Mahfuza Farooque - Academia.edu (original) (raw)
Papers by Mahfuza Farooque
In this paper we explore a proof-theoretic approach to the com-bination of generic proof-search a... more In this paper we explore a proof-theoretic approach to the com-bination of generic proof-search and theory-specific procedures, in presence of quantifiers. Forming the basis of tableaux methods, the sequent calculus is re-fined with meta-variables (a.k.a existential variables or free variables) to delay the choice of witnesses, and parameterised by theory-specific features based on a system of constraints for meta-variables. An axiomatisation of these features is given, so that the soundness and completeness of the sequent calculus with meta-variables can be generically proved (with respect to the sequent calculus where the choice of witnesses is not delayed). We then describe a theory-generic proof-search implementation, that is parameterised by a theory-specific mod-ule whose specification is given by the above axiomatisation. 1
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Two simulations about DPLL(T)
In this paper, we introduce two focussed sequent calculi, LK p (T) and LK + (T), that are based o... more In this paper, we introduce two focussed sequent calculi, LK p (T) and LK + (T), that are based on Miller-Liang’s LKF system [LM09] for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly " during proof-search. These features are used in other works [FLM12, FGLM13] to simulate the DPLL(T) procedure [NOT06] as proof-search in the extension of LK p (T) with a cut-rule. In this report we therefore prove cut-elimination in LK p (T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK + (T) that extends LK p (T) with a relaxed focussing discipline, but ...
banner above paper title A bisimulation between DPLL(T) and a proof-search strategy for the focus... more banner above paper title A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus
Automated reasoning techniques as proof-search in sequent calculus Thèse présentée par
ArXiv, 2013
In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Mi... more In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) b...
ArXiv, 2012
In this paper, we extend the sequent calculus LKF [LM09] into a calculus LK(T ), allowing calls t... more In this paper, we extend the sequent calculus LKF [LM09] into a calculus LK(T ), allowing calls to a decision procedure. We prove cut-elimination of LK(T ).
Computer-aided reasoning plays a great role in computer science and mathematical logic, from logi... more Computer-aided reasoning plays a great role in computer science and mathematical logic, from logic programing to automated reasoning, via interactive proof assistants, etc. The general aim of this thesis is to design a general framework where various techniques of Computer-aided reasoning can be implemented, so that they can collaborate, be generalised, and implemented in a safe and trusted way. The framework I propose is a sequent calculus called LKp(T), which generalises an older calculus of the literature to the presence of an arbitrary background theory for which we have a decision procedure, like linear arithmetic. The thesis develops the meta-theory of LKp(T), such as its logical completeness. We then show how it specifies a proof-search procedure that can emulate a well-known technique from the field of Satisfiability-modulo-Theories, namely DPLL(T). Finally, clause and connection tableaux are other widely used techniques of automated reasoning, of a rather different nature f...
ArXiv, 2014
In this paper we explore a proof-theoretic approach to the combination of generic proof-search an... more In this paper we explore a proof-theoretic approach to the combination of generic proof-search and theory-specific procedures, in presence of quantifiers. Forming the basis of tableaux methods, the sequent calculus is refined with meta-variables (a.k.a existential variables or free variables) to delay the choice of witnesses, and parameterised by theory-specific features based on a system of constraints for meta-variables. An axiomatisation of these features is given, so that the soundness and completeness of the sequent calculus with meta-variables can be generically proved (with respect to the sequent calculus where the choice of witnesses is not delayed). We then describe a theory-generic proof-search implementation, that is parameterised by a theory-specific module whose specification is given by the above axiomatisation.
ArXiv, 2012
In this paper we relate different formulations of the DPLL(T ) procedure. The first formulation i... more In this paper we relate different formulations of the DPLL(T ) procedure. The first formulation is based on a system of rewrite rules, which we denote DPLL(T ). The second formulation is an inference system of, which we denote LKDPLL(T ). The third formulation is the application of a standard proof-search mechanism in a sequent calculus LKp(T ) introduced here. We formalise an encoding from DPLL(T ) to LKDPLL(T ) that was, to our knowledge, never explicitly given and, in the case where DPLL(T ) is extended with backjumping and Lemma learning, never even implicitly given. We also formalise an encoding from LKDPLL(T ) to LKp(T ), building on Ivan Gazeau's previous work: we extend his work in that we handle the "-modulo-Theory" aspect of SAT-modulo-theory, by extending the sequent calculus to allow calls to a theory solver (seen as a blackbox). We also extend his work in that we handle advanced features of DPLL such as backjumping and Lemma learning, etc. Finally, we re f...
This paper gives an abstract description of decision procedures for Satisfiability Modulo Theory ... more This paper gives an abstract description of decision procedures for Satisfiability Modulo Theory (SMT) as proof search procedures in a sequent calculus with polarities and focusing. In particular, we show how to simulate the execution of standard techniques based on the Davis-Putnam- Logemann-Loveland (DPLL) procedure modulo theory as the gradual construction of a proof tree in sequent calculus. The construction mimicking a run of DPLL-modulo-Theory can be obtained by a meta-logical control on the proof-search in sequent calculus. This control is provided by polarities and focusing features, which there- fore narrow the corresponding search space in a sense we discuss. This simulation can also account for backjumping and learning steps, which correspond to the use of general cuts in sequent calculus.
We describe how the Davis-Putnam-Logemann-Loveland procedure, in its version DPLL(T ) for Satisfi... more We describe how the Davis-Putnam-Logemann-Loveland procedure, in its version DPLL(T ) for Satisfiability Modulo Theory (SMT), can be interpreted as a proof-search mechanism for the incremental construction of a proof tree in sequent calculus. For this we use a sequent calculus with polarities and focusing and show how its meta-logical control features allow for a precise simulation of runs of DPLL(T ). This simulation also accounts for backjumping and learning steps. Finally we describe Psyche, an extensible prototype for this sequent calculus, that includes the implementation of a proof-search component based on this simulation of DPLL(T ).
In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Mi... more In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T .
Lecture Notes in Computer Science, 2015
In this paper we explore a proof-theoretic approach to the com-bination of generic proof-search a... more In this paper we explore a proof-theoretic approach to the com-bination of generic proof-search and theory-specific procedures, in presence of quantifiers. Forming the basis of tableaux methods, the sequent calculus is re-fined with meta-variables (a.k.a existential variables or free variables) to delay the choice of witnesses, and parameterised by theory-specific features based on a system of constraints for meta-variables. An axiomatisation of these features is given, so that the soundness and completeness of the sequent calculus with meta-variables can be generically proved (with respect to the sequent calculus where the choice of witnesses is not delayed). We then describe a theory-generic proof-search implementation, that is parameterised by a theory-specific mod-ule whose specification is given by the above axiomatisation. 1
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Two simulations about DPLL(T)
In this paper, we introduce two focussed sequent calculi, LK p (T) and LK + (T), that are based o... more In this paper, we introduce two focussed sequent calculi, LK p (T) and LK + (T), that are based on Miller-Liang’s LKF system [LM09] for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly " during proof-search. These features are used in other works [FLM12, FGLM13] to simulate the DPLL(T) procedure [NOT06] as proof-search in the extension of LK p (T) with a cut-rule. In this report we therefore prove cut-elimination in LK p (T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK + (T) that extends LK p (T) with a relaxed focussing discipline, but ...
banner above paper title A bisimulation between DPLL(T) and a proof-search strategy for the focus... more banner above paper title A bisimulation between DPLL(T) and a proof-search strategy for the focused sequent calculus
Automated reasoning techniques as proof-search in sequent calculus Thèse présentée par
ArXiv, 2013
In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Mi... more In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) b...
ArXiv, 2012
In this paper, we extend the sequent calculus LKF [LM09] into a calculus LK(T ), allowing calls t... more In this paper, we extend the sequent calculus LKF [LM09] into a calculus LK(T ), allowing calls to a decision procedure. We prove cut-elimination of LK(T ).
Computer-aided reasoning plays a great role in computer science and mathematical logic, from logi... more Computer-aided reasoning plays a great role in computer science and mathematical logic, from logic programing to automated reasoning, via interactive proof assistants, etc. The general aim of this thesis is to design a general framework where various techniques of Computer-aided reasoning can be implemented, so that they can collaborate, be generalised, and implemented in a safe and trusted way. The framework I propose is a sequent calculus called LKp(T), which generalises an older calculus of the literature to the presence of an arbitrary background theory for which we have a decision procedure, like linear arithmetic. The thesis develops the meta-theory of LKp(T), such as its logical completeness. We then show how it specifies a proof-search procedure that can emulate a well-known technique from the field of Satisfiability-modulo-Theories, namely DPLL(T). Finally, clause and connection tableaux are other widely used techniques of automated reasoning, of a rather different nature f...
ArXiv, 2014
In this paper we explore a proof-theoretic approach to the combination of generic proof-search an... more In this paper we explore a proof-theoretic approach to the combination of generic proof-search and theory-specific procedures, in presence of quantifiers. Forming the basis of tableaux methods, the sequent calculus is refined with meta-variables (a.k.a existential variables or free variables) to delay the choice of witnesses, and parameterised by theory-specific features based on a system of constraints for meta-variables. An axiomatisation of these features is given, so that the soundness and completeness of the sequent calculus with meta-variables can be generically proved (with respect to the sequent calculus where the choice of witnesses is not delayed). We then describe a theory-generic proof-search implementation, that is parameterised by a theory-specific module whose specification is given by the above axiomatisation.
ArXiv, 2012
In this paper we relate different formulations of the DPLL(T ) procedure. The first formulation i... more In this paper we relate different formulations of the DPLL(T ) procedure. The first formulation is based on a system of rewrite rules, which we denote DPLL(T ). The second formulation is an inference system of, which we denote LKDPLL(T ). The third formulation is the application of a standard proof-search mechanism in a sequent calculus LKp(T ) introduced here. We formalise an encoding from DPLL(T ) to LKDPLL(T ) that was, to our knowledge, never explicitly given and, in the case where DPLL(T ) is extended with backjumping and Lemma learning, never even implicitly given. We also formalise an encoding from LKDPLL(T ) to LKp(T ), building on Ivan Gazeau's previous work: we extend his work in that we handle the "-modulo-Theory" aspect of SAT-modulo-theory, by extending the sequent calculus to allow calls to a theory solver (seen as a blackbox). We also extend his work in that we handle advanced features of DPLL such as backjumping and Lemma learning, etc. Finally, we re f...
This paper gives an abstract description of decision procedures for Satisfiability Modulo Theory ... more This paper gives an abstract description of decision procedures for Satisfiability Modulo Theory (SMT) as proof search procedures in a sequent calculus with polarities and focusing. In particular, we show how to simulate the execution of standard techniques based on the Davis-Putnam- Logemann-Loveland (DPLL) procedure modulo theory as the gradual construction of a proof tree in sequent calculus. The construction mimicking a run of DPLL-modulo-Theory can be obtained by a meta-logical control on the proof-search in sequent calculus. This control is provided by polarities and focusing features, which there- fore narrow the corresponding search space in a sense we discuss. This simulation can also account for backjumping and learning steps, which correspond to the use of general cuts in sequent calculus.
We describe how the Davis-Putnam-Logemann-Loveland procedure, in its version DPLL(T ) for Satisfi... more We describe how the Davis-Putnam-Logemann-Loveland procedure, in its version DPLL(T ) for Satisfiability Modulo Theory (SMT), can be interpreted as a proof-search mechanism for the incremental construction of a proof tree in sequent calculus. For this we use a sequent calculus with polarities and focusing and show how its meta-logical control features allow for a precise simulation of runs of DPLL(T ). This simulation also accounts for backjumping and learning steps. Finally we describe Psyche, an extensible prototype for this sequent calculus, that includes the implementation of a proof-search component based on this simulation of DPLL(T ).
In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Mi... more In this paper, we introduce two focussed sequent calculi, LKp(T) and LK+(T), that are based on Miller-Liang's LKF system for polarised classical logic. The novelty is that those sequent calculi integrate the possibility to call a decision procedure for some background theory T, and the possibility to polarise literals "on the fly" during proof-search. These features are used in our other works to simulate the DPLL(T) procedure as proof-search in the extension of LKp(T) with a cut-rule. In this report we therefore prove cut-elimination in LKp(T). Contrary to what happens in the empty theory, the polarity of literals affects the provability of formulae in presence of a theory T. On the other hand, changing the polarities of connectives does not change the provability of formulae, only the shape of proofs. In order to prove this, we introduce a second sequent calculus, LK+(T) that extends LKp(T) with a relaxed focussing discipline, but we then show an encoding of LK+(T) back into the more restrictive system LK(T). We then prove completeness of LKp(T) (and therefore of LK+(T)) with respect to first-order reasoning modulo the ground propositional lemmas of the background theory T .
Lecture Notes in Computer Science, 2015