. Das liegt daran, dass die A-Übersetzung viele doppelte Negationen hinzufügt und alle diese Negationen durch die rechnerisch relevante Form der Ziel-Formel substituiert werden. In dieser Doktorarbeit werden Methoden dargestellt, um Teile der redundante Information in den extrahierten Programen zu entfernen. Einerseits wird das Einfügen der Negationen minimal gehalten und anderseits werden die nicht rechnerischen Teile als solche indentifiziert und ausgezeichnet. Wir bemerken zuerst, dass in der Minimallogik das Einfügen der doppelten Negationen nicht mehr nötig ist. Darüber hinaus, um das Ersetzen aller Negationen zu vermeiden, identifizieren (Berger et al., 2002) diejenigen, wo die Substitution nicht nötig ist. Diese verfeinerte A-Übersetzung hat aber den Nachteil, dass sie den Anwendungsbereich begrenzt. Um das zu beseitigen, wird in dieser Dissertation eine verfeinerte Doppel-Negation angewandt, die bestimmte Formeln so umsetzt, dass die verfeinerte A-Übersetzung darauf anwendbar ist. Als Zugabe kann diese Methode auch benutzt werden, um konstruktive Beweise mancher \Pi^0_2-Formeln in der Minimallogik durchzuführen. Dieses Verfahren wird durch Anwendung der verfeinerten A-Übersetzung auf eine Reihe von bedeutenden Fallstudien illustriert. Es werden das Lemma von Dickson, das unendliche Schubfachprinzip und das Erdös-Szekeres Theorem betrachtet. Dabei wird es festgestellt, dass ein Zusammenhang zu der Endrekursion und dem Rechnen mit Fortsezungen besteht. Ferner, um möglichst viel der überflüssigen Information zu entfernen, wird ein Dekorationsalgorithmus vorgelegt. Dadurch werden die rechnerisch irrelevanten Komponenten identifiziert und entsprechend annotiert, so dass sie während der Extraktion nicht berücksichtigt werden. Es wird gezeigt, dass das vorgeschlagene Dekorationsverfahren, das auf Beweisebene eingesetzt wird, optimal, korrekt und terminierend ist

Proof search issues in some non-classical logics

This thesis develops techniques and ideas on proof search. Proof search is used with one of two meanings. Proof search can be thought of either as the search for a yes/no answer to a query (theorem proving), or as the search for all proofs of a formula (proof enumeration). This thesis is an investigation into issues in proof search in both these senses for some non-classical logics. Gentzen systems are well suited for use in proof search in both senses. The rules of Gentzen sequent calculi are such that implementations can be directed by the top level syntax of sequents, unlike other logical calculi such as natural deduction. All the calculi for proof search in this thesis are Gentzen sequent calculi. In Chapter 2, permutation of inference rules for Intuitionistic Linear Logic is studied. A focusing calculus, ILLF, in the style of Andreoli ([And92]) is developed. This calculus allows only one proof in each equivalence class of proofs equivalent up to permutations of inferences. The issue here is both theorem proving and proof enumeration. For certain logics, normal natural deductions provide a proof-theoretic semantics. Proof enumeration is then the enumeration of all these deductions. Herbelin's cut- free LJT ([Her95], here called MJ) is a Gentzen system for intuitionistic logic allowing derivations that correspond in a 1-1 way to the normal natural deductions of intuitionistic logic. This calculus is therefore well suited to proof enumeration. Such calculi are called 'permutation-free' calculi. In Chapter 3, MJ is extended to a calculus for an intuitionistic modal logic (due to Curry) called Lax Logic. We call this calculus PFLAX. The proof theory of MJ is extended to PFLAX. Chapter 4 presents work on theorem proving for propositional logics using a history mechanism for loop-checking. This mechanism is a refinement of one developed by Heuerding et al ([HSZ96]). It is applied to two calculi for intuitionistic logic and also to two modal logics; Lax Logic and intuitionistic S4. The calculi for intuitionistic logic are compared both theoretically and experimentally with other decision procedures for the logic. Chapter 5 is a short investigation of embedding intuitionistic logic in Intuitionistic Linear Logic. A new embedding of intuitionistic logic in Intuitionistic Linear Logic is given. For the hereditary Harrop fragment of intuitionistic logic, this embedding induces the calculus MJ for intuitionistic logic. In Chapter 6 a 'permutation-free' calculus is given for Intuitionistic Linear Logic. Again, its proof-theoretic properties are investigated. The calculus is proved to be sound and complete with respect to a proof-theoretic semantics and (weak) cut- elimination is proved. Logic programming can be thought of as proof enumeration in constructive logics. All the proof enumeration calculi in this thesis have been developed with logic programming in mind. We discuss at the appropriate points the relationship between the calculi developed here and logic programming. Appendix A contains presentations of the logical calculi used and Appendix B contains the sets of benchmark formulae used in Chapter 4

Partial Order Reduction for Security Protocols

Security protocols are concurrent processes that communicate using cryptography with the aim of achieving various security properties. Recent work on their formal verification has brought procedures and tools for deciding trace equivalence properties (e.g., anonymity, unlinkability, vote secrecy) for a bounded number of sessions. However, these procedures are based on a naive symbolic exploration of all traces of the considered processes which, unsurprisingly, greatly limits the scalability and practical impact of the verification tools. In this paper, we overcome this difficulty by developing partial order reduction techniques for the verification of security protocols. We provide reduced transition systems that optimally eliminate redundant traces, and which are adequate for model-checking trace equivalence properties of protocols by means of symbolic execution. We have implemented our reductions in the tool Apte, and demonstrated that it achieves the expected speedup on various protocols

Automated Reasoning

This volume, LNAI 13385, constitutes the refereed proceedings of the 11th International Joint Conference on Automated Reasoning, IJCAR 2022, held in Haifa, Israel, in August 2022. The 32 full research papers and 9 short papers presented together with two invited talks were carefully reviewed and selected from 85 submissions. The papers focus on the following topics: Satisfiability, SMT Solving,Arithmetic; Calculi and Orderings; Knowledge Representation and Jutsification; Choices, Invariance, Substitutions and Formalization; Modal Logics; Proofs System and Proofs Search; Evolution, Termination and Decision Prolems. This is an open access book